RAA-PINN: Residual Anti-Aliasing for Physics-Informed Neural Networks

Physics-informed neural networks (PINNs) enforce differential equations at finitely many collocation points, but small residuals at those points can conceal unresolved errors between them. We study this discrepancy as residual aliasing, focusing on high-frequency structure in the PDE residual that pointwise training may miss. We propose RAA-PINN, a frequency-aware framework that couples residual spectral regularization with spectrum-aware adaptive sampling. The regularizer penalizes high-frequency residual energy rather than the solution spectrum, while the sampler combines residual magnitude with local spectral energy to allocate new collocation points. Across five PDE benchmarks covering Allen–Cahn, convection–diffusion, and Kovasznay flow, RAA-PINN achieves the lowest raw high-frequency residual energy among the evaluated methods under both full-core and equal-time settings. A strict spectral guard admits RAA-PINN candidates to auxiliary-metric comparisons only when their raw high-frequency residual energy is lower than that of the strongest applicable non-RAA baseline. Under this protocol, the selected candidates also achieve the lowest relative L2 error, worst-region error, and dense residual on all five tasks. Ablation studies examine the contributions of spectral regularization, adaptive sampling, and guarded refinement. Independent two- and three-dimensional FEM/COMSOL reference comparisons further show lower aggregate solution errors. These results support residual-spectrum control as an effective way to improve physics-informed learning beyond pointwise residual minimization.