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.

翻译：尽管机器学习在科学问题上的许多应用前景看似广阔，但视觉呈现可能具有误导性。通过运用数值分析技术，我们严格量化了应用于线性微分方程进行参数发现或求解的特定机器学习模型的精度、收敛速率及泛化界。除数据量与离散化程度外，我们发现数据的函数空间对模型泛化能力具有决定性影响。通过实证研究，我们在包括物理专用技术在内的常用模型中同样观察到类似的泛化能力缺失现象。反直觉的是，我们发现不同类别的模型可能表现出截然相反的泛化行为。基于理论分析，我们进一步提出一种新的科学模型机制可解释性视角——通过该视角可从黑盒模型的权重中提取格林函数表示。我们的研究成果为衡量物理系统泛化能力的新型交叉验证技术提供了理论依据，该技术可成为相关领域的基准测试方法。