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

This paper is a step-by-step, self-contained guide to the complete training cycle of a Physics-Informed Neural Network (PINN) -- a topic that existing tutorials and guides typically delegate to automatic differentiation libraries without exposing the underlying algebra. Using a first-order initial value problem with a known analytical solution as a running example, we walk through every stage of the process: forward propagation of both the network output and its temporal derivative, evaluation of a composite loss function built from the ODE residual and the initial condition, backpropagation of gradients -- with particular attention to the product rule that arises in hidden layers -- and a gradient descent parameter update. Every calculation is presented with explicit, verifiable numerical values using a 1-3-3-1 multilayer perceptron with two hidden layers and 22 trainable parameters. From these concrete examples, we derive general recursive formulas -- expressed as sensitivity propagation relations -- that extend the gradient computation to networks of arbitrary depth, and we connect these formulas to the automatic differentiation engines used in practice. The trained network is then validated against the exact solution, achieving a relative $L^2$ error of $4.290 \times 10^{-4}$ using only the physics-informed loss, without any data from the true solution. A companion Jupyter/PyTorch notebook reproduces every manual calculation and the full training pipeline, providing mutual validation between hand-derived and machine-computed gradients.

We systematically investigate finite-difference (FD) derivative computation in Physics-Informed Neural Networks (PINNs) as an alternative to automatic differentiation (AD). On three benchmark PDEs we show that, with a properly calibrated step size, FD matches AD in accuracy on every problem while running faster across the full tested batch-size range and using substantially less GPU memory, and that a stochastic variant we propose outperforms AD on a stationary problem. We further show that for neural architectures with inter-sample dependencies (e.g. BatchNorm, self-attention) the standard PyTorch autograd idiom is silently incorrect; the correct per-sample alternative is computationally infeasible at PINN-relevant batch sizes, while FD provides a forward-only approximation that is empirically an order of magnitude closer to the true per-sample derivative.

Physics-informed neural networks (PINNs) train a single neural approximation by minimizing multiple physics- and data-derived losses, but the gradients of these losses often interfere and can stall optimization. Existing remedies typically treat this pathology either through scalar loss balancing or full-parameter-space gradient surgery, leaving it unclear which intervention is most appropriate. We show that PINN gradient conflict is not a uniform failure mode with one universal remedy. Instead, we identify distinct PINN gradient-conflict regimes, each associated with a different intervention class. Persistent directional conflict may require separate loss-indexed parameter subspaces, magnitude imbalance often favors scalar reweighting, and low or transient conflict may require no extra mitigation. To select between scalar reweighting and a lightweight architectural intervention, we propose a diagnostic-first framework. It profiles a 1000-step unmodified PINN run and, when intervention is warranted, uses one low-rank adapter per loss to create explicit loss-indexed parameter subspaces attached to a shared PINN trunk, providing each loss with a direct gradient pathway. Across more than 60 PDE configurations, including forward, inverse, multi-physics, parameter-varying, and high-dimensional problems up to 50D, persistent directional conflict dominates standard forward $K=3$ benchmarks and a natural $K=4$ thermoelastic system, where adapters combined with reweighting yield significant improvements. In contrast, $K=3$ inverse problems and natural $K=5$ and $K=6$ multi-physics systems are largely magnitude-dominated and often favor reweighting alone, while full-parameter-space gradient surgery can fail on heterogeneous parameter spaces.