Chebyshev-Augmented One-Shot Transfer Learning for PINNs on Nonlinear Differential Equations

arXiv cs.LG / 5/5/2026

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Key Points

Physics-Informed Neural Networks (PINNs) often require retraining for each new set of forcing terms, boundary/initial conditions, or parameters, limiting reuse.
The paper extends one-shot transfer learning (OTL) to nonlinear differential equations by using Chebyshev polynomial surrogates to approximate smooth weakly nonlinear terms within a chosen solution range.
It turns nonlinearity into a polynomial form that can be handled via a perturbative decomposition into linear subproblems, enabling closed-form output-layer adaptation.
A multi-head PINN is trained to learn a reusable latent representation tied to the dominant linear operator, while new problem instances are solved through a sequence of closed-form linear solves without retraining.
Experiments across ODE and PDE benchmarks—including non-polynomial and singular nonlinearities and a reaction-diffusion PDE—show accurate and fast online adaptation in many-query settings.

Abstract

Physics-Informed Neural Networks (PINNs) offer a flexible paradigm for solving differential equations by embedding governing laws into the training objective. A persistent limitation is instance specificity: standard PINNs typically require retraining for each new forcing term, boundary/initial condition, or parameter setting. One-shot transfer learning (OTL) addresses this bottleneck for linear operators by freezing a pretrained latent representation and computing optimal output weights in closed form, but for nonlinear problems closed-form adaptation is generally unavailable because the loss is nonconvex in the output layer. 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.