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

• URL: http://arxiv.org/abs/2605.01634v1
• Date: Sat, 02 May 2026 22:49:37 GMT
• Title: Chebyshev-Augmented One-Shot Transfer Learning for PINNs on Nonlinear Differential Equations
• Abstract summary: Physics-Informed Neural Networks (PINNs) offer a flexible paradigm for solving differential equations by governing governing laws into the network.<n>A limitation is instance specificity: standard PINNs typically require retraining for each new forcing term, latent/initial condition, or nonlinear problems closed-form adaptation.<n>In this paper we address the class of nonlinearities amenable to one-shot PINN transfer by combining OTL with Chebyshev surrogates.
• Score: 0.794957965474334
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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