LocRes–PINN: A Physics–Informed Neural Network with Local Awareness and Residual Learning

Physics–Informed Neural Networks (PINNs) have demonstrated efficacy in solving both forward and inverse problems for nonlinear partial differential equations (PDEs). However, they frequently struggle to accurately capture multiscale physical features, particularly in regions exhibiting sharp local variations such as shock waves and discontinuities, and often suffer from optimization difficulties in complex loss landscapes. To address these issues, we propose LocRes–PINN, a physics–informed neural network framework that integrates local awareness mechanisms with residual learning. This framework integrates a radial basis function (RBF) encoder to enhance the perception of local variations and embeds it within a residual backbone to facilitate stable gradient propagation. Furthermore, we incorporate a residual–based adaptive refinement strategy and an adaptive weighted loss scheme to dynamically focus training on high–error regions and balance multi–objective constraints. Numerical experiments on the Extended Korteweg–de Vries, Navier–Stokes, and Burgers equations demonstrate that LocRes–PINN reduces relative prediction errors by approximately 12% to 67% compared to standard benchmarks. The results also verify the model’s robustness in parameter identification and noise resilience.

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