Physics-Informed Neural Networks (PINNs)

Putting a PDE into the loss function. The idea, the gotchas, and a working example for the heat equation.

A PINN is a neural network with a physics term in the loss. That’s the whole idea; the rest is engineering. This tutorial is about the engineering — what to put in the loss, how to weight it, and why your first PINN will train to a flat function unless you do a few specific things.

What’s in here

• the residual loss: what nabla u - f = 0 looks like as a PyTorch loss term
• soft vs hard boundary conditions
• a working PINN for the 1D heat equation, end to end
• why naïve PINNs collapse to a constant — and three fixes (loss balancing, causal training, NTK reweighting)
• when PINNs are the right tool, and when a finite-element solver wins
• pointers to the modern PINN literature (and what’s hype)

Prerequisites

• Tutorial 3 (NN intro)
• Some familiarity with PDEs — knowing what the heat equation describes is enough