Hierarchical rank-evolving representation for physics-informed neural networks

2026-08-10

AI Summary

This work addresses the limitations of existing tensor-based physics-informed neural networks (PINNs), which rely on manually prescribed low-rank structures and struggle to adaptively capture the intrinsic complexity of multivariate functions. To overcome this, the authors propose a hierarchical rank evolution (HRE) representation that decomposes multivariate functions into small-scale core tensors coupled with univariate functions along each mode. A tailored tensor network architecture is introduced to enable automatic rank adaptation and discovery of optimal structural configurations. This approach achieves, for the first time in PINNs, adaptive rank learning without manual hyperparameter tuning, substantially enhancing modeling capabilities for high-dimensional, nonlinear, and time-varying systems. Experimental results demonstrate that HRE-PINNs consistently outperform state-of-the-art methods across a range of challenging problems, including high-dimensional static systems, nonlinear time-dependent dynamics, and complex fluid flow scenarios.

Abstract

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.

Problem

• physics-informed neural networks
• tensor decomposition
• low-rank representation
• rank selection
• multivariate functions

Innovation

• hierarchical rank-evolving
• physics-informed neural networks
• tensor decomposition
• automatic rank determination
• multivariate function representation