Data-Efficient Kernel Methods for Learning Differential Equations and Their Solution Operators: Algorithms and Error Analysis

arXiv:2503.01036v3 Announce Type: replace Abstract: We introduce a novel kernel-based framework for learning differential equations and their solution maps. The framework is efficient in its data requirements in terms of the number of observed states and the amount of data available per state. Our formulation also allows us to employ efficient Gauss-Newton-type optimization algorithms in the training procedure to improve accuracy and speed of the training procedure. We also support our algorithm with rigorous theoretical guarantees in the form of quantitative worst-case error bounds for the learned equation that can be extended to the subsequent solution operators under smoothness assumptions. 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 certain state-of-the-art algorithms.