假设、评估、优化：用于空间系数场未知的PDE发现的科学智能体
Hypothesize, Evaluate, Refine: A Scientific Agent for PDE Discovery with Unknown Spatial Coefficient Fields
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中文总结 AI 辅助
HER-PDE科学智能体框架分析含噪轨迹提出PDE结构假设，经HEI评估优化后，在5%噪声的二维系统中成功恢复PDE算子，9个未知系数场的中位数皮尔逊相关达0.85，可无参数形式约束发现非均匀PDE。
中文摘要 AI 辅助
在非均匀介质中发现偏微分方程（PDE）需要联合识别控制算子及其参数化的未知空间场，这些任务相互耦合：场的位置变化会改变微分规律，而足够灵活的场可在单条轨迹上掩盖结构误差。本文提出用于PDE发现的假设、评估、优化框架HER-PDE，该科学智能体框架可同时发现组合PDE结构与非参数、时不变系数场。智能体分析由不同激励生成的两条含噪轨迹，提出完整的表达式树假设，将创造性结构探索与局部候选优化相结合；其假设评估接口（HEI）仅估计每个假设中明确声明的场，不添加缺失项，并通过双向跨激励迁移对结构打分，所选规律随后在密封时间区间上接受审计。在5%相对高斯状态噪声观测的5个受控二维系统中，智能体在所有5个案例中均恢复了生成算子，包括等效符号场与乘积规则参数化；在9个未知系数场上，恢复场的中位数皮尔逊相关系数约为0.85，中位数相对L2误差约为0.28。这些结果表明，智能体引导的假设优化可在不为空间系数指定参数形式的情况下恢复非均匀控制规律。
英文摘要
Discovering PDEs in heterogeneous media requires jointly identifying the governing operator and the unknown spatial fields that parameterize it. These tasks are coupled: changing field placement changes the differential law, while a sufficiently flexible field can conceal structural error on a single trajectory. We present Hypothesize, Evaluate, Refine for PDE Discovery (HER-PDE), a scientific-agent framework that discovers compositional PDE structure together with nonparametric, time-invariant coefficient fields. The Agent analyzes two noisy trajectories generated by different excitations, proposes complete expression-tree hypotheses, and combines creative structural exploration with local candidate refinement. Its Hypothesis Evaluation Interface (HEI) estimates only the fields explicitly declared in each hypothesis, never adds missing terms, and scores structures by bidirectional cross-excitation transfer. The selected law is subsequently audited on a sealed temporal interval. Across five controlled two-dimensional systems observed with 5 percent relative Gaussian state noise, the Agent recovers the generating operator in all five cases, including equivalent signed-field and product-rule parameterizations. Across nine unknown coefficient fields, the recovered fields attain a median Pearson correlation of approximately 0.85 and a median relative L2 error of approximately 0.28. These results show that agent-guided hypothesis refinement can recover heterogeneous governing laws without prescribing a parametric form for their spatial coefficients.
发表机构
- Fujian University of Technology（福建工程学院）
- Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences（中国科学院深圳先进技术研究院）
机构由 AI 辅助整理，请以论文原文为准。
