One-Shot Transfer Learning for Nonlinear PDEs with Perturbative PINNs

• URL: http://arxiv.org/abs/2511.11137v1
• Date: Fri, 14 Nov 2025 10:12:50 GMT
• Title: One-Shot Transfer Learning for Nonlinear PDEs with Perturbative PINNs
• Authors: Samuel Auroy, Pavlos Protopapas,
• Abstract summary: We propose a framework for solving nonlinear partial differential equations.<n>Our contributions are: (i) extending one-shot transfer learning from nonlinear ODEs to PDEs, (ii) deriving a closed-form solution for adapting to new PDE instances, and (iii) demonstrating accuracy and efficiency on canonical nonlinear PDEs.
• Score: 0.794957965474334
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We propose a framework for solving nonlinear partial differential equations (PDEs) by combining perturbation theory with one-shot transfer learning in Physics-Informed Neural Networks (PINNs). Nonlinear PDEs with polynomial terms are decomposed into a sequence of linear subproblems, which are efficiently solved using a Multi-Head PINN. Once the latent representation of the linear operator is learned, solutions to new PDE instances with varying perturbations, forcing terms, or boundary/initial conditions can be obtained in closed form without retraining. We validate the method on KPP-Fisher and wave equations, achieving errors on the order of 1e-3 while adapting to new problem instances in under 0.2 seconds; comparable accuracy to classical solvers but with faster transfer. Sensitivity analyses show predictable error growth with epsilon and polynomial degree, clarifying the method's effective regime. Our contributions are: (i) extending one-shot transfer learning from nonlinear ODEs to PDEs, (ii) deriving a closed-form solution for adapting to new PDE instances, and (iii) demonstrating accuracy and efficiency on canonical nonlinear PDEs. We conclude by outlining extensions to derivative-dependent nonlinearities and higher-dimensional PDEs.

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