PINN

Burgers Equation Using PINNs

Problem Setup

The 1D viscous Burgers equation is solved using a Physics-Informed Neural Network (PINN):

$$ u_t + u u_x = \nu u_{xx} $$

where

$$ \nu = \frac{0.01}{\pi} $$

The spatial and temporal domains are:

• $x \in [-1,1]$
• $t \in [0,1]$

---

Initial Condition

$$ u(x,0) = -\sin(\pi x) $$

---

Boundary Conditions

$$ u(-1,t) = 0 $$

$$ u(1,t) = 0 $$

This configuration is commonly used in PINN benchmarks because the solution develops steep gradients during evolution, making it a challenging nonlinear PDE problem.

---

Standard PINN

Model

A fully connected neural network with tanh activation functions is used to approximate the solution

$$ u(x,t) $$

The PINN is trained by minimizing three components:

• PDE residual loss
• Initial condition loss
• Boundary condition loss

Collocation Points

• Interior collocation points enforce the PDE.
• Initial points enforce the initial condition.
• Boundary points enforce the boundary conditions.

---

Training Strategy

Training uses a two-stage optimization scheme:

1. Adam optimizer – coarse optimization
2. L-BFGS optimizer – refinement

This hybrid strategy improves convergence when minimizing the physics residual.

---

Results

The standard PINN reproduces the global structure of the Burgers solution:

• Sinusoidal initial condition
• Wave steepening over time
• Symmetric Burgers dynamics

The PDE residual remains small across most of the domain, with slightly higher values near the shock region around $x \approx 0$.

This region contains steep gradients, which are challenging for standard PINNs.

---

Fourier Feature PINN

Intended Hypotheses

Standard neural networks tend to learn low-frequency components more easily than high-frequency structures.

Since Burgers dynamics produce sharp spatial gradients, this can limit the model's representation capacity.

---

Method

To address this limitation, Fourier feature embeddings are introduced before the neural network.

The input coordinates are mapped as

$$ (x,t) \rightarrow [\sin(Bx), \cos(Bx)] $$

This expands the input representation using periodic basis functions.

---

Results

The Fourier PINN produces:

• Sharper representation of the evolving wave
• Better modeling of steep gradients near the shock region

The PDE residual heatmap also shows a more consistent distribution of error across the domain.

---

Diagnostics

Two visualizations are used to evaluate the model.

Solution Heatmap

Shows the predicted Burgers solution across space and time.

PDE Residual Heatmap

Displays the magnitude of the physics residual:

$$ \left| u_t + u u_x - \nu u_{xx} \right| $$

This highlights regions where the physics constraint is hardest for the network to satisfy.

Residual values are lowest across most of the domain and increase slightly near the shock region, which is expected due to the large gradients present there.

---

Summary

Both models successfully solve the Burgers equation using physics-informed training.

Standard PINN

• Captures the overall dynamics
• Correctly models the global solution

Fourier PINN

• Improves representation of sharp spatial features
• Better handles high-frequency gradients

These results demonstrate how feature embeddings can enhance PINN performance when modeling nonlinear PDEs with steep gradients.