一种基于混合物理信息神经网络的多尺度求解器，作为偏微分方程约束优化问题
A hybrid physics-informed neural network based multiscale solver as a partial differential equation constrained optimization problem
AI总结：
本文提出一种混合物理信息神经网络多尺度求解器，将PINN与有限元结合，通过粗尺度约束和弱收敛正则化项改善细尺度PDE求解的收敛性能。
摘要：
在这项工作中，我们研究了受偏微分方程（PDEs）约束的物理信息神经网络（PINNs）及其在逼近具有两个特征尺度的偏微分方程中的应用。从连续视角来看，我们的公式对应于一个具有PINN型目标的非标准PDE约束优化问题。从离散角度来看，该公式代表了一种同时利用神经网络和有限元的混合数值求解器。为了进行问题分析，我们引入了一个适当的函数空间，并开发了一种数值求解算法。后者将基于伴随方法的高效梯度计算技术与自动微分相结合。这种新的多尺度方法随后被示例性地应用于一个具有振荡系数的热传导问题。在此背景下，神经网络逼近一个细尺度问题，而一个粗尺度问题约束着相关的学习过程。我们证明了通过基于弱收敛的正则化项将粗尺度信息纳入神经网络训练过程是有益的。事实上，在保持升尺度一致性的同时，该项鼓励非平凡的PINN解，并且还充当细尺度PDE低频分量的预处理器，从而改善了PINN方法的收敛性能。
英文摘要：
In this work, we study physics-informed neural networks (PINNs) constrained by partial differential equations (PDEs) and their application in approximating PDEs with two characteristic scales. From a continuous perspective, our formulation corresponds to a non-standard PDE-constrained optimization problem with a PINN-type objective. From a discrete standpoint, the formulation represents a hybrid numerical solver that utilizes both neural networks and finite elements. For the problem analysis, we introduce a proper function space, and we develop a numerical solution algorithm. The latter combines an adjoint-based technique for the efficient gradient computation with automatic differentiation. This new multiscale method is then applied exemplarily to a heat transfer problem with oscillating coefficients. In this context, the neural network approximates a fine-scale problem, and a coarse-scale problem constrains the associated learning process. We demonstrate that incorporating coarse-scale information into the neural network training process via a weak convergence-based regularization term is beneficial. Indeed, while preserving upscaling consistency, this term encourages non-trivial PINN solutions and also acts as a preconditioner for the low-frequency component of the fine-scale PDE, resulting in improved convergence properties of the PINN method.
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