AI Writes Its Own Physics Equations for Materials from Raw Data

For more than a century, the way engineers describe how materials deform, harden, and fail has followed a familiar ritual. A researcher with deep physical intuition proposes a mathematical formula, guesses its general shape, and then spends weeks or months calibrating its adjustable constants against experiments. The formula works, more or less, until a new material, a new temperature range, or a new loading condition pushes it beyond the assumptions baked into its structure. Now a team at the Eastern Institute of Technology, Ningbo, has built an artificial intelligence framework that breaks this ritual apart. Instead of beginning with a human-chosen equation, the method starts purely from experimental data and autonomously searches for the optimal constitutive law itself, producing compact, explicit, human-readable equations that outperform the empirical models engineers have relied on for decades. The work, published in Science Advances, demonstrates the approach on alloy steels, lithium metal, and filled rubbers, three material classes whose mechanical behavior has long resisted clean mathematical description.

Constitutive models are the connective tissue of solid mechanics. They are the mathematical rules that translate stress, strain, temperature, and strain rate into predictions of how a real object will bend, dent, stretch, or shatter. Every crash simulation of a car body, every structural analysis of a bridge, and every design study for a battery electrode depends on them. Hao Xu, a postdoctoral researcher at EIT and lead author of the study, explains that the traditional paradigm, while enormously successful, carries an inherent ceiling. Researchers derive a mathematical form based on physical intuition and then calibrate its parameters with data, but the predetermined equation structure restricts what the model can ultimately describe and predict. If the true material behavior involves a functional dependence the chosen formula cannot express, no amount of parameter fitting will recover it. The new framework, called GraphED, inverts the workflow: data comes first, and the equation emerges from it.

The central technical obstacle to such data-driven discovery is representation. Machine learning systems can only optimize what they can encode, and mathematical equations are notoriously awkward objects for algorithms to manipulate. Yuntian Chen, EIT associate professor and co-corresponding author of the study, describes the challenge as efficiently encoding both equation architectures and material-specific parameters into a searchable format for computational algorithms. Conventional symbolic regression approaches typically represent expressions as trees, with operators branching from a root node down to leaves. Trees, however, make it difficult to share structure across different materials and to embed tunable parameters cleanly. The EIT team’s solution is to abandon the tree entirely and encode each candidate equation as a directed graph. In this graph architecture, nodes correspond to mathematical operators and physical variables, while directed edges define the logical and computational connections between them.

The graph representation is more than a cosmetic change, because the edges themselves become carriers of physical meaning. An edge can hold a fixed physical constant, such as a universal exponent, or a tunable material-dependent parameter that varies from one alloy or polymer to another. This design allows GraphED to do two things simultaneously that earlier methods treated separately. It can identify a universal mathematical structure shared across diverse materials and experimental conditions, and it can adaptively calibrate personalized parameters for each individual material scenario. The framework then iterates: it generates candidate graph-structured equations, evaluates how well each one reproduces the experimental data, and optimizes both the topology of the graph and the values of its parameters. The output is a constitutive law that is physically consistent, mathematically compact, and, crucially, interpretable by a human engineer who can read the equation and understand what each term contributes.

That last quality, interpretability, is what separates this work from the black-box models that dominate much of modern machine learning. A neural network trained on material data may predict well, but it offers no equation, no insight, and no transferable knowledge. GraphED delivers the opposite: an explicit formula that a researcher can inspect, question, embed in existing simulation software, and carry into new contexts. The team validated this promise across three very different solid material systems, each chosen for its distinct mechanical character and its historical resistance to empirical modeling. The breadth of the test cases matters, because a discovery method that only works on one material family is a curiosity; one that generalizes across steels, metals, and elastomers is a tool.

The first test case was alloy steel, the workhorse of structural engineering, whose behavior under high strain rates and large plastic deformation is governed by strain-rate dependence and strain hardening. These are precisely the phenomena captured by the Johnson–Cook model, one of the most widely adopted constitutive formulations in impact and crashworthiness analysis. GraphED discovered explicit equations governing both strain-rate dependence and strain hardening directly from experimental data, and when the team integrated these data-driven equations into a complete constitutive model, the result delivered more precise mechanical predictions than the Johnson–Cook benchmark. For a field in which the incumbent model has been refined over decades, the demonstration that an autonomously discovered equation can beat it on predictive accuracy is a significant result.

The second material pushed the method into far harder territory. Lithium metal is a critical component of next-generation energy devices, including high-energy-density batteries, but its mechanical behavior is notoriously difficult to model. Its plastic flow is highly sensitive to both temperature and strain rate, and the interplay of these dependencies has made traditional empirical formulations unreliable. GraphED yielded concise plastic-flow constitutive equations that achieve superior agreement with experimental measurements compared with conventional empirical models. For battery designers, whose simulations of electrode deformation and dendrite-related failure depend on accurate mechanical descriptions, the ability to extract a trustworthy equation directly from data could shorten development cycles and improve the fidelity of electrochemical-mechanical coupled models.

The third validation targeted filled rubbers, a class of elastomers whose hyperelastic response changes with filler composition and temperature. Rubber components in tires, seals, and vibration isolators experience large, reversible deformations, and their stiffness depends on both what they are made of and how hot they are. GraphED captured a compact hyperelastic constitutive equation that maintains robust accuracy across varying material compositions and temperature conditions, meaning a single discovered formula could serve a family of rubber compounds rather than requiring a bespoke model for each formulation. Together, the three case studies span rate-dependent plasticity, temperature-sensitive metal plasticity, and finite-strain hyperelasticity, covering a remarkably wide slice of solid mechanics with one framework.

The researchers see the implications reaching well beyond computational mechanics. Dongxiao Zhang, Chair Professor at EIT and corresponding author of the study, notes that many complex material mechanical behaviors cannot be well described by existing empirical constitutive models, and that 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. The phrase that matters is rigorous mathematical description: the goal is not a surrogate that mimics data, but a law, expressed in symbols, that captures the underlying behavior. Because the framework requires only experimental or observational data as input, its potential applications extend to any discipline seeking interpretable physical laws, from soft matter physics and biomechanics to geophysics and beyond, wherever data exists but the governing equation does not.

The study also signals a broader shift in how science may handle the equation-discovery step itself. Symbolic regression and equation learning have been growing fields, but the graph-based encoding developed here resolves a key bottleneck by enabling simultaneous optimization of equation topology and material parameterization, a combination that makes the search tractable for realistic, multi-material datasets. If the paradigm spreads, the role of the mechanician may evolve from proposing formulas to curating data, specifying physical constraints, and interpreting the equations that the algorithm returns. The authors declare no competing interest, and the research was published as a peer-reviewed contribution in Science Advances. For a field built on equations handed down through generations of researchers, the prospect of machines that read raw experimental data and hand back compact, readable laws of material behavior is a genuinely new way of doing physics, and the steels, lithium, and rubber that proved it are only the beginning.

Subject of Research: Graph-based AI equation discovery of constitutive laws in solid materials from experimental data

Article Title: AI discovers interpretable constitutive laws in solids directly from data

Article References: AI discovers interpretable constitutive laws in solids directly from data. (n.d.). Original publication

Image Credits: AI Generated

DOI: Not provided

Keywords: constitutive models, solid mechanics, graph-based equation discovery, symbolic regression, alloy steels, lithium metal, filled rubbers, machine learning, Science Advances, interpretable AI, strain hardening, hyperelasticity