See, Hypothesize, Validate: Multimodal Agentic Framework for Discovering Governing PDEs

Discovering governing partial differential equations (PDEs) from observational data remains a core challenge across the sciences. Existing sparse-regression, symbolic-regression, and LLM-based approaches can be constrained by predefined libraries, noise sensitivity, hallucination, or limited iterative refinement. We introduce MAGE (Multimodal Agentic Governing Equation Discovery), an agentic framework that organizes PDE discovery as a textitconfidence governed hypothesis validation loop inspired by the scientific cycle of observation, hypothesis, and falsification. Four role-specialized agents collaborate: a textitDifferential Observer computing derivatives and diagnostic visualizations; a VLM-powered textitPhenomenology Extractor distilling qualitative cues from multimodal diagnostics; an LLM-driven textitGoverning Law Synthesizer proposing candidates without a predefined library; and an textitEquation Arbiter fitting coefficients and assigning confidence scores. Discovery iterates until the top candidate clears a user-specified threshold, providing a structured process with an explicit accept-reject protocol. On the evaluated canonical PDE suite, MAGE obtains 8/8 exact structural recovery and the lowest coefficient error among the compared methods on 7/8 systems, with improvements of up to 4 orders of magnitude and a geometric-mean improvement of approximately 3 orders of magnitude. The pipeline also recovers the expected operators in two complex geometries and, on one laboratory sensor record, selects a cubic restoring-force model with held-out R2=0.98538. These results support further study of structured agentic reasoning for library-free governing-law discovery, while broader generalization remains to be evaluated.