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Peer-reviewedCC BY-NCScience Advances

Beyond empirical models: Discovering constitutive laws in solids with graph-based equation discovery

  • Hao Jun Xu1,2
  • Yuntian Chen3,2
  • Dongxiao Zhang4,2
4 affiliations
  1. 1Department of Electrical Engineering, Tsinghua University, Beijing 100084, P. R. China
  2. 2Zhejiang Key Laboratory of Industrial Intelligence and Digital Twin, Eastern Institute of Technology, Ningbo 315200, Zhejiang, China
  3. 3Ningbo Institute of Digital Twin, Eastern Institute of Technology, Ningbo 315200, Zhejiang, P. R. China
  4. 4Institute for Advanced Study, Lingnan University, Tuen Mun, Hong Kong, China
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Abstract

Constitutive models are fundamental to solid mechanics and materials science, underpinning the quantitative description of material behaviors. Traditional phenomenological models are often built on expert intuition and empirical fitting, which limits their generalizability. In this work, we propose a graph-based equation discovery framework for automated discovery of constitutive laws directly from multicase experimental data. This framework expresses equations as directed graphs, where nodes represent operators and variables, edges denote computational relations, and edge features encode parametric dependencies. This enables the generation and optimization of free-form symbolic expressions with undetermined material-specific parameters. Through the framework, we have found constitutive models for strain-rate effects in alloy steel materials, deformation behavior of lithium metal, and hyperelastic behavior of filled rubbers. The discovered models exhibit compact analytical structures and achieve higher accuracy than empirical models. The proposed framework provides a generalizable and interpretable approach for data-driven scientific modeling, particularly in contexts where traditional empirical models are inadequate for representing complex physical phenomena.

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Cite

BibTeXRIS
BibTeX
@article{xu2026beyond,
  title = {{Beyond empirical models: Discovering constitutive laws in solids with graph-based equation discovery}},
  author = {Hao Jun Xu and Yuntian Chen and Dongxiao Zhang},
  journal = {Science Advances},
  year = {2026},
  month = sep,
  doi = {10.1126/sciadv.aec0989},
  url = {https://doi.org/10.1126/sciadv.aec0989},
  note = {CC BY-NC},
}
RIS
TY  - JOUR
TI  - Beyond empirical models: Discovering constitutive laws in solids with graph-based equation discovery
AU  - Hao Jun Xu
AU  - Yuntian Chen
AU  - Dongxiao Zhang
PY  - 2026
DA  - 2026/09/11
JO  - Science Advances
DO  - 10.1126/sciadv.aec0989
UR  - https://doi.org/10.1126/sciadv.aec0989
AB  - Constitutive models are fundamental to solid mechanics and materials science, underpinning the quantitative description of material behaviors. Traditional phenomenological models are often built on expert intuition and empirical fitting, which limits their generalizability. In this work, we propose a graph-based equation discovery framework for automated discovery of constitutive laws directly from multicase experimental data. This framework expresses equations as directed graphs, where nodes represent operators and variables, edges denote computational relations, and edge features encode parametric dependencies. This enables the generation and optimization of free-form symbolic expressions with undetermined material-specific parameters. Through the framework, we have found constitutive models for strain-rate effects in alloy steel materials, deformation behavior of lithium metal, and hyperelastic behavior of filled rubbers. The discovered models exhibit compact analytical structures and achieve higher accuracy than empirical models. The proposed framework provides a generalizable and interpretable approach for data-driven scientific modeling, particularly in contexts where traditional empirical models are inadequate for representing complex physical phenomena.
SN  - 2375-2548
ER  -