RBAP and RBAC: Two Novel Types in Nonlinear Residual Weighting for Physics-Informed Neural Networks

Physics-informed neural networks (PINNs) face challenges in prediction accuracy and robustness due to loss weight imbalance. We propose two gradient-free nonlinear weighting schemes: Residual-based Attention Power-law (RBAP) and Residual-based Attention Constrained (RBAC). RBAP applies power-law amplification to residuals, intensifying focus on high-error regions. RBAC introduces clipping for robustness, applying uniform weighting to the high-residual points while preserving fine-grained weighting for low-residual ones. Experiments on 1D Reaction, 1D-Wave and Convection equations show about 50% accuracy improvement over baseline RBA without additional computational cost.