HyResPINNs: Hybrid Residual Networks for Adaptive Neural and RBF Integration in Solving PDEs

Abstract: Physics-informed neural networks (PINNs) have emerged as a powerful approach for solving partial differential equations (PDEs) by training neural networks with loss functions that incorporate physical constraints. In this work, we introduce HyResPINNs, a novel class of PINNs featuring adaptive hybrid residual blocks that integrate standard neural networks and radial basis function (RBF) networks. A distinguishing characteristic of HyResPINNs is the use of adaptive combination parameters within each residual block, enabling dynamic weighting of the neural and RBF network contributions. Our empirical evaluation of a diverse set of challenging PDE problems demonstrates that HyResPINNs consistently achieve superior accuracy to baseline methods. These results highlight the potential of HyResPINNs to bridge the gap between classical numerical methods and modern machine learning-based solvers, paving the way for more robust and adaptive approaches to physics-informed modeling.