One-Shot Transfer Learning for Nonlinear PDEs with Perturbative PINNs

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 $10^{-3}$ 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.