Physics-informed machine learning (PIML) has emerged as a promising alternative to conventional numerical methods for solving partial differential equations (PDEs). PIML models are increasingly built via deep neural networks (NNs) whose architecture and training process are designed such that the network satisfies the PDE system. While such PIML models have substantially advanced over the past few years, their performance is still very sensitive to the NN's architecture and loss function. Motivated by this limitation, we introduce kernel-weighted Corrective Residuals (CoRes) to integrate the strengths of kernel methods and deep NNs for solving nonlinear PDE systems. To achieve this integration, we design a modular and robust framework which consistently outperforms competing methods in solving a broad range of benchmark problems. This performance improvement has a theoretical justification and is particularly attractive since we simplify the training process while negligibly increasing the inference costs. Additionally, our studies on solving multiple PDEs indicate that kernel-weighted CoRes considerably decrease the sensitivity of NNs to factors such as random initialization, architecture type, and choice of optimizer. We believe our findings have the potential to spark a renewed interest in leveraging kernel methods for solving PDEs.

翻译：物理信息机器学习（PIML）已成为求解偏微分方程（PDE）传统数值方法的有前途替代方案。PIML模型越来越多地通过深度神经网络构建，其架构和训练过程设计使网络满足PDE系统。尽管此类PIML模型在过去几年取得了显著进展，但其性能仍对神经网络架构和损失函数非常敏感。受此局限性启发，我们引入了核加权修正残差（CoRes）以整合核方法与深度神经网络的优势，用于求解非线性PDE系统。为实现这一整合，我们设计了一个模块化且稳健的框架，该框架在求解广泛基准问题中持续优于竞争方法。这一性能提升具有理论依据，尤其吸引人的是我们在简化训练过程的同时，仅略微增加了推理成本。此外，我们对多个PDE的求解研究表明，核加权CoRes显著降低了神经网络对随机初始化、架构类型和优化器选择等因素的敏感性。我们相信这一研究成果有望激发利用核方法求解PDE的新一轮兴趣。