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

• URL: http://arxiv.org/abs/2503.01036v2
• Date: Fri, 04 Apr 2025 15:13:38 GMT
• Title: Data-Efficient Kernel Methods for Learning Differential Equations and Their Solution Operators: Algorithms and Error Analysis
• Authors: Yasamin Jalalian, Juan Felipe Osorio Ramirez, Alexander Hsu, Bamdad Hosseini, Houman Owhadi,
• Abstract summary: We introduce a novel kernel-based framework for learning differential equations and their solution maps that is efficient in data requirements.<n>Our approach is mathematically interpretable and backed by rigorous theoretical guarantees in the form of quantitative worst-case error bounds for the learned equation.
• Score: 40.72119156403829
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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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