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

TL;DR

PTL-PINN introduces a perturbation-guided transfer learning framework that unifies classical perturbation theory with Physics-Informed Neural Networks to solve weakly nonlinear ODEs and PDEs. By learning a shared latent representation with a Multi-Headed-PINN and performing closed-form updates for each perturbation order, it reduces the problem to a sequence of linear subproblems solvable with matrix-vector operations, achieving accuracy comparable to RK methods while being faster than gradient-based transfer learning. The approach demonstrates strong performance across damped/undamped oscillators, Lotka–Volterra dynamics, and PDEs like KPP–Fisher and the Wave equation, and it provides a clear analysis of perturbation limitations and mitigation via Lindstedt--Poincaré methods. The framework offers practical speedups and broad applicability to nonlinear dynamical systems, with public code and potential extensions to higher-dimensional problems and alternative perturbation strategies.

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.

Figures (14)

• Figure 1: Multi-Headed-PINN architecture used for one-shot transfer learning. A shared neural backbone learns a latent representation $\textbf{H}(x,t)$ across $k$ training equations, while linear output heads represented by matrices $\textbf{W}$ (head weights) map this representation to solution components for each equation. At inference time, new linear problems are solved by computing closed-form output weights without gradient-based optimization.
• Figure 2: Comparison of initial-condition handling strategies for the unforced overdamped oscillator ($\zeta = 10$). PTL-PINN solutions are shown using 15 perturbation corrections for a cubic nonlinearity $0.5x^3$. The leading-order approach enforces the full initial condition at zeroth-order only, while the uniform approach distributes the initial condition evenly across all perturbation orders. Results are shown for increasing initial positions $x(0) \in \{0.5, 1, 1.5, 2, 3\}$ with zero initial velocity.
• Figure 3: Comparison of the solution of various nonlinear overdamped oscillators using the baseline RK45 and PTL-PINN undamped, underdamped and overdamped models. Each row corresponds to a fixed PTL-PINN model, while each column represents a different damping value from $\{5, 10, 30, 60\}$. The PTL-PINN overdamped model has greatest accuracy.
• Figure 4: Lindstedt--Poincaré frequency convergence for the undamped and unforced oscillator with $x(0) = 1$ and $x'(0) = 0$ using an undamped (twelve-headed), underdamped and vanilla (one-headed) model, showing absolute relative frequency contribution (right) and frequency series MAE (left)
• Figure 5: Comparison of the first, fifth and eighth-order corrections when using RK45 and a twelve-headed and one-headed PTL-PINN model.