用智能体循环验证法，从数据中自动发现物理方程，无需预设公式库。
See, Hypothesize, Validate: Multimodal Agentic Framework for Discovering Governing PDEs
- 四类智能体协作：观测导数、提取现象、生成方程、评估置信度。
- 8个经典方程全正确复现，系数误差降低达4个数量级。
- 适合做物理规律挖掘的科研人员，尤其关注无预设公式的发现方法。
从观测数据中发现控制性偏微分方程（PDE）仍是科学领域核心挑战。现有稀疏回归、符号回归及基于大语言模型的方法受限于预定义公式库、对噪声敏感、易产生幻觉或缺乏迭代优化能力。我们提出 extbf{MAGE}（ extbf{M}ultimodal extbf{A}gentic extbf{G}overning extbf{E}quation Discovery），一个受科学探索周期启发的智能体框架，将 PDE 发现过程组织为以置信度驱动的假设验证循环。四个角色专用智能体协同工作： extit{微分观测者}计算导数与诊断可视化；基于视觉语言模型的 extit{现象提取器}从多模态诊断中提炼定性线索；基于大语言模型的 extit{控制律合成器}在无预设库情况下生成候选方程； extit{方程仲裁者}拟合系数并赋予置信度评分。发现过程持续迭代，直至最优候选通过用户设定阈值，提供结构化流程与显式接受/拒绝机制。在评估的标准 PDE 套件上，MAGE 实现 8/8 的精确结构恢复，在 7/8 系统上系数误差最低，最高提升达 4 个数量级，几何均值提升约 3 个数量级。该流程还成功恢复了两个复杂几何中的预期算子，并在一项实验传感器数据中选出立方恢复力模型，保留数据 $R^2=0.98538$。这些结果支持进一步研究无公式库下的结构化智能体推理，但更广泛泛化能力仍待评估。
原文摘要 · Abstract (English)
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.
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