One-Shot Transfer Learning for Nonlinear PDEs with Perturbative PINNs
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arXiv
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| Format: | Preprint |
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2025
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| author | Auroy, Samuel Protopapas, Pavlos |
| author_facet | Auroy, Samuel Protopapas, Pavlos |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_11137 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | One-Shot Transfer Learning for Nonlinear PDEs with Perturbative PINNs Auroy, Samuel Protopapas, Pavlos Numerical Analysis Machine Learning 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. |
| title | One-Shot Transfer Learning for Nonlinear PDEs with Perturbative PINNs |
| topic | Numerical Analysis Machine Learning |
| url | https://arxiv.org/abs/2511.11137 |