Learning approximate solutions of partial differential equations within the physics-informed framework

Abstract

Variants of physics-informed neural networks (PINNs) often treat neural networks as the main solution representation, yet solutions are encoded implicitly in high-dimensional weights, making training behavior and outcomes difficult to predict or analyze across different partial differential equation (PDE) settings. In this study, we propose Physics-Informed Spectral Coefficient Learning (PISCL), a network-free physics-informed formulation that represents PDE solutions using a finite set of spatial and temporal spectral modes and learns the corresponding coefficients by minimizing a residual loss. This coefficient-based representation is explicit and interpretable, since each trainable parameter corresponds to a specific mode. We evaluated PISCL on two benchmark PDEs (1-D linear wave and viscous Burgers). Over repeated independent runs, it achieves MAE $0.0080\pm0.0062$ in the wave equation and MAE $0.0578\pm0.0003$ in the most challenging Burgers setting, matching or outperforming standard PINN baselines under the same sampling budget. Overall, these results indicate that neural networks are not the only viable solution representation and that explicit coefficient-based representations can provide a simple and interpretable alternative for physics-informed PDE solving.