Physics-Informed Neural Network for the Heat Equation

Problem 1 (100 points, 12 parts) considers u_t − α u_xx = 0 on x, t in [0, 1] with u(0, x) = sin(πx) and zero boundary values. The contestant proves the analytic solution u = e^{−απ²t} sin(πx), builds a fully connected network HeatPINN with tanh activations, creates PDE collocation (500 random points), initial-condition (101 points) and boundary-condition (202 points) datasets, uses torch.autograd.grad for first and second derivatives, trains with Adam (learning rate 1e-3, α = 0.1, 1,000 epochs, mini-batches of 32 for the PDE set), explains the batching choice, and evaluates the network on a 101×101 grid.