Abstract:Solving partial differential equations (PDEs) with machine learning typically requires training a new neural network for every new equation. This optimization is slow. We introduce MetaColloc. It is an optimization-free and data-free framework that removes this bottleneck completely. We decouple basis discovery from the solving process. We meta-train a dual-branch neural network on diverse Gaussian Random Fields. This offline process creates a universal dictionary of neural basis functions. At test time, we freeze the network. We solve the PDE by assembling a collocation matrix. We find the solution through a single linear least squares step. For non-linear PDEs, we apply the Newton-Raphson method to achieve fast quadratic convergence. Our experiments across six 2D and 3D PDEs show massive improvements. MetaColloc reaches state-of-the-art accuracy on smooth and non-linear problems. It also reduces test-time computation by several orders of magnitude. Finally, we provide a detailed frequency sweep analysis. This analysis reveals a critical mismatch between function approximation and operator stability at extremely high frequencies. This profound finding opens a clear path toward future operator-aware meta-learning.
| Comments: | 21 pages, 5 figures, 6 tables |
| Subjects: | Machine Learning (cs.LG) |
| Cite as: | arXiv:2605.12368 [cs.LG] |
| (or arXiv:2605.12368v1 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2605.12368 arXiv-issued DOI via DataCite (pending registration) |
From: Zichuan Yang [view email]
[v1]
Tue, 12 May 2026 16:36:12 UTC (15,708 KB)
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