张量列压缩可分离PINN：面向高维参数化偏微分方程的曲率感知优化框架

中文摘要 AI 辅助

本文针对应用于高维参数化偏微分方程（PDE）的物理信息神经网络（PINN），开发了一个二阶优化框架。该框架基于高斯-牛顿拉回度量构建，该度量在参数空间中提供了算子感知的曲率概念，并将该方法与更广泛的自然梯度方案家族联系起来。我们证明，对于坐标可分离的神经架构和允许有限可分离表示的线性微分算子（或非线性情况下的线性化算子），残差雅可比矩阵继承了结构化的可分离分解。这产生了在约化空间中高斯-牛顿步的精确压缩公式，无需在指数大的张量积配置网格上组装完整的残差雅可比矩阵。约化空间的维度（有效压缩维度）由局部配置网格大小、可分离算子结构和架构的收缩模式决定，从而避免了对完整张量积网格大小的依赖，并将参数空间中的稠密线性代数替换为规模小得多的结构化问题。在我们的框架内，我们研究了典型多面体分解和张量列参数化，并推导了它们与高斯-牛顿方法相关的完整代数特征，包括残差雅可比矩阵的结构、所得压缩系统及其有效压缩维度。在高维PDE（包括参数化问题）上的数值实验证明了所提出的压缩高斯-牛顿方法的高效率，该方法以少几个数量级的迭代次数和仅一小部分计算时间，实现了比张量压缩一阶基线低得多的误差。

英文摘要

In this work, we develop a second-order optimization framework for physics-informed neural networks (PINNs) applied to high-dimensional parametric partial differential equations (PDEs). The framework is built on the Gauss--Newton pullback metric, which provides an operator-informed notion of curvature in parameter space and connects the method to the broader family of natural gradient schemes. We show that, for coordinate-separable neural architectures and linear differential operators (or linearized operators in the nonlinear case) admitting a finite separable representation, the residual Jacobian inherits a structured separable factorization. This yields an exact compressed formulation of the Gauss--Newton step in a reduced space, without assembling the full residual Jacobian on the exponentially large tensor-product collocation grid. The dimension of the reduced space (the effective compressed dimension) is determined by the local collocation grid sizes, the separable operator structure, and the contraction pattern of the architecture, thereby avoiding dependence on the full tensor-product grid size and replacing dense linear algebra in parameter space by a substantially smaller structured problem. Within our framework, we investigate canonical polyadic and tensor-train parametrizations and derive their full algebraic characterization relevant to the Gauss--Newton method, including the structure of the residual Jacobian, the resulting compressed system, and its effective compressed dimension. Numerical experiments on high-dimensional PDEs, including parametric problems, demonstrate the high efficiency of the proposed compressed Gauss--Newton method, which achieves substantially lower errors than tensor-compressed first-order baselines with orders of magnitude fewer iterations and only a fraction of the computing time.

发表机构

• Weierstrass Institute for Applied Analysis and Stochastics (WIAS)（魏尔斯特拉斯应用分析与随机学研究所（WIAS））

机构由 AI 辅助整理，请以论文原文为准。