PTL-PINNs: Perturbation-Guided Transfer Learning with Physics- Informed Neural Networks for Nonlinear Systems

TL;DR

This paper introduces PTL-PINNs, a transfer learning framework guided by perturbation theory that significantly accelerates and improves the accuracy of physics-informed neural networks in solving nonlinear differential equations.

Contribution

It presents a novel perturbation-guided transfer learning approach for PINNs that uses closed-form solutions for linear perturbations, enabling faster and more accurate solutions for nonlinear systems.

Findings

Achieves accuracy comparable to Runge-Kutta methods.

Speeds up computations by up to ten times.

Successfully solves diverse nonlinear problems.

Abstract

Accurately and efficiently solving nonlinear differential equations is crucial for modeling dynamic behavior across science and engineering. Physics-Informed Neural Networks (PINNs) have emerged as a powerful solution that embeds physical laws in training by enforcing equation residuals. However, these struggle to model nonlinear dynamics, suffering from limited generalization across problems and long training times. To address these limitations, we propose a perturbation-guided transfer learning framework for PINNs (PTL-PINN), which integrates perturbation theory with transfer learning to efficiently solve nonlinear equations. Unlike gradient-based transfer learning, PTL-PINNs solve an approximate linear perturbative system using closed-form expressions, enabling rapid generalization with the time complexity of matrix-vector multiplication. We show that PTL-PINNs achieve accuracy comparable to various Runge-Kutta methods, with computational speeds up to one order of magnitude faster. To benchmark performance, we solve a broad set of problems, including nonlinear oscillators across various damping regimes, the equilibrium-centered Lotka-Volterra system, the KPP-Fisher and the Wave equation. Since perturbation theory sets the accuracy bound of PTL-PINNs, we systematically evaluate its practical applicability. This work connects long-standing perturbation methods with PINNs, demonstrating how perturbation theory can guide foundational models to solve nonlinear systems with speeds comparable to those of classical solvers.