diff --git a/diffrax/__init__.py b/diffrax/__init__.py
index 9b13acdf..3ce7f85d 100644
--- a/diffrax/__init__.py
+++ b/diffrax/__init__.py
@@ -7,6 +7,7 @@
)
from .autocitation import citation, citation_rules
from .brownian import AbstractBrownianPath, UnsafeBrownianPath, VirtualBrownianTree
+from .delays import Delays
from .event import (
AbstractDiscreteTerminatingEvent,
DiscreteTerminatingEvent,
diff --git a/diffrax/adjoint.py b/diffrax/adjoint.py
index b13c5903..20e2c7de 100644
--- a/diffrax/adjoint.py
+++ b/diffrax/adjoint.py
@@ -13,9 +13,8 @@
from .ad import implicit_jvp
from .heuristics import is_sde, is_unsafe_sde
-from .saveat import SaveAt, SubSaveAt, save_y
-from .solver import (AbstractItoSolver, AbstractRungeKutta,
- AbstractStratonovichSolver)
+from .saveat import save_y, SaveAt, SubSaveAt
+from .solver import AbstractItoSolver, AbstractRungeKutta, AbstractStratonovichSolver
from .term import AbstractTerm, AdjointTerm
@@ -532,6 +531,7 @@ def _loop_backsolve_bwd(
max_steps,
throw,
init_state,
+ y0_history,
):
#
diff --git a/diffrax/delays.py b/diffrax/delays.py
new file mode 100644
index 00000000..036dab3a
--- /dev/null
+++ b/diffrax/delays.py
@@ -0,0 +1,383 @@
+from typing import Callable, Optional, Sequence, Type, Union
+
+import equinox as eqx
+import jax
+import jax.lax as lax
+import jax.numpy as jnp
+import jax.tree_util as jtu
+from equinox.internal import unvmap_any
+from optimistix import fixed_point, FixedPointIteration
+
+from .custom_types import Array, Bool, Int, PyTree, Scalar
+from .global_interpolation import DenseInterpolation
+from .local_interpolation import AbstractLocalInterpolation
+from .nonlinear_solver import NewtonNonlinearSolver
+from .term import VectorFieldWrapper
+
+
+class Delays(eqx.Module):
+ """Module that incorportes all the information needed for integrating DDEs"""
+
+ delays: PyTree[Callable]
+ initial_discontinuities: Union[None, Array, Sequence[Scalar]] = jnp.array([0.0])
+ max_discontinuities: Int = 100
+ recurrent_checking: Bool = False
+ sub_intervals: int = 10
+ max_steps: int = 20
+ rtol: float = 1e-3
+ atol: float = 1e-6
+
+
+class HistoryVectorField(eqx.Module):
+ """VectorField equivalent for a DDE solver that incorporates former
+ estimated values of y(t).
+
+ **Arguments:**
+ - `vector_field`: vector field of the delayed differential equation.
+ - `t0`: global integration start time
+ - `tprev`: start time of current integration step
+ - `tnext`: end time of current integration step
+ - `dense_info` : dense_info from current integration step
+ - `y0_history` : DDE's history function
+ - `delays` : DDE's different deviated arguments
+ """
+
+ vector_field: Callable
+ t0: float
+ tprev: float
+ tnext: float
+ dense_info: PyTree[Array]
+ dense_interp: Optional[DenseInterpolation]
+ interpolation_cls: Type[AbstractLocalInterpolation]
+ y0_history: Callable
+ delays: PyTree[Callable]
+
+ def __call__(self, t, y, args):
+ history_vals = []
+ delays, treedef = jtu.tree_flatten(self.delays)
+ if self.dense_interp is None:
+ assert self.dense_info is None
+ for delay in self.delays:
+ delay_val = delay(t, y, args)
+ alpha_val = t - delay_val
+ y0_val = self.y0_history(alpha_val)
+ history_vals.append(y0_val)
+ else:
+ assert self.dense_info is not None
+ for delay in delays:
+ delay_val = delay(t, y, args)
+ alpha_val = t - delay_val
+
+ is_before_t0 = alpha_val < self.t0
+ is_before_tprev = alpha_val < self.tprev
+ at_most_t0 = jnp.where(alpha_val < self.t0, alpha_val, self.t0)
+ t0_to_tprev = jnp.clip(alpha_val, self.t0, self.tprev)
+ at_least_tprev = jnp.maximum(self.tprev, alpha_val)
+ step_interpolation = self.interpolation_cls(
+ t0=self.tprev, t1=self.tnext, **self.dense_info
+ )
+ switch = jnp.where(is_before_t0, 0, jnp.where(is_before_tprev, 1, 2))
+ history_val = lax.switch(
+ switch,
+ [
+ lambda: self.y0_history(at_most_t0),
+ lambda: self.dense_interp.evaluate(t0_to_tprev),
+ lambda: step_interpolation.evaluate(at_least_tprev),
+ ],
+ )
+ history_vals.append(history_val)
+
+ history_vals = jtu.tree_unflatten(treedef, history_vals)
+ history_vals = tuple(history_vals)
+ return self.vector_field(t, y, args, history=history_vals)
+
+
+def bind_history(
+ terms,
+ delays,
+ dense_info,
+ dense_interp,
+ solver,
+ direction,
+ t0,
+ tprev,
+ tnext,
+ y0_history,
+):
+ delays_fn = jtu.tree_map(
+ lambda x: (lambda t, y, args: x(t, y, args) * direction), delays.delays
+ )
+
+ is_vf_wrapper = lambda x: isinstance(x, VectorFieldWrapper)
+
+ def _apply_history(
+ x,
+ ):
+ if is_vf_wrapper(x):
+ vector_field = HistoryVectorField(
+ x.vector_field,
+ t0,
+ tprev,
+ tnext,
+ dense_info,
+ dense_interp,
+ solver.interpolation_cls,
+ y0_history,
+ delays_fn,
+ )
+ return VectorFieldWrapper(vector_field)
+ else:
+ return x
+
+ terms_ = jtu.tree_map(_apply_history, terms, is_leaf=is_vf_wrapper)
+ return terms_
+
+
+def history_extrapolation_implicit(
+ implicit_step,
+ terms,
+ dense_interp,
+ solver,
+ delays,
+ t0,
+ y0_history,
+ state,
+ args,
+):
+ def fn(dense_info, args):
+ (
+ terms,
+ _,
+ dense_interp,
+ solver,
+ delays,
+ t0,
+ y0_history,
+ state,
+ vf_args,
+ ) = args
+ terms_ = bind_history(
+ terms,
+ delays,
+ dense_info,
+ dense_interp,
+ solver,
+ 1,
+ t0,
+ state.tprev,
+ state.tnext,
+ y0_history,
+ )
+ (y, y_error, new_dense_info, solver_state, solver_result) = solver.step(
+ terms_,
+ state.tprev,
+ state.tnext,
+ state.y,
+ vf_args,
+ state.solver_state,
+ state.made_jump,
+ )
+
+ return new_dense_info, (y, y_error, solver_state, solver_result)
+
+ # unwrapped_buffer = jtu.tree_leaves(
+ # eqx.filter(state.dense_infos, eqx.is_inexact_array),
+ # is_leaf=eqx.is_inexact_array,
+ # )
+ # aux_dense_infos = dict(zip(state.dense_infos.keys(), unwrapped_buffer))
+
+ # def get_dense_info(dense_infos, idx):
+ # return jtu.tree_map(lambda x: x[idx], dense_infos)
+
+ # # dense_info_aux = dict(zip(["k","y0", "y1"],
+ # [jnp.zeros((4,)), jnp.zeros((1,)), jnp.zeros((1,))]))
+ # jax.debug.breakpoint()
+ # struct_dense_info = eqx.filter_eval_shape(get_dense_info, state.dense_infos, 0)
+ # infos = jtu.tree_map(lambda _, x: x[...], struct_dense_info, state.dense_infos)
+ # jax.debug.print("infos integrate {} ", infos)
+ # print("infos,", infos)
+ # print("struct_dense_info", struct_dense_info)
+
+ jax.debug.print("state.dense_infos {}", state.dense_infos)
+ init_guess = jtu.tree_map(
+ lambda x: x[state.dense_save_index - 1], state.dense_infos
+ )
+ alg = FixedPointIteration(rtol=delays.rtol, atol=delays.atol)
+ nonlinear_args = (
+ terms,
+ implicit_step,
+ dense_interp,
+ solver,
+ delays,
+ t0,
+ y0_history,
+ state,
+ args,
+ )
+ sol = fixed_point(fn, alg, init_guess, nonlinear_args, has_aux=True)
+ dense_info, (y, y_error, solver_state, solver_result) = sol.value, sol.aux
+ return y, y_error, dense_info, solver_state, solver_result
+
+
+def maybe_find_discontinuity(
+ tprev,
+ tnext,
+ dense_info,
+ state,
+ delays,
+ solver,
+ args,
+ keep_step,
+ sub_tprev,
+ sub_tnext,
+):
+ dense_discont = solver.interpolation_cls(t0=tprev, t1=tnext, **dense_info)
+ flat_delays = jtu.tree_leaves(delays.delays)
+ _gs = []
+
+ def make_g(delay):
+ # Creating the artifical event functions g that is used to
+ # detect future breaking points.
+ # http://www.cs.toronto.edu/pub/reports/na/hzpEnrightNA09Preprint.pdf
+ # page 7
+ def g(t):
+ return (
+ t
+ - delay(t, dense_discont.evaluate(t), args)
+ - state.discontinuities[...]
+ )
+
+ return g
+
+ for delay in flat_delays:
+ _gs.append(make_g(delay))
+
+ def _find_discontinuity():
+ # Start by doing a cheap bisection search to reduce
+ # over the stored-discontinuity dimension.
+
+ def _cond_fun(_val):
+ _, _, _pred, _ = _val
+ return _pred
+
+ def _body_fun(_val):
+ _ta, _tb, _, _step = _val
+ _step = _step + 1
+ _tmid = _ta + 0.5 * (_tb - _ta)
+ _gas = jnp.stack([jnp.sign(g(_ta)) for g in _gs])
+ _gmids = jnp.stack([jnp.sign(g(_tmid)) for g in _gs])
+ _gbs = jnp.stack([jnp.sign(g(_tb)) for g in _gs])
+ _any_left = jnp.any(_gas != _gmids)
+ _next_ta = jnp.where(_any_left, _ta, _tmid)
+ _next_tb = jnp.where(_any_left, _tmid, _tb)
+ _pred = (
+ jnp.any(jnp.sum(_gas != _gbs, axis=1) > 1) | _step > delays.max_steps
+ )
+ return _next_ta, _next_tb, _pred, _step
+
+ _init_val = (sub_tprev, sub_tnext, True, 0)
+ _final_val = lax.while_loop(_cond_fun, _body_fun, _init_val)
+ _ta, _tb, _, _ = _final_val
+
+ # Then do a more expensive Newton search
+ # to find the first discontinuity.
+ _discont_solver = NewtonNonlinearSolver(rtol=delays.rtol, atol=delays.atol)
+ _disconts = []
+ for g, delay in zip(_gs, flat_delays):
+ changed_sign = jnp.sign(g(_ta)) != jnp.sign(g(_tb))
+ _i = jnp.argmax(changed_sign)
+ _d = state.discontinuities[_i]
+ _h = (
+ lambda t, args, delay=delay, _d=_d: t
+ - delay(t, dense_discont.evaluate(t), args)
+ - _d
+ )
+ _discont = _discont_solver(_h, _tb, args).root
+ _disconts.append(_discont)
+ _disconts = jnp.stack(_disconts)
+
+ best_candidate = jnp.where(
+ (_disconts > sub_tprev) & (_disconts < sub_tnext),
+ _disconts,
+ jnp.inf,
+ )
+ best_candidate = jnp.min(best_candidate)
+ discont_update = jnp.where(
+ jnp.isinf(best_candidate),
+ False,
+ True,
+ )
+ return best_candidate, discont_update
+
+ def _find_discontinuity_wrapper():
+ return lax.cond(
+ jnp.any(init_discont & jnp.invert(keep_step)),
+ _find_discontinuity,
+ lambda: (sub_tnext, False),
+ )
+
+ init_discont = jnp.stack(
+ [jnp.sign(g(sub_tprev)) != jnp.sign(g(sub_tnext)) for g in _gs]
+ )
+ # We might have rejected the step for normal reasons;
+ # skip looking for a discontinuity if so.
+ return lax.cond(
+ unvmap_any((init_discont & jnp.invert(keep_step))),
+ _find_discontinuity_wrapper,
+ lambda: (sub_tnext, False),
+ )
+
+
+Delays.__init__.__doc__ = """
+**Arguments:**
+
+- `delays`: A `PyTree` where the leaves are the DDE's different scalar
+ deviated arguments.
+- `initial_discontinuities`: Discontinuities given by the initial point time
+and history function.
+- `max_discontinuities`: Array length that tracks the discontinuity jumps
+during integration (only relevant when `recurrent_checking` is True). If
+`recurrent checking` is set to `True`, the computation quits unconditionally
+when the total number of discontinuities detected is larger
+than `max_discontinuities`.
+- `recurrent_checking` : If `True`, there will be a systematic check at
+integration step for potential discontinuities (this involves nonlinear solves
+hence expensive). If `False`, discontinuities will only be checked when a step
+is rejected. This allows to integrate faster but can also impact
+the accuracy of the DDE solution.
+- `sub_intervals` : Number of subintervals of the integration step where
+discontinuity tracking is done.
+- `rtol` : Relative tolerance for the nonlinear solver for the DDE's
+implicit stepping and dichotomy for detecting breaking points.
+- `atol` : Absolute tolerance for the nonlinear solver for the DDE's
+implicit stepping and dichotomy for detecting breaking points.
+- `max_steps` : Max iteration of the dichotomy algorithm to
+find a discontinuity.
+
+!!! example
+ To integrate `y'(t) = - y(t-1)`, we need to define the vector
+ field and the `Delays` object.
+ ```py
+ y0 = lambda t: 1.2
+ def vector_field(t, y, args, history):
+ return - history[0]
+
+ delays = Delays(
+ delays=[lambda t, y, args: 1.0],
+ initial_discontinuities=jnp.array([0.0])
+ )
+ t0, t1 = 0.0, 50.0
+ ts = jnp.linspace(t0, t1, 500)
+ sol = diffrax.diffeqsolve(
+ diffrax.ODETerm(vector_field),
+ diffrax.Tsit5(),
+ t0=ts[0],
+ t1=ts[-1],
+ dt0=ts[1] - ts[0],
+ y0 = y0_history,
+ stepsize_controller=diffrax.PIDController(rtol=1e-6, atol=1e-9),
+ saveat=diffrax.SaveAt(ts=ts, dense=True),
+ delays=delays
+ )
+ ```
+"""
diff --git a/diffrax/integrate.py b/diffrax/integrate.py
index 7c40688d..073d71c3 100644
--- a/diffrax/integrate.py
+++ b/diffrax/integrate.py
@@ -1,8 +1,7 @@
import functools as ft
import typing
import warnings
-from typing import (Any, Callable, Optional, Sequence, Tuple, get_args,
- get_origin)
+from typing import Any, Callable, get_args, get_origin, Optional, Tuple, Union
import equinox as eqx
import equinox.internal as eqxi
@@ -13,20 +12,29 @@
from .adjoint import AbstractAdjoint, DirectAdjoint, RecursiveCheckpointAdjoint
from .custom_types import Array, Bool, Int, PyTree, Scalar
+from .delays import bind_history, Delays, history_extrapolation_implicit
from .event import AbstractDiscreteTerminatingEvent
from .global_interpolation import DenseInterpolation
from .heuristics import is_sde, is_unsafe_sde
from .misc import static_select
from .saveat import SaveAt, SubSaveAt
-from .solution import RESULTS, Solution, is_okay, is_successful
-from .solver import (AbstractItoSolver, AbstractSolver,
- AbstractStratonovichSolver, Euler, EulerHeun, ItoMilstein,
- StratonovichMilstein)
-from .step_size_controller import (AbstractAdaptiveStepSizeController,
- AbstractStepSizeController,
- ConstantStepSize, StepTo)
-from .term import (AbstractTerm, MultiTerm, ODETerm, VectorFieldWrapper,
- WrapTerm)
+from .solution import is_okay, is_successful, RESULTS, Solution
+from .solver import (
+ AbstractItoSolver,
+ AbstractSolver,
+ AbstractStratonovichSolver,
+ Euler,
+ EulerHeun,
+ ItoMilstein,
+ StratonovichMilstein,
+)
+from .step_size_controller import (
+ AbstractAdaptiveStepSizeController,
+ AbstractStepSizeController,
+ ConstantStepSize,
+ StepTo,
+)
+from .term import AbstractTerm, MultiTerm, ODETerm, WrapTerm
class SaveState(eqx.Module):
@@ -48,11 +56,15 @@ class State(eqx.Module):
num_steps: Int
num_accepted_steps: Int
num_rejected_steps: Int
+ num_dde_implicit_step: Int
+ num_dde_explicit_step: Int
# Output that is .at[].set() updated during the solve (and their indices)
save_state: PyTree[SaveState]
dense_ts: Optional[Array["times + 1"]] # noqa: F821
dense_infos: Optional[PyTree[Array["times", ...]]] # noqa: F821
dense_save_index: Int
+ discontinuities: Optional[Array["times"]] # noqa: F821
+ discontinuities_save_index: Optional[Int]
def _is_none(x):
@@ -103,6 +115,7 @@ def _outer_buffers(state):
[s.ts for s in save_states]
+ [s.ys for s in save_states]
+ [state.dense_ts, state.dense_infos]
+ + [state.discontinuities]
)
@@ -164,6 +177,7 @@ def loop(
init_state,
inner_while_loop,
outer_while_loop,
+ y0_history,
):
if saveat.dense:
@@ -209,7 +223,6 @@ def body_fun(state):
# Actually do some differential equation solving! Make numerical steps, adapt
# step sizes, all that jazz.
#
-
if delays is None:
(y, y_error, dense_info, solver_state, solver_result) = solver.step(
terms,
@@ -220,42 +233,48 @@ def body_fun(state):
state.solver_state,
state.made_jump,
)
+ num_dde_explicit_step = num_dde_implicit_step = 0
else:
- # TODO: double-check that these are the correct `ts_size` and
- # `direction`.
- history = DenseInterpolation(
- ts=state.dense_ts,
+ min_delay = []
+ flat_delays = jtu.tree_leaves(delays.delays)
+ for delay in flat_delays:
+ min_delay.append(delay(state.tprev, state.y, args))
+ min_delay = jnp.stack(min_delay).min()
+ implicit_step = min_delay < (state.tnext - state.tprev)
+
+ dense_interp = DenseInterpolation(
+ ts=state.dense_ts[...],
ts_size=state.dense_save_index + 1,
interpolation_cls=solver.interpolation_cls,
infos=state.dense_infos,
direction=1,
+ y0_if_trivial=y0_history(t0),
+ t0_if_trivial=t0,
+ )
+
+ (
+ y,
+ y_error,
+ dense_info,
+ solver_state,
+ solver_result,
+ ) = history_extrapolation_implicit(
+ implicit_step,
+ terms,
+ dense_interp,
+ solver,
+ delays,
+ t0,
+ y0_history,
+ state,
+ args,
)
- history_vals = []
- for delay in delays:
- delay_val = delay(state.tprev, state.y, args)
- history_val = history.evaluate(delay_val)
- history_val.append(history_val)
- history_vals = tuple(history_vals)
-
- is_vf_wrapper = lambda x: isinstance(x, VectorFieldWrapper)
-
- def _apply_history(x):
- if is_vf_wrapper(x):
- vector_field = jtu.Partial(x.vector_field, history=history_vals)
- return VectorFieldWrapper(vector_field)
- else:
- return x
-
- terms_ = jtu.tree_map(_apply_history, terms, is_leaf=is_vf_wrapper)
- # TODO: write down implicit problem wrt dense_info, using `terms_`
- (y, y_error, dense_info, solver_state, solver_result) = terms_ # ...
# e.g. if someone has a sqrt(y) in the vector field, and dt0 is so large that
# we get a negative value for y, and then get a NaN vector field. (And then
# everything breaks.) See #143.
y_error = jtu.tree_map(lambda x: jnp.where(jnp.isnan(x), jnp.inf, x), y_error)
- # TODO: handle discontinuity detection for delays
error_order = solver.error_order(terms)
(
keep_step,
@@ -274,12 +293,66 @@ def _apply_history(x):
error_order,
state.controller_state,
)
- assert jnp.result_type(keep_step) is jnp.dtype(bool)
+
+ # Finding all of the potential discontinuity roots
+ # if delays is not None:
+ # _part_maybe_find_discontinuity = ft.partial(
+ # maybe_find_discontinuity,
+ # tprev,
+ # tnext,
+ # dense_info,
+ # state,
+ # delays,
+ # solver,
+ # args,
+ # )
+
+ # tsearch = jnp.linspace(tprev, tnext, delays.sub_intervals)
+ # batch_tprev, batch_tnext = tsearch[:-1], tsearch[1:]
+ # vmap_maybe_find_discontinuity_wrapper = jax.vmap(
+ # _part_maybe_find_discontinuity, (None, 0, 0)
+ # )
+ # if delays.recurrent_checking:
+ # (
+ # tnext_candidate,
+ # batch_discont_update,
+ # ) = vmap_maybe_find_discontinuity_wrapper(
+ # False, batch_tprev, batch_tnext
+ # )
+ # else:
+ # (
+ # tnext_candidate,
+ # batch_discont_update,
+ # ) = vmap_maybe_find_discontinuity_wrapper(
+ # keep_step, batch_tprev, batch_tnext
+ # )
+
+ # proxy_tnext = jnp.where(batch_discont_update, tnext_candidate, jnp.inf)
+ # proxy_tnext = jnp.min(proxy_tnext)
+
+ # tnext, discont_update = jax.lax.cond(
+ # jnp.isinf(proxy_tnext),
+ # lambda: (tnext, False),
+ # lambda: (proxy_tnext, True),
+ # )
+
+ # # Count the number of steps in DDEs, just for statistical purposes
+ # num_dde_implicit_step = state.num_dde_implicit_step + (
+ # keep_step & implicit_step
+ # )
+ # num_dde_explicit_step = state.num_dde_explicit_step + (
+ # keep_step & jnp.invert(implicit_step)
+ # )
+
+ # assert jnp.result_type(discont_update) is jnp.dtype(bool)
+
+ # assert jnp.result_type(keep_step) is jnp.dtype(bool)
#
# Do some book-keeping.
#
-
+ # discont_update = False
+ num_dde_explicit_step = num_dde_implicit_step = 0
tprev = jnp.minimum(tprev, t1)
tnext = _clip_to_end(tprev, tnext, t1, keep_step)
@@ -320,6 +393,8 @@ def _apply_history(x):
dense_ts = state.dense_ts
dense_infos = state.dense_infos
dense_save_index = state.dense_save_index
+ discontinuities = state.discontinuities
+ discontinuities_save_index = state.discontinuities_save_index
def save_ts(subsaveat: SubSaveAt, save_state: SaveState) -> SaveState:
if subsaveat.ts is not None:
@@ -374,6 +449,17 @@ def maybe_inplace(i, u, x):
else:
return x.at[i].set(u, pred=keep_step)
+ # def maybe_inplace_delay(i, u, x):
+ # # Annoying hack. We normally call this with `x` wrapped into a buffer
+ # # (from Equinox's while loops). However we do also first trace through to
+ # # see if we can resolve some values statically, in which case normal JAX
+ # # arrays don't support the extra `pred` argument. We don't then use the
+ # # result of this so we just skip it.
+ # if _filtering:
+ # return x
+ # else:
+ # return x.at[i].set(u, pred=discont_update)
+
def save_steps(subsaveat: SubSaveAt, save_state: SaveState) -> SaveState:
if subsaveat.steps:
ts = maybe_inplace(save_state.save_index, tprev, save_state.ts)
@@ -403,6 +489,21 @@ def save_steps(subsaveat: SubSaveAt, save_state: SaveState) -> SaveState:
)
dense_save_index = dense_save_index + keep_step
+ # Updating discontinuity
+ # if delays is not None:
+ # if delays.recurrent_checking:
+ # eqxi.error_if(
+ # discontinuities_save_index,
+ # discontinuities_save_index >= delays.max_discontinuities,
+ # "the number of discontinuities detected reached the number of"
+ # " `max_discontinuities`, please raise its value.",
+ # )
+
+ # discontinuities = maybe_inplace_delay(
+ # discontinuities_save_index + 1, tnext, discontinuities
+ # )
+ # discontinuities_save_index = discontinuities_save_index + discont_update
+
new_state = State(
y=y,
tprev=tprev,
@@ -415,9 +516,13 @@ def save_steps(subsaveat: SubSaveAt, save_state: SaveState) -> SaveState:
num_accepted_steps=num_accepted_steps,
num_rejected_steps=num_rejected_steps,
save_state=save_state,
+ num_dde_explicit_step=num_dde_explicit_step,
+ num_dde_implicit_step=num_dde_implicit_step,
dense_ts=dense_ts,
dense_infos=dense_infos,
dense_save_index=dense_save_index,
+ discontinuities=discontinuities,
+ discontinuities_save_index=discontinuities_save_index,
)
if discrete_terminating_event is not None:
@@ -496,7 +601,7 @@ def diffeqsolve(
t0: Scalar,
t1: Scalar,
dt0: Optional[Scalar],
- y0: PyTree,
+ y0: Union[PyTree, Callable],
args: Optional[PyTree] = None,
*,
saveat: SaveAt = SaveAt(t1=True),
@@ -504,8 +609,8 @@ def diffeqsolve(
adjoint: AbstractAdjoint = RecursiveCheckpointAdjoint(),
discrete_terminating_event: Optional[AbstractDiscreteTerminatingEvent] = None,
max_steps: Optional[int] = 4096,
- delays: Optional[Sequence[Callable]] = None,
throw: bool = True,
+ delays: Optional[Delays] = None,
solver_state: Optional[PyTree] = None,
controller_state: Optional[PyTree] = None,
made_jump: Optional[Bool] = None,
@@ -631,7 +736,7 @@ def diffeqsolve(
)
with jax.ensure_compile_time_eval():
pred = (t1 - t0) * dt0 < 0
- dt0 = eqxi.error_if(dt0, pred, msg)
+ dt0 = eqxi.error_if(jnp.array(dt0), pred, msg)
# Backward compatibility
if isinstance(
@@ -712,6 +817,18 @@ def _promote(yi):
_dtype = jnp.result_type(yi, *timelikes) # noqa: F821
return jnp.asarray(yi, dtype=_dtype)
+ if delays is None:
+ if callable(y0):
+ raise ValueError("`y0` is passed as a callable and should be an array.")
+ new_discontinuities = None
+ discontinuities_save_index = None
+ y0_history = None
+ else:
+ if not callable(y0):
+ raise ValueError("`y0` should be a callable.")
+ y0_history = y0
+ y0 = y0_history(t0)
+
y0 = jtu.tree_map(_promote, y0)
del timelikes, dtype
@@ -736,9 +853,6 @@ def _wrap(term):
terms,
is_leaf=lambda x: isinstance(x, AbstractTerm) and not isinstance(x, MultiTerm),
)
- if delays is not None:
- delays = [lambda t, y, args, fn=fn: fn(t, y, args) * direction for fn in delays]
-
# Stepsize controller gets an opportunity to modify the solver.
# Note that at this point the solver could be anything so we must check any
# abstract base classes of the solver before this.
@@ -777,9 +891,27 @@ def _check_subsaveat_ts(ts):
else:
tnext = t0 + dt0
tnext = jnp.minimum(tnext, t1)
+
+ if delays is not None:
+ terms_ = bind_history(
+ terms,
+ delays,
+ None,
+ None,
+ solver,
+ direction,
+ t0,
+ tprev,
+ tnext,
+ y0_history,
+ )
+ else:
+ terms_ = terms
+
+ # Got to init the solver
if solver_state is None:
passed_solver_state = False
- solver_state = solver.init(terms, t0, tnext, y0, args)
+ solver_state = solver.init(terms_, t0, tnext, y0, args)
else:
passed_solver_state = True
@@ -814,6 +946,8 @@ def _allocate_output(subsaveat: SubSaveAt) -> SaveState:
num_steps = 0
num_accepted_steps = 0
num_rejected_steps = 0
+ num_dde_explicit_step = 0
+ num_dde_implicit_step = 0
made_jump = False if made_jump is None else made_jump
result = RESULTS.successful
if saveat.dense:
@@ -822,8 +956,22 @@ def _allocate_output(subsaveat: SubSaveAt) -> SaveState:
"`max_steps=None` is incompatible with `saveat.dense=True`"
)
(_, _, dense_info, _, _,) = eqx.filter_eval_shape(
- solver.step, terms, tprev, tnext, y0, args, solver_state, made_jump
+ solver.step, terms_, tprev, tnext, y0, args, solver_state, made_jump
)
+ if delays is not None:
+ if delays.initial_discontinuities is not None:
+ buffer = jnp.full(
+ delays.max_discontinuities - len(delays.initial_discontinuities),
+ jnp.inf,
+ )
+ new_discontinuities = jnp.concatenate(
+ [delays.initial_discontinuities, buffer]
+ )
+ discontinuities_save_index = len(delays.initial_discontinuities) - 1
+ else:
+ new_discontinuities = jnp.full(delays.max_discontinuities, jnp.inf)
+ discontinuities_save_index = 0
+
dense_ts = jnp.full(max_steps + 1, jnp.inf)
_make_full = lambda x: jnp.full((max_steps,) + jnp.shape(x), jnp.inf)
dense_infos = jtu.tree_map(_make_full, dense_info)
@@ -849,6 +997,11 @@ def _allocate_output(subsaveat: SubSaveAt) -> SaveState:
dense_ts=dense_ts,
dense_infos=dense_infos,
dense_save_index=dense_save_index,
+ # dde specific State arguments
+ num_dde_explicit_step=num_dde_explicit_step,
+ num_dde_implicit_step=num_dde_implicit_step,
+ discontinuities=new_discontinuities,
+ discontinuities_save_index=discontinuities_save_index,
)
#
@@ -871,6 +1024,7 @@ def _allocate_output(subsaveat: SubSaveAt) -> SaveState:
throw=throw,
passed_solver_state=passed_solver_state,
passed_controller_state=passed_controller_state,
+ y0_history=y0_history,
)
#
@@ -920,6 +1074,8 @@ def _allocate_output(subsaveat: SubSaveAt) -> SaveState:
"num_accepted_steps": final_state.num_accepted_steps,
"num_rejected_steps": final_state.num_rejected_steps,
"max_steps": max_steps,
+ "num_dde_implicit_step": final_state.num_dde_implicit_step,
+ "num_dde_explicit_step": final_state.num_dde_explicit_step,
}
result = final_state.result
sol = Solution(
diff --git a/docs/api/delays.md b/docs/api/delays.md
new file mode 100644
index 00000000..694d2afa
--- /dev/null
+++ b/docs/api/delays.md
@@ -0,0 +1,42 @@
+# Delays
+
+Delays allow to model a broader class of differential equations, Delay Differential Equations (DDEs).
+Compared to ODEs, DDEs vector fields need a new argument `history` that integrate delayed states.
+
+At the moment only DDEs with known delays is supported.
+
+!!! example
+ The equation's RHS $y'(t) = y(t) + y(t-1) - y(t-2)$ is modelled by
+ ```
+ def vf(t,y,args,history):
+ return y + history[0] - history[1]
+ ```
+
+The first element of a `PyTree` of delayed states in a vector field's definition would be `history[0]`. If the state is multidimensional, then `history[0][i]` refers to the $i$th dimension of the first delayed state.
+
+
+
+::: diffrax.Delays
+ selection:
+ members:
+ - __init__
+
+!!! example
+ Pytree of three `Delays.delays` :
+ ```
+ delays=[lambda t, y, args: 1.0, (lambda t, y ,args: min(t,2), lambda t, y, args : 1/2 * jnp.cos(y))]
+ ```
+
+!!! info
+ If `recurrent_checking=True`, then at integration step, a so-called artificial event function $g$ checks for any new discontinuity jumps unconditionally (which are $g_i$'s new odd multiplicity roots). Let $\lambda_i$ be the combined intial discontinuity jumps and the ones found up to the current integration step.
+
+ $\begin{align}
+ g_i(t,y(t)) = t - \tau(t,y(t)) - \lambda_i, \quad i = \dots, -2,-1,-,1,2,\dots \\
+ \end{align}$
+
+ This can be prohibitely expensive but in some cases can speed up the integration and its accuracy.
+ We refer to this [paper](http://www.cs.toronto.edu/pub/reports/na/hzpEnrightNA09Preprint.pdf) for more information on how to detect new discontinuity jumps in a DDE.
+
+!!! note
+ If the history function is continuous and the initial time point induces a discontinuity `max_discontinuities = jnp.array([t0])`.
+
diff --git a/examples/dde.ipynb b/examples/dde.ipynb
new file mode 100644
index 00000000..5546d4d6
--- /dev/null
+++ b/examples/dde.ipynb
@@ -0,0 +1,261 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Delay Differential Equations : DDEs"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "This example demonstrates the use of DDE solvers to handle delayed system of equations. In this case, we model the dimensionless [delay-logistic equation](https://www.math.fsu.edu/~bertram/lectures/delay.pdf). \n",
+ "\n",
+ "$$ \\frac{dy}{dt} = \\alpha y \\left(1 - y(t-\\tau) \\right) $$\n",
+ "\n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "This example is available as a Jupyter notebook [here](https://github.com/patrick-kidger/diffrax/blob/main/examples/dde.ipynb)."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 64,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "import time\n",
+ "\n",
+ "import diffrax\n",
+ "import jax\n",
+ "import jax.numpy as jnp\n",
+ "import matplotlib.pyplot as plt"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Using 64-bit precision is important when solving problems with tolerances of 1e-8 (or smaller)."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 65,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "jax.config.update(\"jax_enable_x64\", True)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "In order to model a delayed system we need to instantiate its vector field and its history function $\\phi(t) = y(t<0)$ (i.e. initial condition)."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 66,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "def delay_logistic_vf(t, y, args, *, history):\n",
+ " alpha = args\n",
+ " return alpha * y * (1 - history[0])\n",
+ "\n",
+ "\n",
+ "def history_function(t):\n",
+ " return 2.0"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "A DDE's vector field holds an additional argument `history` compared to an ODE's. The `history` variable refers to the delayed terms of your equation (here $y(t-\\tau)$). "
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "We need to instantiate the `Delays` object that encompasses all of the information needed to integrate properly our DDE. The arguments `initial_discontinuities` also need to be given. It accounts for the discontinuities found in the history function. \n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "In our case we only have 0 since $y\\prime(t=0^{-}) \\neq y\\prime(t=0^{+})$ because $y\\prime(t=0^{+}) = - 2 \\alpha$ and $y\\prime(t=0^{-}) = 0$. \n",
+ "We choose $\\tau=1$."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 67,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "delays = diffrax.Delays(\n",
+ " delays=[lambda t, y, args: 1], initial_discontinuities=jnp.array([0.0])\n",
+ ")"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Since DDEs require dense solutions we need to set `diffrax.SaveAt`'s argument `dense=True` in `diffrax.diffeqsolve`. "
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 68,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "@jax.jit\n",
+ "def main(alpha):\n",
+ " terms = diffrax.ODETerm(delay_logistic_vf)\n",
+ " t0 = 0.0\n",
+ " t1 = 20.0\n",
+ " ts = jnp.linspace(0, 20, 200)\n",
+ " solver = diffrax.Bosh3()\n",
+ " stepsize_controller = diffrax.PIDController(rtol=1e-3, atol=1e-6)\n",
+ " sol = diffrax.diffeqsolve(\n",
+ " terms,\n",
+ " solver,\n",
+ " t0,\n",
+ " t1,\n",
+ " ts[1] - ts[0],\n",
+ " y0=history_function,\n",
+ " saveat=diffrax.SaveAt(ts=ts, dense=True),\n",
+ " stepsize_controller=stepsize_controller,\n",
+ " delays=delays,\n",
+ " args=alpha,\n",
+ " )\n",
+ " return sol"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Do one iteration to JIT compile everything. Then time the second iteration."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 69,
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "Integration took in 0.0004687309265136719 seconds.\n"
+ ]
+ },
+ {
+ "data": {
+ "image/png": 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",
+ "text/plain": [
+ "<Figure size 640x480 with 1 Axes>"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "alpha = 2.0\n",
+ "main(alpha)\n",
+ "\n",
+ "start = time.time()\n",
+ "sol = main(alpha)\n",
+ "end = time.time()\n",
+ "print(f\"Integration took in {end - start} seconds.\")\n",
+ "\n",
+ "plt.plot(sol.ts, sol.ys)\n",
+ "plt.show()"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "We can also change the `alpha` term and see its influence on the dynamics."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 70,
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "Integration took in 0.000225067138671875 seconds.\n"
+ ]
+ },
+ {
+ "data": {
+ "image/png": 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",
+ "text/plain": [
+ "<Figure size 640x480 with 1 Axes>"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "main(1)\n",
+ "\n",
+ "start = time.time()\n",
+ "sol = main(1)\n",
+ "end = time.time()\n",
+ "print(f\"Integration took in {end - start} seconds.\")\n",
+ "\n",
+ "plt.plot(sol.ts, sol.ys)\n",
+ "plt.show()"
+ ]
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "Python 3.9.16 ('documentation_diffrax')",
+ "language": "python",
+ "name": "python3"
+ },
+ "language_info": {
+ "codemirror_mode": {
+ "name": "ipython",
+ "version": 3
+ },
+ "file_extension": ".py",
+ "mimetype": "text/x-python",
+ "name": "python",
+ "nbconvert_exporter": "python",
+ "pygments_lexer": "ipython3",
+ "version": "3.9.16"
+ },
+ "orig_nbformat": 4,
+ "vscode": {
+ "interpreter": {
+ "hash": "d644d6de157dae45f4c41ea729963c4364743d9466df2c5e80d490cf71a3f866"
+ }
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}
diff --git a/examples/neural_dde.ipynb b/examples/neural_dde.ipynb
new file mode 100644
index 00000000..990c5e1b
--- /dev/null
+++ b/examples/neural_dde.ipynb
@@ -0,0 +1,319 @@
+{
+ "cells": [
+ {
+ "attachments": {},
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Neural DDE"
+ ]
+ },
+ {
+ "attachments": {},
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "This example demonstrates how to use Diffrax in order to solve a Delay Differential Equation (DDE) with known delays. \n",
+ "Unlike ODEs that are identified by their vector field $f(t, y(t))$ and initial condition $y(0)=y_0$, DDEs are specified by their vector field $f$, deviated arguments $y(t-\\tau)$ and history function $\\phi(t)=y(t<0)$.\n",
+ "\n",
+ "We will model the [Lotka Volterra](https://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equations) (LK) equations with one constant time delay defined as \n",
+ "\n",
+ "$$\n",
+ "\\begin{align}\n",
+ "& y_1'(t) = \\frac{1}{2} y_1(t) ( 1 - y_2(t-0.2)) \\\\\n",
+ "& y_2'(t) = -\\frac{1}{2} y_2(t)( 1 - y_1(t-0.2)) \\\\\n",
+ "& \\phi(t) = y(t<0) = (y_1, y_2) \n",
+ "\\end{align}\n",
+ "$$\n",
+ "\n",
+ "where $x_0, y_0$ are uniformly sampled in $[0.1,2]$."
+ ]
+ },
+ {
+ "attachments": {},
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "This example is available as a Jupyter notebook [here](url)."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "import time\n",
+ "\n",
+ "import diffrax\n",
+ "import equinox as eqx # https://github.com/patrick-kidger/equinox\n",
+ "import jax\n",
+ "import jax.nn as jnn\n",
+ "import jax.numpy as jnp\n",
+ "import jax.random as jrandom\n",
+ "import matplotlib.pyplot as plt\n",
+ "import optax # https://github.com/deepmind/optax"
+ ]
+ },
+ {
+ "attachments": {},
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "In order to model our problem as a DDE $y'(t) = f_{\\theta}(t, y(t), y(t-\\tau_1), \\dots, y(t-\\tau_d))$, we first need to define a `Delays` object that incorporates deviated arguments in our vector field $f$. \n",
+ "\n",
+ "LK's initial time point $t=0$ has a derivative jump because $\\phi^{\\prime}(t=0^{-}) \\neq y^{\\prime}(t=0^{+})$ and the history function $\\phi(t)$ has `None`. \n",
+ "The DDE model only has one time delay so $d=1$ and our vector field will be $y'(t) = f_{\\theta}(t, y(t), y(t-\\tau))$"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "delays = diffrax.Delays(\n",
+ " delays=[lambda t, y, args: 0.2], initial_discontinuities=jnp.array([0.0])\n",
+ ")"
+ ]
+ },
+ {
+ "attachments": {},
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Below is defined the vector field $f_{\\theta}$. "
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "class Func(eqx.Module):\n",
+ " mlp: eqx.nn.MLP\n",
+ "\n",
+ " def __init__(self, data_size, width_size, depth, *, key, **kwargs):\n",
+ " super().__init__(**kwargs)\n",
+ " self.mlp = eqx.nn.MLP(\n",
+ " in_size=data_size,\n",
+ " out_size=data_size,\n",
+ " width_size=width_size,\n",
+ " depth=depth,\n",
+ " activation=jnn.relu,\n",
+ " key=key,\n",
+ " )\n",
+ "\n",
+ " def __call__(self, t, y, args, history):\n",
+ " return self.mlp(y, history[0])"
+ ]
+ },
+ {
+ "attachments": {},
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "The `history` variable inside the network's `__call__` is a tuple of deviated arguments. For example, if we possess a `Delays` object with 2 delays then the first element of tuple would be the first deviated argument $y(t-\\tau_1)$ and the second one $y(t-\\tau_1)$. \n",
+ "In our case, `history[0]` corresponds to $y(t-0.2)$ and by extension `history[0][0]` is $y_1(t-0.2)$."
+ ]
+ },
+ {
+ "attachments": {},
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Here we wrap up the entire DDE solve into a model."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "class NeuralDDE(eqx.Module):\n",
+ " func: Func\n",
+ "\n",
+ " def __init__(self, data_size, width_size, depth, *, key, **kwargs):\n",
+ " super().__init__(**kwargs)\n",
+ " self.func = Func(data_size, width_size, depth, key=key)\n",
+ "\n",
+ " def __call__(self, ts, y0):\n",
+ " solution = diffrax.diffeqsolve(\n",
+ " diffrax.ODETerm(self.func),\n",
+ " diffrax.Tsit5(),\n",
+ " t0=ts[0],\n",
+ " t1=ts[-1],\n",
+ " dt0=ts[1] - ts[0],\n",
+ " y0=y0,\n",
+ " delays=delays,\n",
+ " stepsize_controller=diffrax.PIDController(rtol=1e-3, atol=1e-6),\n",
+ " saveat=diffrax.SaveAt(ts=ts, dense=True),\n",
+ " )\n",
+ " return solution.ys"
+ ]
+ },
+ {
+ "attachments": {},
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "We generate the LK dataset."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "def _get_data(ts, *, key):\n",
+ " y0 = jrandom.uniform(key, (2,), minval=0.1, maxval=2.0)\n",
+ "\n",
+ " def vector_field(t, y, args, history):\n",
+ " return jnp.array(\n",
+ " [\n",
+ " 1 / 2 * y[0] * (1 - history[0][1]),\n",
+ " -1 / 2 * y[1] * (1 - history[0][0]),\n",
+ " ]\n",
+ " )\n",
+ "\n",
+ " sol = diffrax.diffeqsolve(\n",
+ " diffrax.ODETerm(vector_field),\n",
+ " diffrax.Dopri5(),\n",
+ " t0=ts[0],\n",
+ " t1=ts[-1],\n",
+ " dt0=ts[1] - ts[0],\n",
+ " y0=lambda t: y0,\n",
+ " adjoint=diffrax.NoAdjoint(),\n",
+ " stepsize_controller=diffrax.PIDController(rtol=1e-6, atol=1e-9),\n",
+ " saveat=diffrax.SaveAt(ts=ts, dense=True),\n",
+ " delays=delays,\n",
+ " )\n",
+ "\n",
+ " return sol.ys\n",
+ "\n",
+ "\n",
+ "def get_data(dataset_size, *, key):\n",
+ " ts = jnp.linspace(0, 15, 200)\n",
+ " key = jrandom.split(key, dataset_size)\n",
+ " ys = jax.vmap(lambda key: _get_data(ts, key=key))(key)\n",
+ " return ts, ys\n",
+ "\n",
+ "\n",
+ "def dataloader(arrays, batch_size, *, key):\n",
+ " dataset_size = arrays[0].shape[0]\n",
+ " assert all(array.shape[0] == dataset_size for array in arrays)\n",
+ " indices = jnp.arange(dataset_size)\n",
+ " while True:\n",
+ " perm = jrandom.permutation(key, indices)\n",
+ " (key,) = jrandom.split(key, 1)\n",
+ " start = 0\n",
+ " end = batch_size\n",
+ " while end < dataset_size:\n",
+ " batch_perm = perm[start:end]\n",
+ " yield tuple(array[batch_perm] for array in arrays)\n",
+ " start = end\n",
+ " end = start + batch_size"
+ ]
+ },
+ {
+ "attachments": {},
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Main entry point. Try runnning `main()`."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "def main(\n",
+ " dataset_size=256,\n",
+ " batch_size=32,\n",
+ " width_size=32,\n",
+ " depth=2,\n",
+ " tot_steps=500,\n",
+ " lr=10e-3,\n",
+ " seed=5678,\n",
+ " plot=True,\n",
+ " print_every=100,\n",
+ "):\n",
+ " key = jrandom.PRNGKey(seed)\n",
+ " data_key, model_key, loader_key = jrandom.split(key, 3)\n",
+ "\n",
+ " ts, ys = get_data(dataset_size, key=data_key)\n",
+ " _, _, data_size = ys.shape\n",
+ "\n",
+ " model = NeuralDDE(data_size, width_size, depth, key=model_key)\n",
+ "\n",
+ " @eqx.filter_value_and_grad\n",
+ " def grad_loss(model, ti, yi):\n",
+ " y_pred = jax.vmap(model, in_axes=(None, 0))(ti, yi[:, 0])\n",
+ " return jnp.mean((yi - y_pred) ** 2)\n",
+ "\n",
+ " @eqx.filter_jit\n",
+ " def make_step(ti, yi, model, opt_state):\n",
+ " loss, grads = grad_loss(model, ti, yi)\n",
+ " updates, opt_state = optim.update(grads, opt_state)\n",
+ " model = eqx.apply_updates(model, updates)\n",
+ " return loss, model, opt_state\n",
+ "\n",
+ " optim = optax.adabelief(lr)\n",
+ " opt_state = optim.init(eqx.filter(model, eqx.is_inexact_array))\n",
+ " for step, (yi,) in zip(\n",
+ " range(tot_steps), dataloader((ys,), batch_size, key=loader_key)\n",
+ " ):\n",
+ " start = time.time()\n",
+ " loss, model, opt_state = make_step(ts, yi, model, opt_state)\n",
+ " end = time.time()\n",
+ " if (step % print_every) == 0 or step == tot_steps - 1:\n",
+ " print(f\"Step: {step}, Loss: {loss}, Computation time: {end - start}\")\n",
+ "\n",
+ " if plot:\n",
+ " plt.plot(ts, ys[0, :, 0], c=\"dodgerblue\", label=\"Real\")\n",
+ " plt.plot(ts, ys[0, :, 1], c=\"dodgerblue\")\n",
+ " model_y = model(ts, ys[0, 0])\n",
+ " plt.plot(ts, model_y[:, 0], c=\"crimson\", label=\"Model\")\n",
+ " plt.plot(ts, model_y[:, 1], c=\"crimson\")\n",
+ " plt.legend()\n",
+ " plt.tight_layout()\n",
+ " plt.savefig(\"neural_ode.png\")\n",
+ " plt.show()\n",
+ "\n",
+ " return ts, ys, model"
+ ]
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "Python 3.9.13 ('dde')",
+ "language": "python",
+ "name": "python3"
+ },
+ "language_info": {
+ "codemirror_mode": {
+ "name": "ipython",
+ "version": 3
+ },
+ "file_extension": ".py",
+ "mimetype": "text/x-python",
+ "name": "python",
+ "nbconvert_exporter": "python",
+ "pygments_lexer": "ipython3",
+ "version": "3.9.13"
+ },
+ "orig_nbformat": 4,
+ "vscode": {
+ "interpreter": {
+ "hash": "85a82c4ca03817695851f28a9a8c882825d82c078a665316024297a0baa050af"
+ }
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}
diff --git a/mkdocs.yml b/mkdocs.yml
index 4d8d3fa5..50a7e4cc 100644
--- a/mkdocs.yml
+++ b/mkdocs.yml
@@ -98,6 +98,7 @@ nav:
- 'usage/extending.md'
- Examples:
- Basic ODE/SDE/CDE examples: 'other_examples/basic-examples.md'
+ # - DDE : 'examples/dde.ipynb' to add in Basic examples
- Coupled ODEs: 'examples/coupled_odes.ipynb'
- Stiff ODE: 'examples/stiff_ode.ipynb'
- Forcing terms: 'examples/forcing.ipynb'
@@ -106,6 +107,7 @@ nav:
- Neural CDE: 'examples/neural_cde.ipynb'
- Neural SDE: 'examples/neural_sde.ipynb'
- Latent ODE: 'examples/latent_ode.ipynb'
+ # - Neural DDE: 'examples/neural_dde.ipynb'
- Continuous normalising flow: 'examples/continuous_normalising_flow.ipynb'
- Symbolic regression: 'examples/symbolic_regression.ipynb'
- Steady state: 'examples/steady_state.ipynb'
@@ -131,6 +133,7 @@ nav:
- 'api/interpolation.md'
- 'api/brownian.md'
- 'api/nonlinear_solver.md'
+ # - 'api/delays.md'
- Further details:
- 'further_details/faq.md'
- 'further_details/acknowledgements.md'
diff --git a/test/julia_dde/dde.jl b/test/julia_dde/dde.jl
new file mode 100644
index 00000000..acf20d7c
--- /dev/null
+++ b/test/julia_dde/dde.jl
@@ -0,0 +1,315 @@
+# File that generates DDE dynamics of several systems
+
+using DifferentialEquations
+using DelimitedFiles
+using PyPlot
+
+# Basic check 1 =========================================
+function basic_check_1(du, u, h, p, t)
+ tau = p
+ hist1 = h(p, t-tau)[1]
+ du[1] = u[1] * (1 - hist1)
+end
+
+h(p,t) = 1.2 * ones(1)
+u0 = [1.2]
+tau = 1.0
+lags = [tau]
+p = (tau)
+tspan = (0.0, 50.0)
+
+
+prob = DDEProblem(basic_check_1, u0, h, tspan, p; constant_lags=lags)
+alg = MethodOfSteps(Tsit5()) # doesn't work with DP5 DP8 but works with Tsit5 and Bosh3
+sol = solve(prob,alg, saveat=0.05)
+usol = transpose(hcat(sol.u...))
+time = sol.t
+
+plot(sol.t, usol)
+plt.xlabel("Time")
+plt.show()
+
+writedlm("test_basic_check_1.txt", [sol.t usol])
+
+# Basic check 2 =========================================
+function basic_check_2(du, u, h, p, t)
+ tau = p
+ hist1 = h(p, t-tau)[1]
+ du[1] = u[1] * (1 - hist1)
+end
+
+h(p,t) = 1.2 * ones(1)
+u0 = [1.2]
+tau = 2.0
+lags = [tau]
+p = (tau)
+tspan = (0.0, 50.0)
+
+
+prob = DDEProblem(basic_check_2, u0, h, tspan, p; constant_lags=lags)
+alg = MethodOfSteps(BS3()) # doesn't work with DP5 DP8 but works with Tsit5 and Bosh3
+sol = solve(prob,alg, saveat=0.05)
+usol = transpose(hcat(sol.u...))
+time = sol.t
+
+plot(sol.t, usol)
+plt.xlabel("Time")
+plt.show()
+
+writedlm("test_basic_check_2.txt", [sol.t usol])
+
+
+# Basic check 3 & 4 =========================================
+function basic_check_3(du, u, h, p, t)
+ tau = p
+ hist1 = h(p, t-tau)[1]
+ du[1] = u[1] * (1 - hist1)
+end
+
+h(p,t) = 1.2 * ones(1)
+u0 = [1.2]
+tau = 3.0
+lags = [tau]
+p = (tau)
+tspan = (0.0, 50.0)
+
+
+prob = DDEProblem(basic_check_3, u0, h, tspan, p; constant_lags=lags)
+alg = MethodOfSteps(BS3()) # doesn't work with DP5 DP8 but works with Tsit5 and Bosh3
+sol = solve(prob,alg, saveat=0.05)
+usol = transpose(hcat(sol.u...))
+
+alg = MethodOfSteps(Tsit5()) # doesn't work with DP5 DP8 but works with Tsit5 and Bosh3
+sol2 = solve(prob,alg, saveat=0.05)
+usol2 = transpose(hcat(sol2.u...))
+time = sol.t
+
+plot(sol.t, usol2)
+plot(sol.t, usol)
+plt.xlabel("Time")
+plt.show()
+
+writedlm("test_basic_check_3.txt", [sol.t usol])
+writedlm("test_basic_check_4.txt", [sol.t usol2])
+
+# Basic check 5 & 6 =========================================
+function basic_check_5(du, u, h, p, t)
+ tau = p
+ hist1 = h(p, t-tau)[1]
+ du[1] = u[1] * (1 - hist1)
+end
+
+h(p,t) = 1.2 * ones(1)
+u0 = [1.2]
+tau = 4.0
+lags = [tau]
+p = (tau)
+tspan = (0.0, 50.0)
+
+
+prob = DDEProblem(basic_check_5, u0, h, tspan, p; constant_lags=lags)
+alg = MethodOfSteps(BS3()) # doesn't work with DP5 DP8 but works with Tsit5 and Bosh3
+sol = solve(prob,alg, saveat=0.05)
+usol = transpose(hcat(sol.u...))
+
+alg = MethodOfSteps(Tsit5()) # doesn't work with DP5 DP8 but works with Tsit5 and Bosh3
+sol2 = solve(prob,alg, saveat=0.05)
+usol2 = transpose(hcat(sol2.u...))
+time = sol.t
+
+plot(sol.t, usol2)
+plot(sol.t, usol)
+plt.xlabel("Time")
+plt.show()
+
+writedlm("test_basic_check_5.txt", [sol.t usol])
+writedlm("test_basic_check_6.txt", [sol.t usol2])
+
+# Basic check 7 =========================================
+function basic_check_7(du, u, h, p, t)
+ tau = p
+ hist1 = h(p, t-tau)[1]
+ du[1] = u[1] * (1 - hist1)
+end
+
+h(p,t) = 1.2 * ones(1)
+u0 = [1.2]
+tau = 4.0
+lags = [tau]
+p = (tau)
+tspan = (0.0, 50.0)
+
+
+prob = DDEProblem(basic_check_7, u0, h, tspan, p; constant_lags=lags)
+alg = MethodOfSteps(Kvaerno5()) # doesn't work with DP5 DP8 but works with Tsit5 and Bosh3
+sol = solve(prob,alg, saveat=0.05)
+usol = transpose(hcat(sol.u...))
+time = sol.t
+
+plot(sol.t, usol)
+plt.xlabel("Time")
+plt.show()
+
+writedlm("test_basic_check_7.txt", [sol.t usol])
+
+# Basic check 8 =========================================
+function basic_check_8(du, u, h, p, t)
+ tau1, tau2 = p
+ hist1 = h(p, t-tau1)[1]
+ hist2 = h(p, t-tau2)[1]
+ du[1] = - hist1 - hist2
+end
+
+h(p,t) = 1.2 * ones(1)
+lags = [1.0/3 1.0/5]
+p = (1.0/3, 1.0/5)
+tspan = (0.0, 10.0)
+u0 = [1.2]
+
+prob = DDEProblem(basic_check_8, u0, h, tspan, p; constant_lags=lags)
+alg = MethodOfSteps(Tsit5())
+sol = solve(prob,alg, saveat=0.1)
+usol = transpose(hcat(sol.u...))
+time = sol.t
+
+plot(sol.t, usol)
+plt.xlabel("Time")
+plt.show()
+
+writedlm("test_basic_check_8.txt", [sol.t usol])
+
+# Basic check 9 =========================================
+function basic_check_9(du, u, h, p, t)
+ tau = p
+ hist1 = h(p, t-tau)[1]
+ du[1] = 0.2 * hist1 / (1+ hist1^10) - 0.1 * u[1]
+end
+
+
+h(p,t) = 1.2 * ones(1)
+u0 = [1.2]
+tau = 6.0
+lags = [tau]
+p = (tau)
+tspan = (0.0, 50.0)
+
+prob = DDEProblem(basic_check_9, u0, h, tspan, p; constant_lags=lags)
+alg = MethodOfSteps(Tsit5())
+sol = solve(prob,alg, saveat=0.1)
+usol = transpose(hcat(sol.u...))
+time = sol.t
+
+plot(sol.t, usol)
+plt.xlabel("Time")
+plt.show()
+
+writedlm("test_basic_check_9.txt", [sol.t usol])
+
+# Basic check 10 =========================================
+function basic_check_10(du, u, h, p, t)
+ hist1 = h(p, t- 2 -sin(t))[1]
+ du[1] = u[1] * (1 - hist1)
+end
+
+function h_basic_check_10(p, t)
+ 1.2 * ones(1)
+end
+
+prob = DDEProblem(basic_check_10, h_basic_check_10, (0.0, 40.0) ; dependent_lags = ((u, p, t) -> 2 + sin(t),))
+alg = MethodOfSteps(BS3())
+sol = solve(prob,alg, saveat=0.1)
+usol = transpose(hcat(sol.u...))
+time = sol.t
+
+plot(sol.t, usol)
+plt.xlabel("Time")
+plt.show()
+
+writedlm("test_basic_check_10.txt", [sol.t usol])
+
+Basic check 11 =========================================
+function basic_check_11(du, u, h, p, t)
+ hist1 = h(p, t- 1/2*(exp(-u[1]^2) + 1))[1]
+ du[1] = -10* hist1
+end
+
+function h_basic_check_11(p, t)
+ 1.0 * ones(1)
+end
+
+prob = DDEProblem(basic_check_11, h_basic_check_11, (0.0, 5.0) ; dependent_lags = ((u, p, t) -> 1/2*(exp(-u[1]^2) + 1),))
+alg = MethodOfSteps(Kvaerno5())
+sol = solve(prob,alg, saveat=0.01)
+usol = transpose(hcat(sol.u...))
+
+alg = MethodOfSteps(Kvaerno4())
+sol = solve(prob,alg, saveat=0.01)
+usol2 = transpose(hcat(sol.u...))
+
+time = sol.t
+
+plot(sol.t, usol2, label="Kv4")
+plot(sol.t, usol, label="Kv5")
+plt.xlabel("Time")
+plt.legend()
+plt.show()
+
+writedlm("test_basic_check_11.txt", [sol.t usol])
+
+
+# Numerical check 1 =========================================
+function numerical_check_1(du, u, h, p, t)
+ hist1 = h(p, u[1])[1]
+ du[1] = hist1
+end
+
+function h_numerical_check_1(p, t)
+ if t < 2.0
+ 1/2 * ones(1)
+ else
+ 1 * ones(1)
+ end
+end
+
+prob = DDEProblem(numerical_check_1, h_numerical_check_1, u0=[1.0 * ones(1)], (2.0, 5.5) ; dependent_lags = ((u, p, t) -> u[1],))
+alg = MethodOfSteps(BS3())
+sol = solve(prob,alg, saveat=0.01)
+usol = transpose(hcat(sol.u...))
+
+alg = MethodOfSteps(Tsit5())
+sol = solve(prob,alg, saveat=0.01)
+usol2 = transpose(hcat(sol.u...))
+
+time = sol.t
+
+plot(sol.t, usol2, label="Kv4")
+plot(sol.t, usol, label="Kv5")
+plt.xlabel("Time")
+plt.legend()
+plt.show()
+
+writedlm("test_basic_numerical_check_1.txt", [sol.t usol])
+
+# Numerical check 2 =========================================
+function numerical_check_2(du, u, h, p, t)
+ hist1 = h(p, log(u[1]))[1]
+ du[1] = hist1 * u[1] / t
+end
+
+function h_numerical_check_2(p, t)
+ 1 * ones(1)
+end
+
+prob = DDEProblem(numerical_check_2, h_numerical_check_2, (1.0, 10.0) ; dependent_lags = ((u, p, t) -> log(u[1]),))
+alg = MethodOfSteps(BS3())
+sol = solve(prob,alg, saveat=0.01)
+usol = transpose(hcat(sol.u...))
+time = sol.t
+
+plot(sol.t, usol2, label="Kv4")
+plot(sol.t, usol, label="Kv5")
+plt.xlabel("Time")
+plt.legend()
+plt.show()
+
+writedlm("test_basic_numerical_check_2.txt", [sol.t usol])
diff --git a/test/julia_dde/test_basic_check_1.txt b/test/julia_dde/test_basic_check_1.txt
new file mode 100644
index 00000000..ea119434
--- /dev/null
+++ b/test/julia_dde/test_basic_check_1.txt
@@ -0,0 +1,1001 @@
+0.0 1.2
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diff --git a/test/julia_dde/test_basic_check_10.txt b/test/julia_dde/test_basic_check_10.txt
new file mode 100644
index 00000000..0e15e27c
--- /dev/null
+++ b/test/julia_dde/test_basic_check_10.txt
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diff --git a/test/julia_dde/test_basic_check_11.txt b/test/julia_dde/test_basic_check_11.txt
new file mode 100644
index 00000000..c1948729
--- /dev/null
+++ b/test/julia_dde/test_basic_check_11.txt
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diff --git a/test/julia_dde/test_basic_check_2.txt b/test/julia_dde/test_basic_check_2.txt
new file mode 100644
index 00000000..66a0924a
--- /dev/null
+++ b/test/julia_dde/test_basic_check_2.txt
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diff --git a/test/julia_dde/test_basic_check_3.txt b/test/julia_dde/test_basic_check_3.txt
new file mode 100644
index 00000000..20ab7e04
--- /dev/null
+++ b/test/julia_dde/test_basic_check_3.txt
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diff --git a/test/julia_dde/test_basic_check_4.txt b/test/julia_dde/test_basic_check_4.txt
new file mode 100644
index 00000000..e61f2232
--- /dev/null
+++ b/test/julia_dde/test_basic_check_4.txt
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diff --git a/test/julia_dde/test_basic_check_5.txt b/test/julia_dde/test_basic_check_5.txt
new file mode 100644
index 00000000..db42a42d
--- /dev/null
+++ b/test/julia_dde/test_basic_check_5.txt
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diff --git a/test/julia_dde/test_basic_check_6.txt b/test/julia_dde/test_basic_check_6.txt
new file mode 100644
index 00000000..eb94fce5
--- /dev/null
+++ b/test/julia_dde/test_basic_check_6.txt
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diff --git a/test/julia_dde/test_basic_check_7.txt b/test/julia_dde/test_basic_check_7.txt
new file mode 100644
index 00000000..cb4d1299
--- /dev/null
+++ b/test/julia_dde/test_basic_check_7.txt
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diff --git a/test/julia_dde/test_basic_check_8.txt b/test/julia_dde/test_basic_check_8.txt
new file mode 100644
index 00000000..03be058d
--- /dev/null
+++ b/test/julia_dde/test_basic_check_8.txt
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diff --git a/test/julia_dde/test_basic_check_9.txt b/test/julia_dde/test_basic_check_9.txt
new file mode 100644
index 00000000..36c43baf
--- /dev/null
+++ b/test/julia_dde/test_basic_check_9.txt
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diff --git a/test/julia_dde/test_basic_numerical_check_1.txt b/test/julia_dde/test_basic_numerical_check_1.txt
new file mode 100644
index 00000000..a0f079db
--- /dev/null
+++ b/test/julia_dde/test_basic_numerical_check_1.txt
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diff --git a/test/julia_dde/test_basic_numerical_check_2.txt b/test/julia_dde/test_basic_numerical_check_2.txt
new file mode 100644
index 00000000..3cc1ca7b
--- /dev/null
+++ b/test/julia_dde/test_basic_numerical_check_2.txt
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diff --git a/test/test_delays.py b/test/test_delays.py
new file mode 100644
index 00000000..da85900d
--- /dev/null
+++ b/test/test_delays.py
@@ -0,0 +1,1109 @@
+import diffrax
+import jax.lax as lax
+import jax.numpy as jnp
+import jax.random as jrandom
+import pytest
+from diffrax.delays import Delays
+
+
+def open_process_file(path):
+ ts, ys = [], []
+ with open(path, "r", encoding="utf-8") as infile:
+ for line in infile:
+ data = line.split()
+ ts.append(float(data[0])), ys.append(float(data[1]))
+ return jnp.array(ts), jnp.array(ys)
+
+
+def test_dde_solver_with_ode():
+ # testing that dde solver solves ode
+ # when history not specific
+ key = jrandom.PRNGKey(5678)
+ akey, ykey = jrandom.split(key, 2)
+
+ A = jrandom.normal(akey, (10, 10), dtype=jnp.float64) * 0.5
+
+ def dde_f(t, y, args, history):
+ return A @ y
+
+ def ode_f(t, y, args):
+ return A @ y
+
+ dde_term = diffrax.ODETerm(dde_f)
+ ode_term = diffrax.ODETerm(ode_f)
+ t0 = 0
+ t1 = 4
+ ts = jnp.linspace(t0, t1, int(10 * (t1 - t0)))
+ y0 = jrandom.normal(ykey, (10,), dtype=jnp.float64)
+ delays = diffrax.Delays(
+ delays=[lambda t, y, args: 0.2],
+ initial_discontinuities=jnp.array([0.0]),
+ max_discontinuities=100,
+ recurrent_checking=True,
+ rtol=10e-3,
+ atol=10e-6,
+ )
+
+ dt0 = 0.1
+ dde_sol = diffrax.diffeqsolve(
+ dde_term,
+ diffrax.Dopri5(),
+ t0,
+ t1,
+ dt0,
+ y0=lambda t: y0,
+ delays=delays,
+ saveat=diffrax.SaveAt(ts=ts, dense=True),
+ )
+ ode_sol = diffrax.diffeqsolve(
+ ode_term,
+ diffrax.Dopri5(),
+ t0,
+ t1,
+ dt0,
+ y0,
+ saveat=diffrax.SaveAt(ts=ts, dense=True),
+ )
+
+ error = jnp.mean(jnp.abs(dde_sol.ys - ode_sol.ys))
+ assert error < 10**-5
+
+
+def test_jump_ts_dde_solver():
+ # test jump ts with dde solver
+ # when t=2.0 the vf changes
+
+ key = jrandom.PRNGKey(5678)
+
+ def vf(t, y, args, history):
+ sign = jnp.where(t < 2, 1, -1)
+ return sign * history[0]
+
+ def first_part_vf(t, y, args, history):
+ return history[0]
+
+ def second_part_vf(t, y, args, history):
+ return -history[0]
+
+ t0, t1 = 0.0, 4.0
+ dt0 = 0.1
+ ts_first = jnp.linspace(0, 2.0, 20)
+ ts_second = jnp.linspace(2.0, 4.0, 20)
+ ts = jnp.concatenate([ts_first, ts_second[1:]])
+ y0 = jrandom.normal(key, (1,), dtype=jnp.float64)
+ delays = diffrax.Delays(
+ delays=[lambda t, y, args: 1.0],
+ initial_discontinuities=jnp.array([0.0]),
+ max_discontinuities=10,
+ recurrent_checking=False,
+ rtol=10e-3,
+ atol=10e-6,
+ )
+
+ delays2 = diffrax.Delays(
+ delays=[lambda t, y, args: 1.0],
+ initial_discontinuities=jnp.array([2.0]),
+ max_discontinuities=10,
+ recurrent_checking=False,
+ rtol=10e-3,
+ atol=10e-6,
+ )
+
+ first_part_dde = diffrax.diffeqsolve(
+ diffrax.ODETerm(first_part_vf),
+ diffrax.Dopri5(),
+ t0,
+ ts_first[-1],
+ dt0,
+ y0=lambda t: y0,
+ delays=delays,
+ saveat=diffrax.SaveAt(ts=ts_first, dense=True),
+ )
+ second_part_dde = diffrax.diffeqsolve(
+ diffrax.ODETerm(second_part_vf),
+ diffrax.Dopri5(),
+ ts_first[-1],
+ t1,
+ dt0,
+ y0=lambda t: first_part_dde.interpolation.evaluate(t),
+ delays=delays2,
+ stepsize_controller=diffrax.PIDController(rtol=1e-6, atol=1e-3),
+ saveat=diffrax.SaveAt(ts=ts_second, dense=True),
+ )
+ complete_dde = diffrax.diffeqsolve(
+ diffrax.ODETerm(vf),
+ diffrax.Dopri5(),
+ t0,
+ t1,
+ dt0,
+ y0=lambda t: y0,
+ delays=delays,
+ stepsize_controller=diffrax.PIDController(
+ rtol=1e-9, atol=1e-6, jump_ts=jnp.array([2.0])
+ ),
+ saveat=diffrax.SaveAt(ts=ts, dense=True),
+ )
+
+ error = jnp.mean(
+ jnp.abs(
+ complete_dde.ys
+ - jnp.concatenate([first_part_dde.ys, second_part_dde.ys[1:]])
+ )
+ )
+ assert error < 1**-5
+
+
+def test_smooth_dde():
+ # testing a smooth dde with no initial discontinuities
+ # y' = y(t-1), phi = t + 1
+ # for t in [0,1], y(t) = t**2/2 + 1
+ # for t in [1,2], y(t) = (t-1)**3/(2*3) + t + 1/2
+ # we compare the values at t=1,2 with their analytical
+ # solution
+ def dde_f(t, y, args, history):
+ return history[0]
+
+ dde_term = diffrax.ODETerm(dde_f)
+ t0, t1 = 0.0, 2.0
+ ts = jnp.linspace(t0, t1, 100)
+ delays = diffrax.Delays(
+ delays=[lambda t, y, args: 1.0],
+ initial_discontinuities=None,
+ max_discontinuities=10,
+ recurrent_checking=True,
+ rtol=10e-3,
+ atol=10e-6,
+ )
+
+ dt0 = 0.1
+ dde_sol = diffrax.diffeqsolve(
+ dde_term,
+ diffrax.Dopri5(),
+ t0,
+ t1,
+ dt0,
+ y0=lambda t: t + 1,
+ delays=delays,
+ saveat=diffrax.SaveAt(ts=ts, dense=True),
+ )
+
+ error1 = jnp.mean(jnp.abs(dde_sol.ys[50] - 3 / 2))
+ error2 = jnp.mean(jnp.abs(dde_sol.ys[100] - 8 / 3))
+
+ assert error1 < 10**-1
+ assert error2 < 10**-3
+
+
+def _test_exceed_max_discontinuities():
+ # we recurrent_checking to True and
+ # integrate a DDE with a delay equal
+ # to 1 and hence from t > 10.0 we
+ # should have RunTimeError picked
+ # up
+ def dde_f(t, y, args, history):
+ return -history[0]
+
+ t0, t1 = 0.0, 12.0
+ ts = jnp.linspace(t0, t1, 120)
+ delays = diffrax.Delays(
+ delays=[lambda t, y, args: 1.0],
+ initial_discontinuities=jnp.array([0.0]),
+ max_discontinuities=10,
+ recurrent_checking=True,
+ rtol=10e-3,
+ atol=10e-6,
+ )
+
+ dt0 = 0.1
+
+ return diffrax.diffeqsolve(
+ diffrax.ODETerm(dde_f),
+ diffrax.Dopri5(),
+ t0,
+ t1,
+ dt0,
+ y0=lambda t: 1.0,
+ delays=delays,
+ saveat=diffrax.SaveAt(ts=ts, dense=True),
+ )
+
+
+def test_exceed_max_discontinuities():
+ with pytest.raises(RuntimeError):
+ _test_exceed_max_discontinuities()
+
+
+def test_only_explicit_stepping():
+ # we check that we only do explicit
+ # stepping here by putting
+ # dt = 0.9 < delays
+ def dde_f(t, y, args, history):
+ return -history[0]
+
+ t0, t1 = 0.0, 12.0
+ ts = jnp.linspace(t0, t1, 120)
+ delays = diffrax.Delays(
+ delays=[lambda t, y, args: 1.0],
+ initial_discontinuities=jnp.array([0.0]),
+ max_discontinuities=10,
+ recurrent_checking=False,
+ rtol=10e-3,
+ atol=10e-6,
+ )
+
+ dt0 = 0.1
+ sol = diffrax.diffeqsolve(
+ diffrax.ODETerm(dde_f),
+ diffrax.Bosh3(),
+ t0=ts[0],
+ t1=ts[-1],
+ dt0=dt0,
+ y0=lambda t: 1.0,
+ stepsize_controller=diffrax.PIDController(rtol=1e-6, atol=1e-9, dtmax=0.9),
+ saveat=diffrax.SaveAt(ts=ts, dense=True),
+ delays=delays,
+ )
+
+ assert sol.stats["num_dde_implicit_step"] == 0
+ assert sol.stats["num_dde_explicit_step"] > 0
+
+
+def test_hit_explicit_and_implicit_stepping():
+ # we check that we only do implicit
+ # stepping here by putting
+ # dt=1.1 > delays
+ def dde_f(t, y, args, history):
+ return -history[0]
+
+ t0, t1 = 0.0, 25.0
+ ts = jnp.linspace(t0, t1, 120)
+ delays = diffrax.Delays(
+ delays=[lambda t, y, args: 1.0],
+ initial_discontinuities=jnp.array([0.0]),
+ max_discontinuities=10,
+ recurrent_checking=False,
+ rtol=10e-3,
+ atol=10e-6,
+ )
+
+ sol = diffrax.diffeqsolve(
+ diffrax.ODETerm(dde_f),
+ diffrax.Dopri5(),
+ t0=ts[0],
+ t1=ts[-1],
+ dt0=0.1,
+ y0=lambda t: 1.0,
+ stepsize_controller=diffrax.PIDController(rtol=1e-3, atol=1e-6),
+ saveat=diffrax.SaveAt(ts=ts, dense=True),
+ delays=delays,
+ )
+ assert sol.stats["num_dde_implicit_step"] > 0
+ assert sol.stats["num_dde_explicit_step"] > 0
+
+
+def test_basic_check_1():
+ def vector_field(t, y, args, *, history):
+ return y * (1 - history[0])
+
+ y0_history = lambda t: 1.2
+
+ delays = Delays(
+ delays=[lambda t, y, args: 1.0],
+ initial_discontinuities=jnp.array([0.0]),
+ max_discontinuities=100,
+ recurrent_checking=False,
+ rtol=10e-3,
+ atol=10e-6,
+ )
+ made_jump = delays.initial_discontinuities is None
+ t0, t1 = 0.0, 50.0
+ ts = jnp.linspace(t0, t1, 1001)
+ sol = diffrax.diffeqsolve(
+ diffrax.ODETerm(vector_field),
+ diffrax.Tsit5(),
+ t0=ts[0],
+ t1=ts[-1],
+ dt0=ts[1] - ts[0],
+ y0=y0_history,
+ stepsize_controller=diffrax.PIDController(rtol=1e-6, atol=1e-9),
+ saveat=diffrax.SaveAt(ts=ts, dense=True),
+ delays=delays,
+ made_jump=made_jump,
+ )
+
+ delays = Delays(
+ delays=[lambda t, y, args: 1.0],
+ initial_discontinuities=jnp.array([0.0]),
+ max_discontinuities=100,
+ recurrent_checking=True,
+ rtol=10e-3,
+ atol=10e-6,
+ )
+ sol2 = diffrax.diffeqsolve(
+ diffrax.ODETerm(vector_field),
+ diffrax.Tsit5(),
+ t0=ts[0],
+ t1=ts[-1],
+ dt0=ts[1] - ts[0],
+ y0=y0_history,
+ stepsize_controller=diffrax.PIDController(rtol=1e-6, atol=1e-9),
+ saveat=diffrax.SaveAt(ts=ts, dense=True),
+ delays=delays,
+ made_jump=made_jump,
+ )
+
+ _, juliays = open_process_file("test/julia_dde/test_basic_check_1.txt")
+ error1, error2, error3 = (
+ jnp.mean(jnp.abs(juliays - sol.ys)),
+ jnp.mean(jnp.abs(juliays - sol2.ys)),
+ jnp.mean(jnp.abs(sol.ys - sol2.ys)),
+ )
+
+ assert error1 < 1e-4
+ assert error2 < 1e-4
+ assert error3 < 1e-5
+
+
+def test_basic_check_2():
+ def vector_field(t, y, args, *, history):
+ return y * (1 - history[0])
+
+ y0_history = lambda t: 1.2
+
+ delays = Delays(
+ delays=[lambda t, y, args: 2.0],
+ initial_discontinuities=jnp.array([0.0]),
+ max_discontinuities=100,
+ recurrent_checking=False,
+ rtol=10e-3,
+ atol=10e-6,
+ )
+ made_jump = delays.initial_discontinuities is None
+ t0, t1 = 0.0, 50.0
+ ts = jnp.linspace(t0, t1, 1001)
+ sol = diffrax.diffeqsolve(
+ diffrax.ODETerm(vector_field),
+ diffrax.Bosh3(),
+ t0=ts[0],
+ t1=ts[-1],
+ dt0=ts[1] - ts[0],
+ y0=y0_history,
+ stepsize_controller=diffrax.PIDController(rtol=1e-6, atol=1e-9),
+ saveat=diffrax.SaveAt(ts=ts, dense=True),
+ delays=delays,
+ made_jump=made_jump,
+ )
+
+ delays = Delays(
+ delays=[lambda t, y, args: 2.0],
+ initial_discontinuities=jnp.array([0.0]),
+ max_discontinuities=100,
+ recurrent_checking=True,
+ rtol=10e-3,
+ atol=10e-6,
+ )
+ sol2 = diffrax.diffeqsolve(
+ diffrax.ODETerm(vector_field),
+ diffrax.Bosh3(),
+ t0=ts[0],
+ t1=ts[-1],
+ dt0=ts[1] - ts[0],
+ y0=y0_history,
+ stepsize_controller=diffrax.PIDController(rtol=1e-6, atol=1e-9),
+ saveat=diffrax.SaveAt(ts=ts, dense=True),
+ delays=delays,
+ made_jump=made_jump,
+ )
+
+ _, juliays = open_process_file("test/julia_dde/test_basic_check_2.txt")
+ error1, error2, error3 = (
+ jnp.mean(jnp.abs(juliays - sol.ys)),
+ jnp.mean(jnp.abs(juliays - sol2.ys)),
+ jnp.mean(jnp.abs(sol.ys - sol2.ys)),
+ )
+
+ assert error1 < 1e-2
+ assert error2 < 1e-2
+ assert error3 < 1e-5
+
+
+def test_basic_check_3():
+ # same test as test_basic_check_2 but we
+ # have a larger delay =3
+ def vector_field(t, y, args, *, history):
+ return y * (1 - history[0])
+
+ y0_history = lambda t: 1.2
+
+ delays = Delays(
+ delays=[lambda t, y, args: 3.0],
+ initial_discontinuities=jnp.array([0.0]),
+ max_discontinuities=100,
+ recurrent_checking=False,
+ rtol=10e-3,
+ atol=10e-6,
+ )
+ made_jump = delays.initial_discontinuities is None
+ t0, t1 = 0.0, 50.0
+ ts = jnp.linspace(t0, t1, 1001)
+ sol = diffrax.diffeqsolve(
+ diffrax.ODETerm(vector_field),
+ diffrax.Bosh3(),
+ t0=ts[0],
+ t1=ts[-1],
+ dt0=ts[1] - ts[0],
+ y0=y0_history,
+ stepsize_controller=diffrax.PIDController(rtol=1e-6, atol=1e-9),
+ saveat=diffrax.SaveAt(ts=ts, dense=True),
+ delays=delays,
+ made_jump=made_jump,
+ )
+
+ delays = Delays(
+ delays=[lambda t, y, args: 3.0],
+ initial_discontinuities=jnp.array([0.0]),
+ max_discontinuities=100,
+ recurrent_checking=True,
+ rtol=10e-3,
+ atol=10e-6,
+ )
+ sol2 = diffrax.diffeqsolve(
+ diffrax.ODETerm(vector_field),
+ diffrax.Bosh3(),
+ t0=ts[0],
+ t1=ts[-1],
+ dt0=ts[1] - ts[0],
+ y0=y0_history,
+ stepsize_controller=diffrax.PIDController(rtol=1e-6, atol=1e-9),
+ saveat=diffrax.SaveAt(ts=ts, dense=True),
+ delays=delays,
+ made_jump=made_jump,
+ )
+
+ _, juliays = open_process_file("test/julia_dde/test_basic_check_3.txt")
+ error1, error2, error3 = (
+ jnp.mean(jnp.abs(juliays - sol.ys)),
+ jnp.mean(jnp.abs(juliays - sol2.ys)),
+ jnp.mean(jnp.abs(sol.ys - sol2.ys)),
+ )
+
+ assert error1 < 0.15
+ assert error2 < 0.15
+ assert error3 < 1e-5
+
+
+def test_basic_check_4():
+ # same experiment as test_basic_check_3
+ # but solver is Tsit5
+ def vector_field(t, y, args, *, history):
+ return y * (1 - history[0])
+
+ y0_history = lambda t: 1.2
+
+ delays = Delays(
+ delays=[lambda t, y, args: 3.0],
+ initial_discontinuities=jnp.array([0.0]),
+ max_discontinuities=100,
+ recurrent_checking=False,
+ rtol=10e-3,
+ atol=10e-6,
+ )
+ made_jump = delays.initial_discontinuities is None
+ t0, t1 = 0.0, 50.0
+ ts = jnp.linspace(t0, t1, 1001)
+ sol = diffrax.diffeqsolve(
+ diffrax.ODETerm(vector_field),
+ diffrax.Tsit5(),
+ t0=ts[0],
+ t1=ts[-1],
+ dt0=ts[1] - ts[0],
+ y0=y0_history,
+ stepsize_controller=diffrax.PIDController(rtol=1e-6, atol=1e-9),
+ saveat=diffrax.SaveAt(ts=ts, dense=True),
+ delays=delays,
+ made_jump=made_jump,
+ )
+
+ delays = Delays(
+ delays=[lambda t, y, args: 3.0],
+ initial_discontinuities=jnp.array([0.0]),
+ max_discontinuities=100,
+ recurrent_checking=True,
+ rtol=10e-3,
+ atol=10e-6,
+ )
+ sol2 = diffrax.diffeqsolve(
+ diffrax.ODETerm(vector_field),
+ diffrax.Tsit5(),
+ t0=ts[0],
+ t1=ts[-1],
+ dt0=ts[1] - ts[0],
+ y0=y0_history,
+ stepsize_controller=diffrax.PIDController(rtol=1e-6, atol=1e-9),
+ saveat=diffrax.SaveAt(ts=ts, dense=True),
+ delays=delays,
+ made_jump=made_jump,
+ )
+
+ _, juliays = open_process_file("test/julia_dde/test_basic_check_4.txt")
+ error1, error2, error3 = (
+ jnp.mean(jnp.abs(juliays - sol.ys)),
+ jnp.mean(jnp.abs(juliays - sol2.ys)),
+ jnp.mean(jnp.abs(sol.ys - sol2.ys)),
+ )
+
+ assert error1 < 1e-2
+ assert error2 < 1e-2
+ assert error3 < 1e-5
+
+
+def test_basic_check_5():
+ # same test as test_basic_check_3 but we
+ # have a larger delay =4
+ def vector_field(t, y, args, *, history):
+ return y * (1 - history[0])
+
+ y0_history = lambda t: 1.2
+
+ delays = Delays(
+ delays=[lambda t, y, args: 4.0],
+ initial_discontinuities=jnp.array([0.0]),
+ max_discontinuities=10,
+ recurrent_checking=False,
+ rtol=10e-3,
+ atol=10e-6,
+ )
+ made_jump = delays.initial_discontinuities is None
+ t0, t1 = 0.0, 50.0
+ ts = jnp.linspace(t0, t1, 1001)
+ sol = diffrax.diffeqsolve(
+ diffrax.ODETerm(vector_field),
+ diffrax.Bosh3(),
+ t0=ts[0],
+ t1=ts[-1],
+ dt0=ts[1] - ts[0],
+ y0=y0_history,
+ stepsize_controller=diffrax.PIDController(rtol=1e-6, atol=1e-9),
+ saveat=diffrax.SaveAt(ts=ts, dense=True),
+ delays=delays,
+ made_jump=made_jump,
+ )
+
+ delays = Delays(
+ delays=[lambda t, y, args: 4.0],
+ initial_discontinuities=jnp.array([0.0]),
+ max_discontinuities=100,
+ recurrent_checking=True,
+ rtol=10e-3,
+ atol=10e-6,
+ )
+ sol2 = diffrax.diffeqsolve(
+ diffrax.ODETerm(vector_field),
+ diffrax.Bosh3(),
+ t0=ts[0],
+ t1=ts[-1],
+ dt0=ts[1] - ts[0],
+ y0=y0_history,
+ stepsize_controller=diffrax.PIDController(rtol=1e-6, atol=1e-9),
+ saveat=diffrax.SaveAt(ts=ts, dense=True),
+ delays=delays,
+ made_jump=made_jump,
+ )
+
+ _, juliays = open_process_file("test/julia_dde/test_basic_check_5.txt")
+ error1, error2, error3 = (
+ jnp.mean(jnp.abs(juliays - sol.ys)),
+ jnp.mean(jnp.abs(juliays - sol2.ys)),
+ jnp.mean(jnp.abs(sol.ys - sol2.ys)),
+ )
+
+ assert error1 < 0.2
+ assert error2 < 0.2
+ assert error3 < 1e-5
+
+
+def test_basic_check_6():
+ # same experiment as test_basic_check_5
+ # but solver is Tsit5
+ def vector_field(t, y, args, *, history):
+ return y * (1 - history[0])
+
+ y0_history = lambda t: 1.2
+
+ delays = Delays(
+ delays=[lambda t, y, args: 4.0],
+ initial_discontinuities=jnp.array([0.0]),
+ max_discontinuities=100,
+ recurrent_checking=False,
+ rtol=10e-3,
+ atol=10e-6,
+ )
+ made_jump = delays.initial_discontinuities is None
+ t0, t1 = 0.0, 50.0
+ ts = jnp.linspace(t0, t1, 1001)
+ sol = diffrax.diffeqsolve(
+ diffrax.ODETerm(vector_field),
+ diffrax.Tsit5(),
+ t0=ts[0],
+ t1=ts[-1],
+ dt0=ts[1] - ts[0],
+ y0=y0_history,
+ stepsize_controller=diffrax.PIDController(rtol=1e-6, atol=1e-9),
+ saveat=diffrax.SaveAt(ts=ts, dense=True),
+ delays=delays,
+ made_jump=made_jump,
+ )
+
+ delays = Delays(
+ delays=[lambda t, y, args: 4.0],
+ initial_discontinuities=jnp.array([0.0]),
+ max_discontinuities=100,
+ recurrent_checking=True,
+ rtol=10e-3,
+ atol=10e-6,
+ )
+ sol2 = diffrax.diffeqsolve(
+ diffrax.ODETerm(vector_field),
+ diffrax.Tsit5(),
+ t0=ts[0],
+ t1=ts[-1],
+ dt0=ts[1] - ts[0],
+ y0=y0_history,
+ stepsize_controller=diffrax.PIDController(rtol=1e-6, atol=1e-9),
+ saveat=diffrax.SaveAt(ts=ts, dense=True),
+ delays=delays,
+ made_jump=made_jump,
+ )
+
+ _, juliays = open_process_file("test/julia_dde/test_basic_check_6.txt")
+ error1, error2, error3 = (
+ jnp.mean(jnp.abs(juliays - sol.ys)),
+ jnp.mean(jnp.abs(juliays - sol2.ys)),
+ jnp.mean(jnp.abs(sol.ys - sol2.ys)),
+ )
+
+ assert error1 < 1e-2
+ assert error2 < 1e-2
+ assert error3 < 1e-5
+
+
+def test_basic_check_7():
+ # same experiment as test_basic_check_7
+ # but solver is implicit
+ def vector_field(t, y, args, *, history):
+ return y * (1 - history[0])
+
+ y0_history = lambda t: 1.2
+
+ delays = Delays(
+ delays=[lambda t, y, args: 4.0],
+ initial_discontinuities=jnp.array([0.0]),
+ max_discontinuities=100,
+ recurrent_checking=False,
+ rtol=10e-3,
+ atol=10e-6,
+ )
+ made_jump = delays.initial_discontinuities is None
+ t0, t1 = 0.0, 50.0
+ ts = jnp.linspace(t0, t1, 1001)
+ sol = diffrax.diffeqsolve(
+ diffrax.ODETerm(vector_field),
+ diffrax.Kvaerno5(),
+ t0=ts[0],
+ t1=ts[-1],
+ dt0=ts[1] - ts[0],
+ y0=y0_history,
+ stepsize_controller=diffrax.PIDController(rtol=1e-6, atol=1e-9),
+ saveat=diffrax.SaveAt(ts=ts, dense=True),
+ delays=delays,
+ made_jump=made_jump,
+ )
+
+ delays = Delays(
+ delays=[lambda t, y, args: 4.0],
+ initial_discontinuities=jnp.array([0.0]),
+ max_discontinuities=100,
+ recurrent_checking=True,
+ rtol=10e-3,
+ atol=10e-6,
+ )
+ sol2 = diffrax.diffeqsolve(
+ diffrax.ODETerm(vector_field),
+ diffrax.Kvaerno5(),
+ t0=ts[0],
+ t1=ts[-1],
+ dt0=ts[1] - ts[0],
+ y0=y0_history,
+ stepsize_controller=diffrax.PIDController(rtol=1e-6, atol=1e-9),
+ saveat=diffrax.SaveAt(ts=ts, dense=True),
+ delays=delays,
+ made_jump=made_jump,
+ )
+
+ _, juliays = open_process_file("test/julia_dde/test_basic_check_7.txt")
+ error1, error2, error3 = (
+ jnp.mean(jnp.abs(juliays - sol.ys)),
+ jnp.mean(jnp.abs(juliays - sol2.ys)),
+ jnp.mean(jnp.abs(sol.ys - sol2.ys)),
+ )
+
+ assert error1 < 1e-1
+ assert error2 < 1e-1
+ assert error3 < 1e-2
+
+
+def test_basic_check_8():
+ # new system with 2 delays
+
+ def vector_field(t, y, args, *, history):
+ return -history[0] - history[1]
+
+ y0_history = lambda t: 1.2
+
+ delays = Delays(
+ delays=[lambda t, y, args: 1 / 5, lambda t, y, args: 1 / 3],
+ initial_discontinuities=jnp.array([0.0]),
+ max_discontinuities=100,
+ recurrent_checking=False,
+ rtol=10e-3,
+ atol=10e-6,
+ )
+ made_jump = delays.initial_discontinuities is None
+ t0, t1 = 0.0, 10.0
+ ts = jnp.linspace(t0, t1, 101)
+ sol = diffrax.diffeqsolve(
+ diffrax.ODETerm(vector_field),
+ diffrax.Tsit5(),
+ t0=ts[0],
+ t1=ts[-1],
+ dt0=ts[1] - ts[0],
+ y0=y0_history,
+ stepsize_controller=diffrax.PIDController(rtol=1e-6, atol=1e-9),
+ saveat=diffrax.SaveAt(ts=ts, dense=True),
+ delays=delays,
+ made_jump=made_jump,
+ )
+
+ delays = Delays(
+ delays=[lambda t, y, args: 1 / 5, lambda t, y, args: 1 / 3],
+ initial_discontinuities=jnp.array([0.0]),
+ max_discontinuities=1000,
+ recurrent_checking=True,
+ rtol=10e-3,
+ atol=10e-6,
+ )
+ sol2 = diffrax.diffeqsolve(
+ diffrax.ODETerm(vector_field),
+ diffrax.Tsit5(),
+ t0=ts[0],
+ t1=ts[-1],
+ dt0=ts[1] - ts[0],
+ y0=y0_history,
+ stepsize_controller=diffrax.PIDController(rtol=1e-6, atol=1e-9),
+ saveat=diffrax.SaveAt(ts=ts, dense=True),
+ delays=delays,
+ made_jump=made_jump,
+ )
+
+ _, juliays = open_process_file("test/julia_dde/test_basic_check_8.txt")
+ error1, error2, error3 = (
+ jnp.mean(jnp.abs(juliays - sol.ys)),
+ jnp.mean(jnp.abs(juliays - sol2.ys)),
+ jnp.mean(jnp.abs(sol.ys - sol2.ys)),
+ )
+
+ assert error1 < 1e-6
+ assert error2 < 1e-6
+ assert error3 < 1e-6
+
+
+def test_basic_check_9():
+ # new system ie Mackey Glass
+
+ def vector_field(t, y, args, *, history):
+ return 0.2 * (history[0]) / (1 + history[0] ** 10) - 0.1 * y
+
+ y0_history = lambda t: 1.2
+
+ delays = Delays(
+ delays=[lambda t, y, args: 6.0],
+ initial_discontinuities=jnp.array([0.0]),
+ max_discontinuities=100,
+ recurrent_checking=False,
+ rtol=10e-3,
+ atol=10e-6,
+ )
+
+ t0, t1, nb_points = 0.0, 50.0, 501
+ ts = jnp.linspace(t0, t1, nb_points)
+ sol = diffrax.diffeqsolve(
+ diffrax.ODETerm(vector_field),
+ diffrax.Tsit5(),
+ t0=ts[0],
+ t1=ts[-1],
+ dt0=ts[1] - ts[0],
+ y0=y0_history,
+ stepsize_controller=diffrax.PIDController(rtol=1e-6, atol=1e-9),
+ saveat=diffrax.SaveAt(ts=ts, dense=True),
+ delays=delays,
+ )
+
+ delays = Delays(
+ delays=[lambda t, y, args: 6.0],
+ initial_discontinuities=jnp.array([0.0]),
+ max_discontinuities=100,
+ recurrent_checking=True,
+ rtol=10e-3,
+ atol=10e-6,
+ )
+
+ sol2 = diffrax.diffeqsolve(
+ diffrax.ODETerm(vector_field),
+ diffrax.Tsit5(),
+ t0=ts[0],
+ t1=ts[-1],
+ dt0=ts[1] - ts[0],
+ y0=y0_history,
+ stepsize_controller=diffrax.PIDController(rtol=1e-6, atol=1e-9),
+ saveat=diffrax.SaveAt(ts=ts, dense=True),
+ delays=delays,
+ )
+
+ _, juliays = open_process_file("test/julia_dde/test_basic_check_9.txt")
+ error1, error2, error3 = (
+ jnp.mean(jnp.abs(juliays - sol.ys)),
+ jnp.mean(jnp.abs(juliays - sol2.ys)),
+ jnp.mean(jnp.abs(sol.ys - sol2.ys)),
+ )
+
+ assert error1 < 1e-4
+ assert error2 < 1e-4
+ assert error3 < 1e-4
+
+
+def test_basic_check_10():
+ # testing a time dependent equation
+
+ def vector_field(t, y, args, *, history):
+ return y * (1 - history[0])
+
+ y0_history = lambda t: 1.2
+
+ delays = Delays(
+ delays=[lambda t, y, args: 2 + jnp.sin(t)],
+ initial_discontinuities=jnp.array([0.0]),
+ max_discontinuities=100,
+ recurrent_checking=False,
+ rtol=10e-3,
+ atol=10e-6,
+ )
+
+ t0, t1, nb_points = 0.0, 40.0, 401
+ ts = jnp.linspace(t0, t1, nb_points)
+ sol = diffrax.diffeqsolve(
+ diffrax.ODETerm(vector_field),
+ diffrax.Bosh3(),
+ t0=ts[0],
+ t1=ts[-1],
+ dt0=ts[1] - ts[0],
+ y0=y0_history,
+ stepsize_controller=diffrax.PIDController(rtol=1e-6, atol=1e-9),
+ saveat=diffrax.SaveAt(ts=ts, dense=True),
+ delays=delays,
+ )
+
+ delays = Delays(
+ delays=[lambda t, y, args: 2 + jnp.sin(t)],
+ initial_discontinuities=jnp.array([0.0]),
+ max_discontinuities=100,
+ recurrent_checking=True,
+ rtol=10e-3,
+ atol=10e-6,
+ )
+
+ sol2 = diffrax.diffeqsolve(
+ diffrax.ODETerm(vector_field),
+ diffrax.Bosh3(),
+ t0=ts[0],
+ t1=ts[-1],
+ dt0=ts[1] - ts[0],
+ y0=y0_history,
+ stepsize_controller=diffrax.PIDController(rtol=1e-6, atol=1e-9),
+ saveat=diffrax.SaveAt(ts=ts, dense=True),
+ delays=delays,
+ )
+
+ _, juliays = open_process_file("test/julia_dde/test_basic_check_10.txt")
+ error1, error2, error3 = (
+ jnp.mean(jnp.abs(juliays - sol.ys)),
+ jnp.mean(jnp.abs(juliays - sol2.ys)),
+ jnp.mean(jnp.abs(sol.ys - sol2.ys)),
+ )
+
+ assert error1 < 1e-3
+ assert error2 < 1e-3
+ assert error3 < 1e-5
+
+
+def test_basic_numerical_check_1():
+ # testing a dde where we know its analytical value
+ # http://www.cs.toronto.edu/pub/reports/na/hzpEnrightNA09Preprint.pdf
+ # test problem 1
+
+ def vector_field(t, y, args, history):
+ return history[0]
+
+ y0_history = lambda t: lax.cond(t < 2.0, lambda: 0.5, lambda: 1.0)
+
+ delays = Delays(
+ delays=[lambda t, y, args: t - y],
+ initial_discontinuities=jnp.array([2.0]),
+ max_discontinuities=100,
+ recurrent_checking=False,
+ rtol=1e-3,
+ atol=1e-6,
+ )
+
+ t0, t1, nb_points = 2.0, 5.5, 350
+ ts = jnp.linspace(t0, t1, nb_points)
+ sol = diffrax.diffeqsolve(
+ diffrax.ODETerm(vector_field),
+ diffrax.Bosh3(),
+ t0=ts[0],
+ t1=ts[-1],
+ dt0=ts[1] - ts[0],
+ y0=y0_history,
+ stepsize_controller=diffrax.PIDController(rtol=1e-6, atol=1e-9),
+ saveat=diffrax.SaveAt(ts=ts, dense=True),
+ delays=delays,
+ )
+
+ delays = Delays(
+ delays=[lambda t, y, args: t - y],
+ initial_discontinuities=jnp.array([2.0]),
+ max_discontinuities=100,
+ recurrent_checking=True,
+ rtol=1e-3,
+ atol=1e-6,
+ )
+
+ sol2 = diffrax.diffeqsolve(
+ diffrax.ODETerm(vector_field),
+ diffrax.Bosh3(),
+ t0=ts[0],
+ t1=ts[-1],
+ dt0=ts[1] - ts[0],
+ y0=y0_history,
+ stepsize_controller=diffrax.PIDController(rtol=1e-6, atol=1e-9),
+ saveat=diffrax.SaveAt(ts=ts, dense=True),
+ delays=delays,
+ )
+
+ def f1(t):
+ return t / 2
+
+ def f2(t):
+ return 2 * jnp.exp(t / 2 - 2)
+
+ # y(t) = t/2 for 2 <= t <= 4
+ # y(t) = 2 exp(t/2-2) for 4 <= t <= 5.386
+ # y(t) = 4 - 2 ln(1+ 5.386 -t )] for 5.386 <= t <= 5.5
+ error1 = jnp.sum(jnp.abs(sol.ys[:200] - f1(sol.ts[:200])))
+ error2 = jnp.sum(jnp.abs(sol.ys[202:300] - f2(sol.ts[202:300])))
+
+ error11 = jnp.sum(jnp.abs(sol2.ys[:200] - f1(sol2.ts[:200])))
+ error21 = jnp.sum(jnp.abs(sol2.ys[202:300] - f2(sol2.ts[202:300])))
+ juliats, juliays = open_process_file(
+ "test/julia_dde/test_basic_numerical_check_1.txt"
+ )
+
+ error3 = jnp.sum(jnp.abs(juliays[:200] - f1(juliats[:200])))
+ error4 = jnp.sum(jnp.abs(juliays[200:300] - f2(juliats[200:300])))
+ assert error2 < error4
+ assert error1 < 1e-5
+ assert error2 < 1e-2
+ assert error11 < 1e-5
+ assert error21 < 1e-2
+ assert error3 < 1e-5
+
+
+def test_basic_numerical_check_2():
+ # testing a dde where we know its analytical value
+ # http://www.cs.toronto.edu/pub/reports/na/hzpEnrightNA09Preprint.pdf
+ # test problem 3
+
+ def vector_field(t, y, args, history):
+ return y * history[0] / t
+
+ y0_history = lambda t: 1.0
+
+ delays = Delays(
+ delays=[lambda t, y, args: t - jnp.log(y)],
+ initial_discontinuities=jnp.array([1.0]),
+ max_discontinuities=100,
+ recurrent_checking=False,
+ rtol=1e-3,
+ atol=1e-6,
+ )
+
+ t0, t1, nb_points = 1.0, 10.0, 901
+ ts = jnp.linspace(t0, t1, nb_points)
+ sol = diffrax.diffeqsolve(
+ diffrax.ODETerm(vector_field),
+ diffrax.Bosh3(),
+ t0=ts[0],
+ t1=ts[-1],
+ dt0=ts[1] - ts[0],
+ y0=y0_history,
+ stepsize_controller=diffrax.PIDController(rtol=1e-6, atol=1e-9),
+ saveat=diffrax.SaveAt(ts=ts, dense=True),
+ delays=delays,
+ )
+
+ delays = Delays(
+ delays=[lambda t, y, args: t - jnp.log(y)],
+ initial_discontinuities=jnp.array([1.0]),
+ max_discontinuities=100,
+ recurrent_checking=True,
+ rtol=1e-3,
+ atol=1e-6,
+ )
+
+ sol2 = diffrax.diffeqsolve(
+ diffrax.ODETerm(vector_field),
+ diffrax.Bosh3(),
+ t0=ts[0],
+ t1=ts[-1],
+ dt0=ts[1] - ts[0],
+ y0=y0_history,
+ stepsize_controller=diffrax.PIDController(rtol=1e-6, atol=1e-9),
+ saveat=diffrax.SaveAt(ts=ts, dense=True),
+ delays=delays,
+ )
+
+ def f1(t):
+ return t
+
+ def f2(t):
+ return jnp.exp(t / jnp.exp(1))
+
+ # y(t) = t for 1 <= t <= e
+ # y(t) = exp(t/e) for e <= t <= e^2
+ error1 = jnp.sum(jnp.abs(sol.ys[:100] - f1(sol.ts[:100])))
+ error2 = jnp.sum(jnp.abs(sol.ys[300:400] - f2(sol.ts[300:400])))
+
+ error11 = jnp.sum(jnp.abs(sol2.ys[:100] - f1(sol2.ts[:100])))
+ error21 = jnp.sum(jnp.abs(sol2.ys[300:400] - f2(sol2.ts[300:400])))
+ juliats, juliays = open_process_file(
+ "test/julia_dde/test_basic_numerical_check_2.txt"
+ )
+
+ error3 = jnp.sum(jnp.abs(juliays[:100] - f1(juliats[:100])))
+ error4 = jnp.sum(jnp.abs(juliays[300:400] - f2(juliats[300:400])))
+
+ assert error1 < 1e-6
+ assert error11 < 1e-6
+ assert error3 < 1e-6
+ assert error2 < 1e-2
+ assert error21 < 1e-2
+ assert error2 < error4