What Should a Large Language Model See? Physical Invariants as a Data Representation for PDE Discovery

We present an autonomous large language model (LLM) agent for end-to-end, data-driven materials theory development. The model can choose an equation form, generate and run its own code, and test how well the theory matches the data without human intervention. The framework combines step-by-step reasoning with expert-supplied tools, allowing the agent to adjust its approach as needed while keeping a clear record of its decisions. For well-established materials relationships such as the Hall-Petch equation and Paris law, the agent correctly identifies the governing equation and makes reliable predictions on new datasets. For more specialized relationships, such as Kuhn's equation for the HOMO-LUMO gap of conjugated molecules as a function of length, performance depends more strongly on the underlying model, with GPT-5 showing better recovery of the correct equation. Beyond known theories, the agent can also suggest new predictive relationships, illustrated here by a strain-dependent law for changes in the HOMO-LUMO gap. At the same time, the results show that careful validation remains essential, because the agent can still return incorrect, incomplete, or inconsistent equations even when the numerical fit appears strong. Overall, these results highlight both the promise and the current limitations of autonomous LLM agents for AI-assisted scientific modeling and discovery.

Discovering interpretable physical laws from high-dimensional data is a fundamental challenge in scientific research. Traditional methods, such as symbolic regression, often produce complex, unphysical formulas when searching a vast space of possible forms. We introduce a framework that guides the search process by leveraging the embedded scientific knowledge of large language models, enabling efficient identification of physical laws in the data. We validate our approach by modeling key properties of perovskite materials. Our method mitigates the combinatorial explosion commonly encountered in traditional symbolic regression, reducing the effective search space by a factor of approximately $10^5$. A set of novel formulas for bulk modulus, band gap, and oxygen evolution reaction activity are identified, which not only provide meaningful physical insights but also outperform previous formulas in accuracy and simplicity.

Multimodal large language models (MLLMs) are increasingly envisioned as core components of scientific and engineering agents, yet their ability to interpret physical fields remains poorly understood. Existing physics benchmarks largely emphasize textbook problem solving or intuitive physical reasoning, leaving open whether MLLMs can infer physically meaningful information from continuous field observations. We introduce PhysFieldBench, a benchmark comprising 24 tasks and 1,160 evaluation examples across controlled equation fields, simulated physical fields, and observed physical fields. The tasks assess three forms of inference: identifying physical mechanisms, comparing latent control variables, and predicting outcome properties. Across representative open-source and proprietary MLLMs, zero-shot performance is low: the best model achieves a chance-normalized score of 29.3, while several open-source models remain near chance. In contrast, a task-specific supervised vision transformer performs substantially better, demonstrating that the inputs contain learnable physical information. To diagnose these failures, a structured self-explanation analysis attributes most errors to missed visual patterns and incorrect visual-to-physical mappings. Further, to explore whether post-training can improve physical inference and generalize to unseen tasks, we compare supervised fine-tuning with final answers or chain-of-thought supervision and reinforcement learning. Final-answer supervision performs best overall but transfers less effectively, whereas reinforcement learning after chain-of-thought supervision achieves the best generalization. Together, these findings highlight the need to improve visual-to-physical grounding and cross-task generalization for MLLMs to reliably interpret physical fields in scientific and engineering workflows.