September 11, 2026 — Researchers at the Eastern Institute of Technology (EIT), Ningbo, have developed a novel graph-based approach that directly extracts concise and accurate constitutive equations from solid material experimental data. Published in Science Advances, the new method enables the discovery of innovative constitutive models for alloy steels, lithium metal and filled rubbers. It outperforms mainstream empirical models in predictive accuracy while preserving explicit, physically interpretable mathematical formulations.
This breakthrough addresses a longstanding limitation in solid mechanics: the conventional reliance on predefined empirical formulas to characterize the complex mechanical responses of metallic and nonmetallic materials. "Constitutive models are foundational to solid mechanics. Traditionally, researchers derive mathematical forms based on physical intuition and subsequently calibrate model parameters using experimental data," said Hao Xu, EIT postdoctoral researcher and lead author of the study. "Although this paradigm has achieved great success in mechanics research, predetermined equation structures inherently restrict the model's descriptive and predictive capability. Our framework shifts the research paradigm: it starts purely from experimental data and employs artificial intelligence to autonomously search for and identify optimal constitutive equations."
"The core technical challenge lies in efficiently encoding both equation architectures and material-specific parameters into a searchable format for computational algorithms," explained Yuntian Chen, EIT associate professor and co-corresponding author of the study. "By representing mathematical equations as graph structures, we enable simultaneous optimization of equation topology and material parameterization, which resolves this key bottleneck."
Writing equations as graphs
At the core of the proposed GraphED framework lies an innovative graph-based equation representation strategy. Instead of adopting the conventional tree structure for mathematical expressions, the team encodes physical equations as directed graphs. In this graph architecture, nodes correspond to mathematical operators and physical variables, while directed edges define their logical and computational connections. These edges can further accommodate fixed physical constants and tunable material-dependent parameters.
This unique representation empowers GraphED to identify universal mathematical structures across diverse materials and experimental conditions, while adaptively calibrating personalized parameters for individual material scenarios. The framework iteratively generates, evaluates, and optimizes candidate graph-structured equations to output physically consistent, mathematically compact, and human-interpretable constitutive laws with high prediction fidelity.
Discovering new constitutive laws across materials
The research team validated the generality and superiority of GraphED on multiple typical solid material systems with distinct mechanical characteristics.
For alloy steels, the method successfully discovered explicit equations governing strain-rate dependence and strain hardening behaviors. Integrated into a complete constitutive model, these data-driven equations deliver more precise mechanical predictions than the widely adopted Johnson–Cook model.
The team further investigated lithium metal, a critical material for energy devices whose mechanical behaviors are highly sensitive to temperature and strain-rate variations and difficult to model with traditional methods. GraphED yielded concise plastic-flow constitutive equations that achieve superior agreement with experimental measurements compared with conventional empirical models.
In addition, the framework was applied to filled rubbers, capturing a compact hyperelastic constitutive equation that maintains robust accuracy across varying material compositions and temperature conditions.
"Many complex material mechanical behaviors cannot be well described by existing empirical constitutive models," said Dongxiao Zhang, Chair Professor at EIT and corresponding author of the study. "Equation discovery via graph-based AI provides a powerful new paradigm to assist researchers in deriving rigorous mathematical descriptions when traditional model forms fall short." Beyond computational mechanics and material modeling, this framework shows broad potential for a wide range of disciplines that seek interpretable physical laws directly from experimental and observational data.
