Symbolic identification of tensor equations in multidimensional physical fields

2025-07-02

AI Summary

This work addresses the fundamental challenge in data-driven discovery: the inability of existing methods to identify tensor-valued governing equations from multidimensional physical fields. To overcome this, we propose SITE—a symbolic tensor equation identification framework. SITE’s core contributions are threefold: (1) a novel “gene-host–plasmid” structural representation for encoding tensor equation forms; (2) incorporation of dimensional consistency constraints to rigorously prune the hypothesis space; and (3) tight coupling of tensor linear regression for coefficient optimization, thereby enhancing both evolutionary search efficiency and physical interpretability. Built upon multidimensional gene expression programming (M-GEP), SITE integrates genetic information preservation with real-time dimensional validation. It successfully recovers constitutive relations for both compressible and incompressible flows on synthetic and molecular simulation datasets. SITE demonstrates exceptional robustness to noise and limited data, significantly extending the applicability boundary of symbolic regression in tensor-based physical modeling.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Mixed Discrete/Continuous Search
• Cognitive Modeling & Cognitive Systems: Symbolic Representations

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Web data generation and simulation
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

Recently, data-driven methods have shown great promise for discovering governing equations from simulation or experimental data. However, most existing approaches are limited to scalar equations, with few capable of identifying tensor relationships. In this work, we propose a general data-driven framework for identifying tensor equations, referred to as Symbolic Identification of Tensor Equations (SITE). The core idea of SITE--representing tensor equations using a host-plasmid structure--is inspired by the multidimensional gene expression programming (M-GEP) approach. To improve the robustness of the evolutionary process, SITE adopts a genetic information retention strategy. Moreover, SITE introduces two key innovations beyond conventional evolutionary algorithms. First, it incorporates a dimensional homogeneity check to restrict the search space and eliminate physically invalid expressions. Second, it replaces traditional linear scaling with a tensor linear regression technique, greatly enhancing the efficiency of numerical coefficient optimization. We validate SITE using two benchmark scenarios, where it accurately recovers target equations from synthetic data, showing robustness to noise and small sample sizes. Furthermore, SITE is applied to identify constitutive relations directly from molecular simulation data, which are generated without reliance on macroscopic constitutive models. It adapts to both compressible and incompressible flow conditions and successfully identifies the corresponding macroscopic forms, highlighting its potential for data-driven discovery of tensor equation.

Problem

• Identifying tensor equations from multidimensional physical fields data
• Enhancing robustness in evolutionary algorithms for tensor equation discovery
• Validating framework with noise-resistant tensor equation recovery

Innovation

• Host-plasmid structure for tensor equations
• Dimensional homogeneity check for validity
• Tensor linear regression for coefficient optimization