Engineering-Oriented Symbolic Regression: LLMs as Physics Agents for Discovery of Simulation-Ready Constitutive Laws

This paper introduces an Engineering-Oriented Symbolic Regression framework that leverages Large Language Models as physics-informed agents to autonomously discover simulation-ready, thermodynamically consistent constitutive laws for complex materials, successfully overcoming the numerical instability of traditional models while achieving high predictive accuracy.

The Gist

Imagine you are trying to teach a computer how to predict how a rubber band stretches, squishes, and twists. In the world of engineering, this is called finding a "Constitutive Law." It's basically the mathematical rulebook that tells a simulation software: "If I pull this material this hard, it will stretch that much."

For a long time, scientists have been stuck between two bad options:

1. The "Data-Heavy" Approach: They try to feed the computer millions of measurements from fancy, expensive lab experiments. It's accurate, but it's like trying to build a house by measuring every single grain of sand. It's too expensive and slow for most engineers.
2. The "Old-School Fitting" Approach: They guess a simple formula (like a straight line or a curve) and tweak the numbers until it fits the data perfectly. It's cheap and fast, but it's like fitting a square peg into a round hole. It looks good on the test data, but if you push the material in a weird way (like squishing it sideways), the math breaks, and the computer simulation crashes.

The New Solution: The "Physics-Savvy" AI Agent

This paper introduces a new method called Engineering-Oriented Symbolic Regression (EO-SR). Think of it as hiring a super-smart AI Detective (a Large Language Model) to help find the rulebook, rather than just letting a computer guess numbers.

Here is how it works, using some simple analogies:

1. The "Physics Agent" vs. The "Blind Search"

Imagine you are looking for a lost key in a giant field.

• Old Symbolic Regression is like a robot that runs around randomly, picking up every stick and stone, hoping one of them is the key. It might find a stick that looks like a key, but it's actually just a twig. It fits the data but violates the laws of physics.
• The New EO-SR hires a Physics Agent. Before the robot even starts running, the Agent says: "Wait! The key must be made of metal, it must be shiny, and it must fit in this specific lock. Ignore all the twigs."
• The Agent translates complex physics rules (like "energy can't be created out of nothing" or "materials shouldn't get softer when you squeeze them") into a checklist. The computer only looks for formulas that pass this checklist.

2. The "Rubber Band" Discovery

The researchers tested this on rubber-like materials (like tires or shoe soles).

• The Problem: Rubber gets stiffer the more you stretch it, until it hits a limit where it can't stretch anymore (the chains inside the rubber snap).
• The Old Way: Standard formulas (like the "Ogden" model) are great at stretching but terrible at squishing. When the researchers tried to simulate a rubber part being squished in a crash test, the math exploded, and the simulation crashed. It was like a bridge that held up cars but collapsed if a bird landed on it.
• The New Way: The AI Agent discovered a hybrid formula. It's like a recipe that mixes a simple, smooth sauce (the basic rubber behavior) with a special "safety valve" ingredient.
  • This "safety valve" is a mathematical term that acts like a brick wall. As you stretch the rubber, the formula gets harder and harder to push, eventually becoming infinite. This perfectly mimics the real-world limit where rubber stops stretching.

3. The "Zero-Shot" Magic

Here is the coolest part. The researchers taught the AI using only data from stretching (pulling) and squishing (pulling two ways). They never showed it data for shearing (sliding layers past each other).

• The Old Models: When asked to predict shearing, they guessed wildly and got it wrong. They were just memorizing the stretching data.
• The New Model: Because the AI Agent forced the formula to follow the laws of physics (not just the data points), the model figured out the shearing behavior on its own. It was like teaching a student the rules of grammar and then asking them to write a sentence in a language they've never heard before—they got it right because they understood the logic, not just the words.

4. The "Crash Test"

Finally, they put the new formula into a real-world simulation (Finite Element Analysis) of a rubber part with a notch (a weak spot).

• The Old Model: The simulation crashed immediately. The math got confused when the rubber was squished near the notch, creating a "numerical singularity" (a mathematical error that breaks the computer).
• The New Model: The simulation ran smoothly to the end. The "safety valve" in the formula prevented the math from breaking, ensuring the computer could calculate the stress without crashing.

The Big Takeaway

This paper isn't just about finding a better equation for rubber. It's about a new way of doing science:

• Don't just let AI guess: Don't let AI be a "black box" that spits out answers based on patterns.
• Make AI a "Guardian": Use AI to enforce the rules of the universe (physics) while it searches for answers.

By combining the creativity of AI with the strict rules of physics, the researchers created a tool that finds mathematical laws which are not only accurate but also safe to use in real-world engineering simulations. It bridges the gap between "it looks right on paper" and "it actually works in the real world."

Technical Summary

1. Problem Statement

The discovery of constitutive laws for complex materials faces a critical "validity gap" between two existing paradigms:

• High-Fidelity Data-Driven Approaches: Methods like Physics-Informed Neural Networks (PINNs) or Virtual Fields Method (VFM) offer high accuracy but require expensive full-field experimental data (e.g., DIC) and often produce "black-box" models lacking interpretability.
• Traditional Engineering Fitting: Methods using pre-defined analytical templates (e.g., Ogden, Yeoh) are accessible and interpretable but suffer from a fitting-simulation gap. Models optimized solely for minimizing fitting error (MSE) often violate fundamental physical constraints (e.g., thermodynamic stability, convexity) outside the calibration range. This leads to numerical singularities and convergence failures in Finite Element Analysis (FEA), particularly under complex multi-axial loading.

Core Challenge: How to discover constitutive laws that are both data-accurate (using standard, low-cost uniaxial/biaxial data) and mathematically robust (guaranteeing unconditional stability for simulation) without relying on expensive experimental setups or manual derivation.

2. Methodology: The EO-SR Framework

The authors propose Engineering-Oriented Symbolic Regression (EO-SR), a modular framework that leverages Large Language Models (LLMs) as "Physics-Informed Agents" to constrain the symbolic search process.

A. Architecture

The framework consists of three coupled modules:

1. Data Parser: Standardizes heterogeneous experimental inputs into a consistent tensor representation.
2. Skill Injector (The LLM Agent): The core innovation. An LLM (specifically Gemini 3-pro) acts as a domain expert. Given a natural language descriptor (e.g., "Isotropic Hyperelasticity"), it performs zero-shot reasoning to synthesize a "Skill Tuple" $S = (\mathcal{T}, \mathcal{O}, \mathcal{C})$:
   • $\mathcal{T}$: Coordinate transformations (e.g., mapping stretch ratios to shifted invariants $\tilde{I}_1, \tilde{I}_2$).
   • $\mathcal{O}$: An admissible operator set (e.g., excluding periodic functions like $\sin/\cos$ for monotonic materials; including rational and exponential functions).
   • $\mathcal{C}$: Physical inequality constraints derived from thermodynamics (e.g., Drucker stability, convexity).
3. Symbolic Evolution Engine: A genetic search algorithm that optimizes a composite loss function:
   $$\min_{f} \left( L_{fit}(f, D) + \lambda_{reg}\Omega(f) + \sum_{c \in \mathcal{C}} \lambda_c P(c, f) \right)$$
   Where $P(c, f)$ is a penalty functional (using ReLU) for violating physical constraints.

B. The "Skill" Mechanism

Unlike traditional SR which searches unconstrained mathematical space, the LLM Agent translates abstract physical laws (e.g., "strain energy must be convex") into executable Python logic. It enforces constraints such as:

• Frame Indifference: Ensuring dependence on invariants only.
• Thermodynamic Consistency: Enforcing the Hessian of the strain energy density to be positive semi-definite ($\det(H_W) \geq 0$).
• Agent-Assisted Selection: Post-search, the Agent audits top candidates to reject "mathematically bloated" or non-convex solutions (even if they have lower MSE) in favor of structurally robust forms (e.g., rational functions with singularities representing finite extensibility).

3. Key Contributions

1. Skill-Based Discovery Architecture: A novel framework where LLMs act as domain agents to generate executable physical constraints (zero-shot), transforming SR from a curve-fitter into a physics-governed discovery engine.
2. Bridging the Fitting-Simulation Gap: Demonstrating that injecting thermodynamic constraints (Drucker stability) into the loss function allows the discovery of laws that are unconditionally stable in FEA, eliminating numerical risks associated with traditional fitting.
3. Discovery of Optimal Hybrid Forms: The framework identified a novel constitutive law that combines a Mooney-Rivlin linear base with a rational locking term, outperforming industry standards in both generalization and numerical robustness.

4. Results

The framework was validated on the hyperelastic modeling of rubber-like materials using the standard Treloar dataset (Uniaxial and Equibiaxial tension).

• Model Discovery: The algorithm discovered a hybrid model (Eq16) with a complexity of 16 nodes:
  $$W(\tilde{I}_1, \tilde{I}_2) = 0.031(3.75\tilde{I}_1 + \tilde{I}_2) + \frac{\tilde{I}_1}{77.9 - 1.05\tilde{I}_1}$$

  • The first term represents the linear Mooney-Rivlin behavior.
  • The second term is a rational function with a singularity, physically representing finite extensibility (polymer chain locking) at large strains.

• Performance Metrics:

  • Fitting Accuracy: Achieved a training MSE of ~0.008, comparable to high-order Ogden models (N=3) but with a sparser structure.
  • Zero-Shot Generalization: Successfully predicted Pure Shear behavior (not used in training) with high accuracy (MSE ≈ 0.0048), whereas the Yeoh model failed due to overfitting.
  • Thermodynamic Stability: The discovered model exhibited a strictly convex energy landscape (positive definite Hessian) across the entire domain. In contrast, a lower-error unconstrained solution (Sqrt-Eq) showed non-convex regions, posing a risk of solver divergence.

• Finite Element Validation (The "Torture Test"):

  • Proposed Model: Successfully simulated a double-edge notched specimen under large deformation, handling severe transverse compression without convergence issues.
  • Ogden Model Failure: The industry-standard Ogden model (N=3) failed to converge. Forensic analysis revealed that while it fit tension data well, its negative exponent parameters caused a stiffness singularity ($\lambda^{\alpha-2}$) under transverse compression ($\lambda \approx 0.157$), leading to an ill-conditioned Jacobian matrix.

5. Significance and Future Outlook

• Paradigm Shift: This work redefines the role of LLMs in "AI for Science," moving them from coding assistants to guardians of theoretical consistency. They bridge the semantic gap between high-level physics and low-level optimization.
• Simulation-Ready Discovery: The study proves that numerical robustness is a fundamental discovery criterion, not just an implementation detail. By enforcing convexity and physical limits during the search, the framework produces "simulation-ready" closures.
• Generalizability: The "Skill Injector" architecture is modular. The authors demonstrate its potential for fiber-reinforced tissues (handling tension-compression asymmetry via logical branching constraints like Macaulay brackets), suggesting applicability to plasticity, anisotropy, and other complex systems.
• Low-Barrier Access: The framework enables the discovery of high-fidelity models using only standard, low-cost industrial testing data, removing the need for expensive full-field measurement setups.

In conclusion, EO-SR establishes a generalized pathway for discovering constitutive laws that satisfy both data accuracy and rigorous physical laws, effectively solving the long-standing challenge of generating models that are both accurate and numerically stable.