04 · PINNs · Camilo Rozo

// Overview

About this project

A PINN is a neural network u_θ(x, t) whose loss combines three pieces: initial conditions, boundary conditions, and the PDE residual evaluated at collocation points in the domain. Partial derivatives are computed with torch.autograd.grad — no finite differences, no mesh.

This project implements two classic cases:

1D heat equation: linear case with a known analytical solution. Serves as a sanity check.
1D Burgers equation: classic nonlinear case with shock-like discontinuities that challenge any numerical method.

PINNs are at the active research frontier (Nature, AIAA, JCP papers). A technical blog post explaining the implementation generates visibility and organic contacts.

Mathematical formulation

Generic PDE:

ℱ[u](x, t) = 0

Heat equation:

u_t − α u_xx = 0

Burgers equation:

u_t + u · u_x − ν u_xx = 0

Composite PINN loss:

L = L_PDE + λ_B L_boundary + λ_I L_initial

// Skills demonstrated

What skills it certifies

Advanced autograd

Second-order partial derivatives via torch.autograd.grad with create_graph=True.

Numerical PDEs

Formulate residuals in smooth form, convert physical constraints into loss terms.

Optimization

Balance composite loss weights λ, learning rate schedulers, Adam → L-BFGS.

Paper implementation

Translate Raissi et al. (JCP 2019) directly into clean, testable PyTorch code.

Validation

Relative L² error against the analytical solution (heat) and high-resolution numerical reference (Burgers).

2D visualization

Space-time heatmaps, snapshots of u(x) at various t, side-by-side comparison with classical solver.

Technical writing

Blog post explaining trade-offs: when a PINN beats a classical solver and when it doesn't.

PyTorch NumPy SciPy Matplotlib Autograd PDEs Research Jupyter

// Structure

Project organization

04-pinns/
├── README.md
├── requirements.txt
├── src/
│   ├── pinn.py        # u_theta(x, t) network with Tanh activation
│   ├── physics.py     # heat and Burgers residuals via autograd
│   └── train.py       # loop with composite loss (PDE + BC + IC)
└── notebooks/      # comparison with analytical / numerical solution

// Roadmap

Project status

Implement MLP with Tanh activation (twice-differentiable)
heat_residual(u, x, t, α) function using autograd.grad
burgers_residual(u, x, t, ν) function
Sampling of collocation, boundary, and initial points
Training loop with composite loss and λ weights
Validation: heat vs. analytical solution, Burgers vs. spectral solver
Space-time figures + explanatory blog post

// Metrics

Success criteria

Heat equation

Relative L² error < 1% vs. analytical solution across the entire domain.

Burgers equation

Relative L² error < 5% vs. high-resolution numerical reference.

Correct capture of emergent shock without pathological oscillations.

// Results

Outputs and final metrics

Learned solutions, comparison to analytical/numerical reference, and L² errors.

Pending

No results published yet.

What goes here: space-time heatmaps of learned u(x,t), side-by-side comparison with exact/numerical solution, and relative L² errors.