## Show code cell source

```
import matplotlib.pyplot as plt
%matplotlib inline
import matplotlib_inline
matplotlib_inline.backend_inline.set_matplotlib_formats('svg')
import seaborn as sns
sns.set_context("paper")
sns.set_style("ticks");
!pip show equinox || echo equinox not found. Installing... && pip install equinox 2> /dev/null
```

## Show code cell output

```
Name: equinox
Version: 0.11.2
Summary: Elegant easy-to-use neural networks in JAX.
Home-page:
Author:
Author-email: Patrick Kidger <contact@kidger.site>
License: Apache License
Version 2.0, January 2004
http://www.apache.org/licenses/
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Location: /Users/ibilion/.pyenv/versions/3.11.6/lib/python3.11/site-packages
Requires: jax, jaxtyping, typing-extensions
Required-by: diffrax
Requirement already satisfied: equinox in /Users/ibilion/.pyenv/versions/3.11.6/lib/python3.11/site-packages (0.11.2)
Requirement already satisfied: jax>=0.4.13 in /Users/ibilion/.pyenv/versions/3.11.6/lib/python3.11/site-packages (from equinox) (0.4.19)
Requirement already satisfied: jaxtyping>=0.2.20 in /Users/ibilion/.pyenv/versions/3.11.6/lib/python3.11/site-packages (from equinox) (0.2.23)
Requirement already satisfied: typing-extensions>=4.5.0 in /Users/ibilion/.pyenv/versions/3.11.6/lib/python3.11/site-packages (from equinox) (4.8.0)
Requirement already satisfied: ml-dtypes>=0.2.0 in /Users/ibilion/.pyenv/versions/3.11.6/lib/python3.11/site-packages (from jax>=0.4.13->equinox) (0.3.1)
Requirement already satisfied: numpy>=1.22 in /Users/ibilion/.pyenv/versions/3.11.6/lib/python3.11/site-packages (from jax>=0.4.13->equinox) (1.25.2)
Requirement already satisfied: opt-einsum in /Users/ibilion/.pyenv/versions/3.11.6/lib/python3.11/site-packages (from jax>=0.4.13->equinox) (3.3.0)
Requirement already satisfied: scipy>=1.9 in /Users/ibilion/.pyenv/versions/3.11.6/lib/python3.11/site-packages (from jax>=0.4.13->equinox) (1.11.3)
Requirement already satisfied: typeguard<3,>=2.13.3 in /Users/ibilion/.pyenv/versions/3.11.6/lib/python3.11/site-packages (from jaxtyping>=0.2.20->equinox) (2.13.3)
```

## Show code cell source

```
import jax
import equinox as eqx
class FourierEncoding(eqx.Module):
B: jax.Array
@property
def num_fourier_features(self) -> int:
return self.B.shape[0]
@property
def in_size(self) -> int:
return self.B.shape[1]
@property
def out_size(self) -> int:
return self.B.shape[0] * 2
def __init__(self,
in_size: int,
num_fourier_features: int,
key: jax.random.PRNGKey,
sigma: float = 1.0):
self.B = jax.random.normal(
key, shape=(num_fourier_features, in_size),
dtype=jax.numpy.float32) * sigma
def __call__(self, x: jax.Array, **kwargs) -> jax.Array:
return jax.numpy.concatenate(
[jax.numpy.cos(jax.numpy.dot(self.B, x)),
jax.numpy.sin(jax.numpy.dot(self.B, x))],
axis=0)
```

# Solving Parametric Problems using Physics-informed Neural Networks#

As you have may have noticed, physics-informed neural networks (PINNs) are not a good choice for solving forward problems. Standard numerical methods are much faster and more accurate. One of the places where PINNs can potentially beat traditional approaches is in the solution of \emph{parametric problems}. In these problems, one has an ODE or PDE that depends on some parameters, and one wants to solve the problem for many different values of the parameters. PINNs can help us learn, in one shot, the map from the parameters to the solution of the problem.

It is straightforward to extend the standard PINN approach to parametric problems. Suppose that the parameters are \(\boldsymbol{\xi}\), an \(\mathbb{R}^d\)-valued random vector with PDF \(p(\boldsymbol{\xi})\). The loss function, e.g., from the integrated squared residual, is now dependent on \(\boldsymbol{\xi}\):

The idea, is three-fold. First, try to promote inductive biases by baking in the boundary/initial conditions. Second, parameterize the uknown part using a neural network \(N(\boldsymbol{x}, \boldsymbol{\xi};\theta)\), with parameters \(\theta\) A good choice for the parameterization is this structure:

This is a good choice, because it resembles an expansion in a basis of functions \(\phi_i(\mathbf{x};\theta)\), with coefficients \(b_i(\boldsymbol{\xi};\theta)\). The difference here is that the basis functions are learned - they are not fixed. Third, we find the parameters \(\theta\), by minimizing the expected loss:

We can do this using Adam, or any other optimization algorithm. All we need is to express the loss as an expectation and construct an unbiased estimator of the gradient of the loss with respect to the parameters \(\theta\).

## Example: Parametric Poisson Equation#

Let’s consider the following parametric Poisson equation:

\begin{aligned} -\Delta u(\mathbf{x};\xi) &= f(\mathbf{x};\xi), \quad \mathbf{x} \in \Omega, \ u(\mathbf{x};\xi) &= 0, \quad \mathbf{x}\in \partial\Omega, \end{aligned}

where \(\Omega = [0,1]^2\). The source term is:

The exact solution to this problem is:

Let’s verify that this is indeed the solution to the problem.

```
import sympy as sp
from sympy import symbols
x, y, xi = symbols('x y xi')
f = 2 * sp.pi ** 2 * xi * sp.sin(sp.pi * x) * sp.sin(sp.pi * y)
u = xi * sp.sin(sp.pi * x) * sp.sin(sp.pi * y)
res = sp.simplify(sp.diff(u, x, x) + sp.diff(u, y, y) + f)
res
```

I have purposely chosen a simple problem, so that we don’t have to deal with non-dimensionalization and scaling. A parameterization that satisfies the boundary conditions is:

where \(N(x,y;\xi;\theta)\) is a neural network with parameters \(\theta\). For the neural network, we will use a the structure suggested above.

First, some useful classes to avoid code repetition.

```
import equinox as eqx
class ParametricModel(eqx.Module):
"""This model captures a simple structure made out of branches and trunks."""
branch: list # These are the b's
trunk: list # These are the phi's
def __init__(self, branch, trunk):
self.branch = branch
self.trunk = trunk
def __call__(self, x, xi, **kwargs):
res = 0.0
for b, t in zip(self.branch, self.trunk):
res += b(xi) * t(x)
return res
class EnforceDirichletZeroBoundarySquare(eqx.Module):
"""This is an model that enforces zero Dirichlet boundary conditions on the square."""
neural_net: eqx.Module
def __init__(self, neural_net):
self.neural_net = neural_net
def __call__(self, x, y, xi, **kwargs):
return self.neural_net(jnp.array([x, y]), xi) * (1 - x) * x * (1 - y) * y
```

Let’s now make the actual model we will use:

```
import jax
import jax.numpy as jnp
import jax.random as jrandom
key = jrandom.PRNGKey(0)
key1, key2, key = jrandom.split(key, 3)
# Make the parameterization
m = 10 # number of branches and trunks
# Branches are all MLP's
branch_width = 32
branch_depth = 3
branch = [eqx.nn.MLP('scalar', 'scalar', branch_width, branch_depth, jax.nn.tanh, key=k) for k in jrandom.split(key1, m)]
# Trunks have a FourierEncoding followed by an MLP
trunk_num_fourier_features = 32
trunk_width = 32
trunk_depth = 3
trunk = [eqx.nn.Sequential([
FourierEncoding(2, trunk_num_fourier_features, key=k),
eqx.nn.MLP(trunk_num_fourier_features * 2, 'scalar', trunk_width, trunk_depth, jax.nn.tanh, key=k)
]) for k in jrandom.split(key2, m)]
# Combine the branches and trunks into a ParametricModel
model = EnforceDirichletZeroBoundarySquare(ParametricModel(branch, trunk))
```

Here is how our model looks like as a PyTree:

```
model
```

## Show code cell output

```
EnforceDirichletZeroBoundarySquare(
neural_net=ParametricModel(
branch=[
MLP(
layers=(
Linear(
weight=f32[32,1],
bias=f32[32],
in_features='scalar',
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bias=f32[32],
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weight=f32[32,32],
bias=f32[32],
in_features=32,
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use_bias=True
),
Linear(
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bias=f32[1],
in_features=32,
out_features='scalar',
use_bias=True
)
),
activation=<wrapped function <lambda>>,
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use_bias=True,
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in_size='scalar',
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width_size=32,
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bias=f32[32],
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)
),
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final_activation=<function <lambda>>,
use_bias=True,
use_final_bias=True,
in_size='scalar',
out_size='scalar',
width_size=32,
depth=3
),
MLP(
layers=(
Linear(
weight=f32[32,1],
bias=f32[32],
in_features='scalar',
out_features=32,
use_bias=True
),
Linear(
weight=f32[32,32],
bias=f32[32],
in_features=32,
out_features=32,
use_bias=True
),
Linear(
weight=f32[32,32],
bias=f32[32],
in_features=32,
out_features=32,
use_bias=True
),
Linear(
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MLP(
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Linear(
weight=f32[32,1],
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Linear(
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Linear(
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Linear(
weight=f32[1,32],
bias=f32[1],
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)
),
activation=<wrapped function <lambda>>,
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MLP(
layers=(
Linear(
weight=f32[32,1],
bias=f32[32],
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Linear(
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Linear(
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MLP(
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),
Linear(
weight=f32[1,32],
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)
),
activation=<wrapped function <lambda>>,
final_activation=<function <lambda>>,
use_bias=True,
use_final_bias=True,
in_size='scalar',
out_size='scalar',
width_size=32,
depth=3
),
MLP(
layers=(
Linear(
weight=f32[32,1],
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use_bias=True
),
Linear(
weight=f32[32,32],
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Linear(
weight=f32[32,32],
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),
Linear(
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),
activation=<wrapped function <lambda>>,
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MLP(
layers=(
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use_bias=True
),
Linear(
weight=f32[1,32],
bias=f32[1],
in_features=32,
out_features='scalar',
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)
),
activation=<wrapped function <lambda>>,
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),
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],
trunk=[
Sequential(
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FourierEncoding(B=f32[32,2]),
MLP(
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)
),
activation=<wrapped function <lambda>>,
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use_bias=True,
use_final_bias=True,
in_size=64,
out_size='scalar',
width_size=32,
depth=3
)
)
),
Sequential(
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]
)
)
```

Let’s ensure our model works:

```
model(0.5, 0.5, 0.5)
```

```
Array(0.00502437, dtype=float32)
```

Let’s create our loss function:

```
from jax import vmap, grad
def loss_density(model, x, y, xi):
u_xx = grad(grad(model, 0), 0)(x, y, xi)
u_yy = grad(grad(model, 1), 1)(x, y, xi)
f = 2 * jnp.pi ** 2 * xi * jnp.sin(jnp.pi * x) * jnp.sin(jnp.pi * y)
return (u_xx + u_yy + f) ** 2
# This evaluate the loss at multiple xs and ys, but at a single xi
loss_density_single_xi = vmap(loss_density, in_axes=(None, 0, 0, None))
# Τhis evaluates the loss at multiple xs, ys, and xis.
# But the xs and ys are the same for all xis
loss_density_many_xis = vmap(loss_density_single_xi, in_axes=(None, None, None, 0))
# And the final loss
loss = lambda model, xs, ys, xis: jnp.mean(loss_density_many_xis(model, xs, ys, xis))
```

Let’s make sure it works:

```
xs = jnp.linspace(0, 1, 100)
ys = jnp.linspace(0, 1, 100)
xis = jnp.linspace(0, 1, 20)
res = loss_density_many_xis(model, xs, ys, xis)
res.shape
```

```
(20, 100)
```

And:

```
loss(model, xs, ys, xis)
```

```
Array(48.726887, dtype=float32)
```

We are not going to worry about the Fourier features. We will let the model learn them. Here is the training code:

```
def train_parametric_pinn(
loss,
model,
key,
optimizer,
Lx=1.0,
Ly=1.0,
num_collocation_residual=128,
num_xis = 16,
num_iter=10_000,
freq=1,
):
"""Notice that it assumes the xi's are sclar and uniformly distributed in [0, 1]."""
@eqx.filter_jit
def step(opt_state, model, xs, ys, xis):
value, grads = eqx.filter_value_and_grad(loss)(model, xs, ys, xis)
updates, opt_state = optimizer.update(grads, opt_state)
model = eqx.apply_updates(model, updates)
return model, opt_state, value
opt_state = optimizer.init(eqx.filter(model, eqx.is_inexact_array))
losses = []
for i in range(num_iter):
key1, key2, key3, key = jrandom.split(key, 4)
xs = jrandom.uniform(key1, (num_collocation_residual,), maxval=Lx)
ys = jrandom.uniform(key2, (num_collocation_residual,), maxval=Ly)
xis = jrandom.uniform(key3, (num_xis,))
model, opt_state, value = step(opt_state, model, xs, ys, xis)
if i % freq == 0:
losses.append(value)
print(f"Step {i}, residual loss {value:.3e}")
return model, losses
```

Let’s train. Be patient. It takes a while to compile the model. Once the iterations start, it is faster. It takes about 2.5 minutes to run on my machine.

```
import optax
optimizer = optax.adam(1e-3)
model, losses = train_parametric_pinn(
loss, model, key, optimizer,
num_collocation_residual=128, num_xis=2,
num_iter=1_000, freq=1)
```

## Show code cell output

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```

Let’s visualize the loss:

```
fig, ax = plt.subplots()
ax.plot(losses)
ax.set_yscale('log')
ax.set_xlabel('Iteration')
ax.set_ylabel('Residual Loss')
sns.despine(trim=True);
```

Let’s make some predictions are random \(\xi\)’s:

```
import numpy as np
np.random.seed(1234)
u_true = eqx.filter_jit(vmap(lambda x, y, xi: xi * jnp.sin(jnp.pi * x) * jnp.sin(jnp.pi * y), (0, 0, None)))
u_pred = eqx.filter_jit(vmap(model, (0, 0, None)))
xs = np.linspace(0, 1, 100)
ys = np.linspace(0, 1, 100)
X, Y = np.meshgrid(xs, ys)
X_flat = X.flatten()
Y_flat = Y.flatten()
levels = np.linspace(0, 1.0, 20)
error_levels = np.linspace(0, 0.2, 20)
for xi in np.random.rand(5):
fig, ax = plt.subplots(1, 3, figsize=(12, 4))
ax[0].set_title(f"Prediction for $\\xi$={xi:.2f}")
u = vmap(lambda x, y: model(x, y, xi))(xs, ys)
Z_p = u_pred(X_flat, Y_flat, xi).reshape(X.shape)
c = ax[0].contourf(X, Y, Z_p, levels=levels)
fig.colorbar(c, ax=ax[0])
ax[0].set_xlabel('$x$')
ax[0].set_ylabel('$y$')
sns.despine(trim=True)
ax[1].set_title(f"True for $\\xi$={xi:.2f}")
Z_t = u_true(X_flat, Y_flat, xi).reshape(X.shape)
c = ax[1].contourf(X, Y, Z_t, levels=levels)
fig.colorbar(c, ax=ax[1])
ax[1].set_xlabel('$x$')
ax[1].set_ylabel('$y$')
sns.despine(trim=True)
ax[2].set_title(f"Relative error for $\\xi$={xi:.2f}")
E = np.abs(Z_p - Z_t) / np.max(Z_t)
c = ax[2].contourf(X, Y, E, levels=error_levels)
fig.colorbar(c, ax=ax[2])
ax[2].set_xlabel('$x$')
ax[2].set_ylabel('$y$')
sns.despine(trim=True)
plt.tight_layout()
```

Okay, pretty good! We would have to train for more iterations to get a better fit.