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.

通过设计模块化和强健的框架，引入核加权校正残差（CoRes）将核方法和深度神经网络相结合，提高了求解非线性偏微分方程系统的性能，并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。

使用核加权修正残差的神经网络求解偏微分方程

Neural Networks with Kernel-Weighted Corrective Residuals for Solving  Partial Differential Equations

There have been growing interests in leveraging experimental measurements to
discover the underlying partial differential equations (PDEs) that govern
complex physical phenomena. Although past research attempts have achieved great
success in data-driven PDE discovery, the robustness of the existing methods
cannot be guaranteed when dealing with low-quality measurement data. To
overcome this challenge, we propose a novel physics-encoded discrete learning
framework for discovering spatiotemporal PDEs from scarce and noisy data. The
general idea is to (1) firstly introduce a novel deep convolutional-recurrent
network, which can encode prior physics knowledge (e.g., known PDE terms,
assumed PDE structure, initial/boundary conditions, etc.) while remaining
flexible on representation capability, to accurately reconstruct high-fidelity
data, and (2) perform sparse regression with the reconstructed data to identify
the explicit form of the governing PDEs. We validate our method on three
nonlinear PDE systems. The effectiveness and superiority of the proposed method
over baseline models are demonstrated.

本文提出一种新的基于物理编码离散学习框架，用于从稀缺且有噪声的数据中发现时空偏微分方程，通过引入基于深度卷积 - 循环网络进行先前的物理知识编码，并利用重构数据的稀疏回归来识别控制 PDEs 的显式形式。作者在三个非线性 PDE 系统上进行了验证，展示了该方法的有效性和优越性。