Automatic Differentiation from Scratch: How PyTorch Computes Gradients in Physics-Informed Neural Networks

This paper traces, with explicit numerical values, how PyTorch's automatic differentiation (AD) engine computes gradients for Physics-Informed Neural Network (PINN) training -- a setting that requires two levels of differentiation: computing the physics derivative haty'(t)=dhaty/dt through the network, and computing parameter gradients nablaθ L of a loss that itself depends on haty'(t). Using a 1-3-3-1 multilayer perceptron and the initial value problem y'(t)+y(t)=0, y(0)=1, we trace the complete pipeline at every node: the computational graph built during the forward pass, the reverse-mode backward traversal that computes all 22 parameter gradients in a single pass, and the graph-on-graph mechanism by which textttcreategraph=True enables correct differentiation through the physics-informed residual. Every adjoint value is verified against the hand derivations of Tahimi (2026), connecting the P/Q sensitivity framework to the vector--Jacobian products used by PyTorch's autograd engine.