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In this paper we substantially broaden the class of nonlinearities amenable to one-shot PINN transfer by combining OTL with Chebyshev polynomial surrogates. We approximate general smooth weakly nonlinear terms by truncated Chebyshev expansions over a prescribed solution range, yielding a polynomial nonlinearity that can be handled by a perturbative decomposition into linear subproblems. A multi-head PINN learns a reusable latent space associated with the dominant linear operator; at test time, solutions to new instances are obtained via a sequence of closed-form linear solves in the output layer, without retraining the network body.
We provide a unified derivation of the framework for ODEs and PDEs and demonstrate accuracy and fast online adaptation on nonlinear benchmarks, including non-polynomial and singular ODE nonlinearities as well as a reaction-diffusion PDE with saturating kinetics, demonstrating the method's utility in many-query regimes.
| Comments: | 18 pages, 4 figures, 9 tables, accepted to ICLR 2026 Workshop on Artificial Intelligence and Partial Differential Equations |
| Subjects: | Machine Learning (cs.LG) |
| MSC classes: | 68T07 |
| ACM classes: | I.2 |
| Cite as: | arXiv:2605.01634 [cs.LG] |
| (or arXiv:2605.01634v1 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2605.01634 arXiv-issued DOI via DataCite (pending registration) |
From: Yiqi Rao [view email]
[v1]
Sat, 2 May 2026 22:49:37 UTC (4,476 KB)
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