Tensor-Product Legendre Features for Physics-Informed Neural Networks: A Fixed-Time Evaluation on Nonlinear PDEs

Physics-informed neural networks (PINNs) provide a framework for approximating solutions of partial differential equations (PDEs), but their accuracy depends on the input representation and optimization procedure. We investigated a Legendre-expanded PINN (LE-PINN) that constructs tensor-product Legendre features from space–time inputs before processing them with a multilayer perceptron. LE-PINN and a standard PINN were evaluated on viscous Burgers, nonlinear Klein–Gordon, and nonlinear Schrödinger problems under fixed training-time budgets, with results reported over 10 randomized trials. For the Burgers problem, LE-PINN achieved a lower mean relative L2 error, mean squared error, maximum absolute error, and PDE residual RMS. For Klein–Gordon, it achieved a lower mean relative L2 error and mean squared error, but a higher maximum absolute error and residual RMS. For the nonlinear Schrödinger problem, the standard PINN achieved lower mean values for all four metrics. We additionally reported an Ulam–Hyers-motivated empirical error-to-residual ratio when an exact reference solution is available. Although this ratio was lower for LE-PINN in all three problems, a smaller ratio can result from a larger residual and does not establish an improved accuracy or a certified stability bound. The results demonstrate the problem-dependent benefits and limitations of tensor-product Legendre features under the evaluated computational budgets.