Data-Efficient Kernel Methods for Learning Differential Equations and Their Solution Operators: Algorithms and Error Analysis
This addresses data efficiency and computational cost issues for researchers and practitioners in scientific computing and machine learning, though it appears incremental as an enhancement to kernel methods.
The paper tackles the problem of learning differential equations and their solution operators with a kernel-based framework that is efficient in data and computation, achieving one to two orders of magnitude accuracy improvements over state-of-the-art methods.
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.