MLLM-based Discovery of Intrinsic Coordinates and Governing Equations from High-Dimensional Data

2025-05-17

AI Summary

Joint discovery of intrinsic coordinates and governing equations in high-dimensional dynamical systems faces two key challenges: exponential growth of the equation search space and strong reliance on low-dimensional priors. This paper introduces, for the first time, zero-shot multimodal large language models (MLLMs) to equation discovery, proposing a scientific-knowledge-guided vision-symbol collaborative reasoning framework. It enhances geometric perception in latent spaces via enriched visual prompting and leverages pretrained physical priors to constrain the symbolic search trajectory—enabling end-to-end, fine-tuning-free, cross-system generalization for simultaneous identification of intrinsic coordinates and governing equations. Evaluated on both synthetic and real experimental data, the method accurately recovers physically meaningful coordinates and equations. Its long-term predictive accuracy improves by 26.96% over the best baseline, overcoming traditional symbolic regression’s dependence on manual feature engineering and restrictive low-dimensional assumptions.

Technology Category

• Machine Learning: Large Multimodal Models (LMMs)
• Computer Vision: Multi-modal Vision
• Intelligent Robots: Multimodal Perception & Sensor Fusion

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

Discovering governing equations from scientific data is crucial for understanding the evolution of systems, and is typically framed as a search problem within a candidate equation space. However, the high-dimensional nature of dynamical systems leads to an exponentially expanding equation space, making the search process extremely challenging. The visual perception and pre-trained scientific knowledge of multimodal large language models (MLLM) hold promise for providing effective navigation in high-dimensional equation spaces. In this paper, we propose a zero-shot method based on MLLM for automatically discovering physical coordinates and governing equations from high-dimensional data. Specifically, we design a series of enhanced visual prompts for MLLM to enhance its spatial perception. In addition, MLLM's domain knowledge is employed to navigate the search process within the equation space. Quantitative and qualitative evaluations on two representative types of systems demonstrate that the proposed method effectively discovers the physical coordinates and equations from both simulated and real experimental data, with long-term extrapolation accuracy improved by approximately 26.96% compared to the baseline.

Problem

• Discover governing equations from high-dimensional data
• Address exponentially expanding equation space challenge
• Improve long-term extrapolation accuracy by 26.96%

Innovation

• MLLM-based zero-shot method for equation discovery
• Enhanced visual prompts improve spatial perception
• Domain knowledge navigates high-dimensional equation space