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Chapter 1: A "Hello world!" PINN

index

A quick tour

Let's solve a 1D steady-state heat equation problem where the function u(x) is required to satisfy:

$$\begin{align} u(0)&=&0&\\ u(1)&=&0&\\ u_{xx}(x) + 1 &=&0&, \qquad x\in (0,1) \end{align}$$

Here is the quick version, detailed explanations below

  1. Create a new project by running: mtc create hello-world

  2. Enter the newly created hello-world directory and edit problem.py so it looks like this

    from mtc.problem import *
from cfg import p

[x], [u] = p.add_neural_network(name="NN", inputs=["x"], outputs=["u"])

geom = p.Line1D("geom", 0,1)
interior = p.add_interior_subdomain("interior", geom=geom)
bdry = p.add_boundary_subdomain("bdry", geom=geom)

diff_eq = Eq(u.diff(x,2) + 1, 0)
p.add_constraint("diffusion", enforce(equation=diff_eq, on_domain=interior))
p.add_constraint("bdry", enforce(equation=Eq(u,0), on_domain=bdry))

    Note: As you type the problem definition in problem.py it may be helpful to run mtc show problem in the console. Each time this command runs, it will partially compile the problem and report errors or warnings.

  3. Then save the file and run mtc init-conf in the same directory. This will generate a default project configuration, a summary of which is available with mtc show training

    $ mtc init-conf
$ mtc show training
Stage DAG: 

stage1 ----------------------------------------
[ description] Default stage
[   optimizer] adam, steps < 1000
[         NNs] [training] NN: fully_connected
[ constraints] bdry
               diffusion
========================================

  4. Finally, run mtc train to train the PINN model.

  5. After 1000 steps the model is ready to use!

To inspect the result of training, create a notebook in the project root making sure to use the modulus-python kernel. Run the following

import numpy as np
import matplotlib.pyplot as plt

import training.stage1.infer as infer1

This will load the inference module created for this project by MTC. The type of available inference may be seen by running (in the next cell)

[cell]: infer1.info
[output]: {'stage': 'stage1',
 '__file__': '/workspace/mtc/demo/training/stage1/infer.py',
 'inputs': ['x'],
 'default-outputs': ['u'],
 'possible-outputs': ['u', 'diffusion', 'bdry']}

This shows that both the function u(x) is available. So we can display the value of u(x) with

x = np.linspace(0,1, 10)
plt.plot(x, infer1.infer(x=x)['u'])

s

Now we can go back to the problem and make it a bit more interesting. Let's change the heat source from a constant of 1, to a function that depends on x. Replace the line in problem.py with the diffusion equation with the following one:

diff_eq = Eq(g.diff(x,2) + (x-0.5), 0)

So the heat source is negative near x=0, becomes 0 at the midpoint, and then is positive at x=1. $$g_{xx} + (x-0.5) = 0$$

To retrain, we first have to clear the last results and then train again by running

mtc clean
mtc train

Restarting the kernel and running all the cells again in our inspection notebook will now produce the following figure:

s

What happened?

Now let's take a look at the problem and the tools we used in more detail.

PINN Parameterization

In the introduction we quickly modified the problem into this one: $$ \begin{align} g(0)&=&0&\ g(1)&=&0&\ g_{xx}(x) + (x-0.5) &=&0&, \qquad x\in (0,1) \end{align}$$

And we can also parameterize this problem in a variety of ways. To illustrate the MTC facilities to do so, let's start by parameterizing the source term in the following way: $$ \begin{align} g(0)&=&0&\ g(1)&=&0&\ g_{xx}(x) + (x-a) &=&0&, \qquad x\in (0,1), \quad a\in(0,1) \end{align} $$

We can solve this parameterized set of equations with a single PINN model. The procedure is exactly as above, but adding a as a parameter.

  1. Create a new project with mtc create hello-world-param-a

  2. Modify the problem.py definition to look like

    from cfg import *

[x, a], [g] = p.add_neural_network(name="NN", inputs=["x", 'a'], outputs=["g"])

geom = p.Line1D("geom", 0,1)
interior = p.add_interior_subdomain("interior", geom=geom, params={a:(0,1)})
bdry = p.add_boundary_subdomain("bdry", geom=geom, params={a:(0,1)})

diff_eq = Eq(g.diff(x,2) + (x-a), 0)
p.add_constraint("diffusion", enforce(equation=diff_eq, on_domain=interior))
p.add_constraint("bdry", enforce(equation=Eq(g,0), on_domain=bdry))

    We made three changes:

    1. We added a as a variable to the NN,
    2. Gave it a range in defining the two subdomains
    3. And modified the interior constraint as in the equation above.

  3. Train the model with

    mtc clean # if needed
mtc init-conf
mtc train

  4. Inspect the results in a notebook running the modulus-python kernel; e.g. by running the following code

    import numpy as np
import matplotlib.pyplot as plt

import training.stage1.infer as infer1

x = np.linspace(0,1, 30)

f, axs = plt.subplots(1,2, figsize=(9,4))
for a in np.linspace(0,1, 5):
    X,A = np.meshgrid(x, np.array([a]))
    axs[0].plot(x, infer1.infer(x=X, a=A)['g'], label=f"a={a}")
    axs[1].plot(x, x-a, label=f"a={a} [Q]")
axs[0].set_xlabel("x")
axs[1].set_xlabel("x")
axs[0].set_title("g(x)")
axs[1].set_title("Heat source")

axs[0].axhline(0, lw=1, color='k')
axs[1].axhline(0, lw=1, color='k')
plt.legend()
plt.tight_layout()

    The result should look like this:

    s