HyResPINNs: Adaptive Hybrid Residual Networks for Learning Optimal Combinations of Neural and RBF Components for Physics-Informed Modeling

Physics-informed neural networks (PINNs) are an increasingly popular class of techniques for the numerical solution of partial differential equations (PDEs), where neural networks are trained using loss functions regularized by relevant PDE terms to enforce physical constraints. We present a new class of PINNs called HyResPINNs, which augment traditional PINNs with adaptive hybrid residual blocks that combine the outputs of a standard neural network and a radial basis function (RBF) network. A key feature of our method is the inclusion of adaptive combination parameters within each residual block, which dynamically learn to weigh the contributions of the neural network and RBF network outputs. Additionally, adaptive connections between residual blocks allow for flexible information flow throughout the network. We show that HyResPINNs are more robust to training point locations and neural network architectures than traditional PINNs. Moreover, HyResPINNs offer orders of magnitude greater accuracy than competing methods on certain problems, with only modest increases in training costs. We demonstrate the strengths of our approach on challenging PDEs, including the Allen-Cahn equation and the Darcy-Flow equation. Our results suggest that HyResPINNs effectively bridge the gap between traditional numerical methods and modern machine learning-based solvers.