用于物理信息神经网络的分层秩演化表示

中文摘要 AI 辅助

近来，基于张量的物理信息神经网络（T-PINNs）受到越来越多的关注。然而，现有的T-PINNs仍面临一个根本挑战：它们主要依赖预先指定的低秩张量分解和手动调整的秩，这限制了它们捕捉多元解函数底层结构的能力，并阻碍了其实际部署。为解决这一挑战，我们提出了一种针对多元函数的分层秩演化（缩写为HRE）表示，该表示使我们能够忠实地捕捉目标多元函数的底层结构，同时自动确定秩。具体而言，在HRE表示的分层设计中，目标多元函数被分解为一个小规模内部张量和沿每个模式的一组单变量函数，其中可轻松部署定制的张量网络分解来捕捉小规模内部张量的底层结构。在HRE表示中，关键超参数秩可在分解过程中自适应确定，无需手动调整秩，使HRE可实际应用于现实世界问题。此外，我们相应构建了HRE-PINNs。大量数值实验，包括高维静态问题（亥姆霍兹方程和泊松方程）、非线性时变问题（克莱因-戈登方程）以及复杂流体动力学问题（流动混合方程和纳维-斯托克斯方程），表明HRE-PINNs在精度方面始终优于现有的最先进方法。

英文摘要

Recently, tensor-based physics-informed neural networks (T-PINNs) have received increasing attention. However, existing T-PINNs still face a fundamental challenge: they mainly rely on pre-specified low-rank tensor decompositions with manually tuned ranks, which limits their ability to capture the underlying structures of multivariate solution functions and hinders their practical deployment. To address this challenge, we propose a hierarchical rank-evolving (abbreviated as HRE) representation for multivariate functions, which endows us to faithfully capture the underlying structure of the targeted multivariate function accompanying with automatic rank determination. Concretely, in the hierarchical design of HRE representation, the target multivariate function is decomposed as a small-scale inner tensor with a set of univariate functions along each mode, where a customized tensor network decomposition can be readily deployed to capture the underlying structure of the small-scale inner tensor. In HRE representation, the crucial hyperparameters, ranks, can be adaptively revealed during the decomposition, freeing us from manual rank tuning and making HRE practically applicable to real-world problems. Besides, we build the HRE-PINNs correspondingly. Extensive numerical experiments, including high-dimensional static problems (Helmholtz equation and Poisson equation), nonlinear time-dependent problems (Klein-Gordon equation), and complex fluid-dynamics problems (flow mixing equation and Navier-Stokes equation), demonstrate that HRE-PINNs consistently outperform existing state-of-the-art approaches in terms of accuracy.

发表机构

• University of Electronic Science and Technology of China（电子科技大学）
• Southwestern University of Finance and Economics（西南财经大学）

机构由 AI 辅助整理，请以论文原文为准。