LLM-driven design of physics-constrained constitutive models: two agents are better than one

Technical Summary: LLM-Driven Design of Physics-Constrained Constitutive Models

Problem Statement
Developing constitutive models that accurately capture material deformation under load traditionally demands years of specialized expertise in continuum mechanics, machine learning, and scientific programming. While Large Language Models (LLMs) have recently demonstrated the ability to generate constitutive models on demand, existing single-agent pipelines lack systematic mechanisms to ensure the resulting models adhere to fundamental physical laws. Specifically, prior approaches like GenCANN (Tacke et al., 2025a) could generate accurate models but offered no guarantee that the generated networks satisfied critical hyperelastic constraints such as thermodynamic consistency, objectivity, material symmetry, and polyconvexity. This gap limits the trustworthiness and practical usability of LLM-generated models in engineering applications.

Methodology
The authors introduce a multi-agent LLM-driven framework designed to close the gap between model generation and physical validation. The system employs two specialized agents:

1. The Creator Agent: Tasked with proposing a constitutive model tailored to specific experimental data. It generates a Python implementation of a Constitutive Artificial Neural Network (CANN), focusing on fitting the data while attempting to adhere to physical constraints by construction.
2. The Inspector Agent: Tasked with critically auditing the Creator's proposal against nine specific physical constraints required for admissible hyperelastic models:
   • Thermodynamic consistency
   • Stress tensor symmetry
   • Objectivity
   • Material symmetry
   • Polyconvexity (implying ellipticity/stability)
   • Volumetric growth condition
   • Energy normalization
   • Stress normalization
   • Non-negativity of strain energy

The workflow operates as a refinement loop. The Creator generates a model, which undergoes automated syntactic checks. Valid implementations are passed to the Inspector, which evaluates them against the nine constraints. If violations are detected, the Inspector returns the model with specific justifications to the Creator for refinement. This cycle repeats, with the pair performing two refinement rounds after the initial generation. The system selects the most accurate model from the three inspector-approved candidates produced in a single run.

The study utilizes two LLM backbones: the proprietary Claude Opus 4.7 and the open-source Kimi K2.5. The framework is tested on three benchmark datasets: human brain tissue (challenging, sparse data), experimental rubber (Treloar, 1944), and synthetic rubber (allowing for ground-truth stress calculation for extrapolation testing).

Key Contributions

1. First Multi-Agent Architecture for Constitutive Modeling: The paper presents the first system where a Creator and Inspector collaborate to generate physics-constrained constitutive models. This separates the generative task from the verification task, a design principle shown to improve factual correctness and reasoning in other LLM domains.
2. Quantifiable Improvement in Physical Adherence: The study demonstrates that adding the Inspector significantly increases the reliability of the output. For Claude Opus 4.7, the share of exported models satisfying all physical constraints rose from 91% (Creator alone) to 100%. For Kimi K2.5, it increased from 37% to 56%.
3. Technique-Agnostic Paradigm: While demonstrated using CANNs, the authors formulate the Creator-Inspector loop as a general template applicable to Physics-Augmented Neural Networks (PANNs), Constitutive KANs (CKANs), or symbolic laws, scaling automatically with future LLM capabilities.

Results

• Physical Validity: The multi-agent approach achieved perfect physical adherence (100%) for the Opus backbone and a substantial improvement for Kimi. The Inspector proved highly effective at detecting violations, particularly regarding material symmetry and ellipticity.
• Accuracy and Generalization: The generated models maintained near-baseline accuracy compared to human-designed CANNs. On rubber datasets, they achieved near-perfect predictions on training data. Crucially, they demonstrated remarkable generalization and extrapolation capabilities on unseen loading paths (e.g., Treloar's invariant plane), performing comparably to or better than baselines in extrapolation scenarios.
• Collaboration Dynamics: The Creator-Inspector pair showed effective collaboration. For Opus, 93% of new violations flagged by the Inspector were resolved in the next iteration. Kimi showed a lower resolution rate (69%) and a tendency to get stuck in cycles of repeated violations, though it still improved overall reliability.
• Tool Usage: When the Inspector was granted access to numerical validation tools, Opus used them sparingly and selectively, while Kimi exhibited a bimodal behavior (calling all tools or none). Tool access improved Kimi's adherence to 73% but slightly reduced Opus's performance due to over-confidence in specific constraints.

Significance and Claims
The paper claims that separating generation from inspection transforms LLM-driven constitutive modeling into a "genuinely trustworthy process." The primary significance lies in producing models that are simultaneously physically valid, highly accurate, and capable of reliable extrapolation—a combination previously difficult to achieve with automated methods.

The authors position this work as a step toward automated, physics-aware model discovery. They emphasize that the paradigm is deliberately technique-agnostic and designed to scale with advances in LLM capability. By achieving 100% physical adherence for a frontier model on hyperelastic, isotropic, incompressible materials, the study establishes a milestone for the reliable use of LLMs in constitutive modeling, moving beyond "black-box" generation to "physics-constrained" design. The authors note that while the current setup involves predefined constraints and hard-coded tools, the framework is poised to become more autonomous as LLMs improve in reasoning and code generation.