用数学方法分析机器学习在物理方程中的泛化能力，发现数据函数空间是关键。
Interpretability and Generalization Bounds for Learning Spatial Physics
- 通过数值分析量化模型在微分方程求解中的准确率与收敛速度
- 发现不同模型泛化行为相反，且物理专用模型也存在泛化不足
- 提出从黑箱模型权重中提取格林函数的可解释新视角
尽管机器学习在科学问题中应用前景广阔，但视觉效果可能具有误导性。我们采用数值分析技术，严格量化了某些机器学习模型在求解线性微分方程进行参数发现或解求取时的精度、收敛速率和泛化界。除了数据量和离散化程度外，我们识别出数据的函数空间对模型泛化至关重要。常见模型（包括物理专用方法）在实践中表现出类似的泛化不足。反直觉的是，不同类模型可能呈现相反的泛化行为。基于理论分析，我们引入一种新的机制可解释性视角，可从黑箱模型权重中提取格林函数表示。我们的结果还启发了一种用于测量物理系统泛化的新型交叉验证技术，可作为基准工具。
原文摘要 · Abstract (English)
While there are many applications of ML to scientific problems that look promising, visuals can be deceiving. Using numerical analysis techniques, we rigorously quantify the accuracy, convergence rates, and generalization bounds of certain ML models applied to linear differential equations for parameter discovery or solution finding. Beyond the quantity and discretization of data, we identify that the function space of the data is critical to the generalization of the model. A similar lack of generalization is empirically demonstrated for commonly used models, including physics-specific techniques. Counterintuitively, we find that different classes of models can exhibit opposing generalization behaviors. Based on our theoretical analysis, we also introduce a new mechanistic interpretability lens on scientific models whereby Green's function representations can be extracted from the weights of black-box models. Our results inform a new cross-validation technique for measuring generalization in physical systems, which can serve as a benchmark.
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