Neural Networks with Kernel-Weighted Corrective Residuals for Solving
  Partial Differential Equations
Carlos Mora Amin Yousefpour Shirin Hosseinmardi Ramin Bostanabad
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.