用少量数据高效学习微分方程及其解映射，理论可解释且误差可控。
Data-Efficient Kernel Methods for Learning Differential Equations and Their Solution Operators: Algorithms and Error Analysis
- 基于核方法构建新框架，降低数据与计算成本。
- 相比现有算法，精度提升1到2个数量级，计算复杂度显著下降。
- 适合需高精度、低数据量的科学计算场景，如物理建模与逆问题求解。
我们提出一种新型基于核的方法，用于学习微分方程及其解映射，具有极低的数据需求（包括解例数和每例测量数）和较低的计算成本。该方法数学可解释，并具备严格的定量最坏情况误差界。数值实验表明，在计算复杂度和鲁棒性方面均有显著提升，相比当前最优算法，精度提高一至两个数量级。
原文摘要 · Abstract (English)
We introduce a novel kernel-based framework for learning differential equations and their solution maps that is efficient in data requirements, in terms of solution examples and amount of measurements from each example, and computational cost, in terms of training procedures. Our approach is mathematically interpretable and backed by rigorous theoretical guarantees in the form of quantitative worst-case error bounds for the learned equation. Numerical benchmarks demonstrate significant improvements in computational complexity and robustness while achieving one to two orders of magnitude improvements in terms of accuracy compared to state-of-the-art algorithms.
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