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One-Shot Transfer Learning for Nonlinear PDEs with Perturbative PINNs

Samuel Auroy, Pavlos Protopapas

TL;DR

This work addresses efficient solution of nonlinear PDEs that are perturbed by polynomial terms. It combines perturbation theory with one-shot transfer learning in a Multi-Head PINN to convert nonlinear PDEs into a sequence of linear subproblems while learning a reusable latent representation, enabling closed-form adaptation for new forcing, initial, and boundary data. On canonical problems such as the KPP–Fisher equation and a nonlinear wave equation, the approach achieves relative errors around $10^{-3}$ with fast adaptation (less than $0.2$ s) and performance competitive with traditional solvers, while offering faster transfer. The framework delineates a practical regime of applicability, exposes the sensitivity to perturbation size $\epsilon$ and polynomial degree, and outlines extensions to derivative-dependent nonlinearities and higher-dimensional PDEs, positioning it between single-task PINNs and full operator-learning methods.

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.

One-Shot Transfer Learning for Nonlinear PDEs with Perturbative PINNs

TL;DR

This work addresses efficient solution of nonlinear PDEs that are perturbed by polynomial terms. It combines perturbation theory with one-shot transfer learning in a Multi-Head PINN to convert nonlinear PDEs into a sequence of linear subproblems while learning a reusable latent representation, enabling closed-form adaptation for new forcing, initial, and boundary data. On canonical problems such as the KPP–Fisher equation and a nonlinear wave equation, the approach achieves relative errors around with fast adaptation (less than s) and performance competitive with traditional solvers, while offering faster transfer. The framework delineates a practical regime of applicability, exposes the sensitivity to perturbation size and polynomial degree, and outlines extensions to derivative-dependent nonlinearities and higher-dimensional PDEs, positioning it between single-task PINNs and full operator-learning methods.

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.

Paper Structure

This paper contains 15 sections, 18 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Methodology
  4. Expansion of the solution
  5. Multi-Head Physics-Informed Neural Network (MH-PINN)
  6. One-shot Transfer Learning
  7. Results
  8. Limitations and Extensions
  9. Conclusion
  10. Appendix
  11. Equations used
  12. Graphs of various solutions
  13. Graphs of error
  14. Training Details
  15. Results on the wave equation

Figures (11)

  • Figure 1: Solution to the nonlinear equation computed by PINNs.
  • Figure 2: Relative error versus perturbation order $p$.
  • Figure 3: PINNs solution for $n_1=10$, $n_2=1$.
  • Figure 4: PINNs solution for $n_1=1$, $n_2=10$.
  • Figure 5: PINNs solution for $n_1=2$, $n_2=1$ with various conditions.
  • ...and 6 more figures