Data-Efficient Kernel Methods for Learning Differential Equations and Their Solution Operators: Algorithms and Error Analysis
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | |
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| _version_ | 1866915227321237504 |
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| author | Jalalian, Yasamin Ramirez, Juan Felipe Osorio Hsu, Alexander Hosseini, Bamdad Owhadi, Houman |
| author_facet | Jalalian, Yasamin Ramirez, Juan Felipe Osorio Hsu, Alexander Hosseini, Bamdad Owhadi, Houman |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_01036 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Data-Efficient Kernel Methods for Learning Differential Equations and Their Solution Operators: Algorithms and Error Analysis Jalalian, Yasamin Ramirez, Juan Felipe Osorio Hsu, Alexander Hosseini, Bamdad Owhadi, Houman Machine Learning Numerical Analysis 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. |
| title | Data-Efficient Kernel Methods for Learning Differential Equations and Their Solution Operators: Algorithms and Error Analysis |
| topic | Machine Learning Numerical Analysis |
| url | https://arxiv.org/abs/2503.01036 |