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    First published on Tuesday, Jun 2, 2026 and last modified on Friday, Jun 5, 2026 by François Chaplais.

    A study of Physics-Informed Neural Networks

    François Chaplais WebMagic

    1 Introduction

    \[ \frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2} \]
    \[ u_t - \alpha u_{xx} \neq 0 \]

    \[ \mathcal{L}_{data} = \sum |u_{pred} - u_{true}|^2 \]
    \[ \mathcal{L}_{physics} = \sum |f(x,t)|^2 \]
    \[ f(x,t) = u_t - \alpha u_{xx} \]
    \[ \mathcal{L}_{BC} = \sum |u_{pred} - u_{boundary}|^2 \]
    \[ \mathcal{L} = \mathcal{L}_{data} + \mathcal{L}_{physics} + \mathcal{L}_{BC} \]

    \[ u'(x) = -u(x), ~ u(0)=1 \]
    \[ u(x)=e^{-x} \]

    \[ \frac{du}{dx} + u = 0 \]
    \[ u(0)=1 \] 

    import torch
import torch.nn as nn
import torch.optim as optim
import matplotlib.pyplot as plt

# -----------------------------
# Neural network
# -----------------------------
class PINN(nn.Module):
    def __init__(self):
        super().__init__()
        self.net = nn.Sequential(
            nn.Linear(1, 20),
            nn.Tanh(),
            nn.Linear(20, 20),
            nn.Tanh(),
            nn.Linear(20, 1)
        )

    def forward(self, x):
        return self.net(x)

# -----------------------------
# Create model
# -----------------------------
model = PINN()

# Optimizer
optimizer = optim.Adam(model.parameters(), lr=0.01)

# -----------------------------
# Training loop
# -----------------------------
for epoch in range(5000):
    optimizer.zero_grad()

    # Collocation points inside the domain [0, 1]
    x_phys = torch.rand(100, 1, requires_grad=True)

    # Network prediction
    u = model(x_phys)

    # Compute du/dx using autograd
    du_dx = torch.autograd.grad(
        u,
        x_phys,
        grad_outputs=torch.ones_like(u),
        create_graph=True
    )[0]

    # Physics residual: u' + u = 0
    physics_residual = du_dx + u
    physics_loss = torch.mean(physics_residual**2)

    # Boundary condition u(0) = 1
    x_bc = torch.tensor([[0.0]])
    u_bc = model(x_bc)
    bc_loss = (u_bc - 1.0)**2

    # Total loss
    loss = physics_loss + bc_loss

    loss.backward()
    optimizer.step()

    if epoch         print(f"Epoch {epoch}, Loss: {loss.item():.6f}")

# -----------------------------
# Test and plot
# -----------------------------
x_test = torch.linspace(0, 1, 100).reshape(-1, 1)
u_pred = model(x_test).detach().numpy()

u_exact = torch.exp(-x_test).numpy()

plt.plot(x_test.numpy(), u_pred, label="PINN prediction")
plt.plot(x_test.numpy(), u_exact, "--", label="Exact solution")
plt.legend()
plt.xlabel("x")
plt.ylabel("u(x)")
plt.title("PINN solving u'(x) = -u(x), u(0)=1")
plt.show()

    \[ u'(x) + u(x) = 0 \] 
    physics_residual = du_dx + u

    \[ u(0)=1 \] 
    x_bc = torch.tensor([[0.0]])
u_bc = model(x_bc)
bc_loss = (u_bc - 1.0)**2

    loss = physics_loss + bc_loss

    \[ e^{-x} \]

    \[ u_t = u_{xx} \]

    2 Proposed bibliography

    3 The plan

    @misc{babić2026differentiablechemistrypinnssolving,
      title={Differentiable Chemistry in PINNs for Solving Parameterized and Stiff Reaction Systems}, 
      author={Miloš Babić and Franz M. Rohrhofer and Stefan Posch},
      year={2026},
      eprint={2605.04708},
      archivePrefix={arXiv},
      primaryClass={cs.LG},
      url={https://arxiv.org/abs/2605.04708}, 
}