See, Hypothesize, Validate: Multimodal Agentic Framework for Discovering Governing PDEs
Artificial Intelligence
Summary
The authors present MAGE, a new method that finds the mathematical rules (PDEs) behind observed data by mimicking the scientific process of forming and testing ideas. MAGE uses four specialized agents that work together: one observes data derivatives, another extracts patterns using visual understanding, a third suggests equations without relying on preset options, and the last checks and scores these equations. Their approach successfully recovers correct equations exactly on standard tests and performs much better than existing methods on finding precise coefficients. While promising, the method still needs to be tested further on wider types of problems.
Authors
Sarang Manoj Pekhale, Amartya Roy, Rajat Sarkar, Souvik Chakraborty
Abstract
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 \textbf{MAGE} (\textbf{M}ultimodal \textbf{A}gentic \textbf{G}overning \textbf{E}quation Discovery), an agentic framework that organizes PDE discovery as a \textit{confidence governed hypothesis validation loop} inspired by the scientific cycle of observation, hypothesis, and falsification. Four role-specialized agents collaborate: a \textit{Differential Observer} computing derivatives and diagnostic visualizations; a VLM-powered \textit{Phenomenology Extractor} distilling qualitative cues from multimodal diagnostics; an LLM-driven \textit{Governing Law Synthesizer} proposing candidates without a predefined library; and an \textit{Equation 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 \textbf{8/8} exact structural recovery and the lowest coefficient error among the compared methods on \textbf{7/8} systems, with improvements of up to \textbf{4 orders of magnitude} and a geometric-mean improvement of approximately \textbf{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 $R^2=0.98538$. These results support further study of structured agentic reasoning for library-free governing-law discovery, while broader generalization remains to be evaluated.