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

2026-01-17

AI Summary

This work proposes a novel framework, termed Perturbation Theory–enhanced Physics-Informed Neural Networks (PTL-PINNs), which integrates perturbation theory with transfer learning to address the poor generalization and high computational cost of conventional PINNs when solving nonlinear differential equations. For the first time, perturbation methods are incorporated into the PINN architecture, enabling the analytical derivation of closed-form expressions from an approximately linearized perturbed system to replace portions of the gradient-based optimization. This approach substantially reduces computational complexity while enhancing generalization efficiency. Experimental results across multiple nonlinear dynamical systems demonstrate that PTL-PINNs achieve accuracy comparable to that of classical Runge–Kutta methods, with nearly an order-of-magnitude improvement in computational speed.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Learning to Search
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• nonlinear differential equations
• Physics-Informed Neural Networks
• generalization
• training efficiency
• nonlinear dynamics

Innovation

• Physics-Informed Neural Networks
• Perturbation Theory
• Transfer Learning
• Nonlinear Differential Equations
• Computational Efficiency