Numerical Analysis

Physics-Informed Neural Networks: A Didactic Derivation of the Complete Training Cycle

This paper provides a complete, step-by-step manual derivation of how Physics-Informed Neural Networks (PINNs) are trained, including forward propagation, loss computation, and backpropagation with explicit numerical examples. It bridges the gap between automatic differentiation libraries and the underlying mathematical operations, making PINN training transparent and verifiable.