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.

翻译：物理信息神经网络（PINNs）通过训练包含物理约束的损失函数的神经网络，已成为求解偏微分方程（PDEs）的一种强大方法。本文提出HyResPINNs，这是一类新颖的PINNs，其特点是采用自适应混合残差块，集成了标准神经网络和径向基函数（RBF）网络。HyResPINNs的一个显著特征是在每个残差块中使用自适应组合参数，从而能够动态加权神经网络和RBF网络的贡献。我们对一系列具有挑战性的PDE问题进行的实证评估表明，HyResPINNs始终比基线方法获得更高的精度。这些结果凸显了HyResPINNs在弥合经典数值方法与现代基于机器学习的求解器之间差距的潜力，为开发更鲁棒、更自适应的物理信息建模方法铺平了道路。