analiza aiqbench
The paper proposes a kernel-based method for learning differential equations and their solution operators from limited data, with Gauss-Newton optimization and worst-case error bounds. For AI and ML readers, the notable points are the data-efficiency claims and the theoretical guarantees, which matter for scientific machine learning where training data is expensive. The reported accuracy gains of one to two orders of magnitude over some state-of-the-art baselines are benchmark-specific and would need independent replication to judge.
Data-Efficient Kernel Methods for Learning Differential Equations and Their Solution Operators: Algorithms and Error Analysis
arXiv:2503.01036v3 Announce Type: replace-cross 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.
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.
Machine Learning (stat.ML); Machine Learning (cs.LG); Numerical Analysis (math.NA)
arXiv:2503.01036 [stat.ML]
(or arXiv:2503.01036v3 [stat.ML] for this version)
From: Bamdad Hosseini Dr. [view email] [v1] Sun, 2 Mar 2025 21:48:03 UTC (22,279 KB) [v2] Fri, 4 Apr 2025 15:13:38 UTC (22,187 KB) [v3] Mon, 2 Nov 2026 23:17:07 UTC (30,691 KB)
Text integral sub licența deschisă a publicației · 173 cuvinte