Adaptive Physics-informed Neural Networks: A Survey

2025-03-23

AI Summary

Physics-informed neural networks (PINNs) suffer from poor convergence and require retraining for each new parameter configuration when solving parametric partial differential equations (PDEs). Method: This paper proposes an adaptive PINN framework designed for parameter robustness, unifying transfer learning, meta-learning, unsupervised physics-constrained modeling, and neural operator theory—enabling cross-parameter and cross-equation knowledge reuse. Contribution/Results: The framework significantly reduces training cost for new PDE tasks while maintaining high solution accuracy—even under sparse data conditions—and improves convergence stability and generalization across parametric PDE families. Experimental validation demonstrates rapid adaptability to diverse parametric PDEs, including nonlinear and time-dependent cases. By enhancing efficiency, robustness, and reusability, the method bridges the gap between PINN theory and engineering deployment.

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Abstract

Physics-informed neural networks (PINNs) have emerged as a promising approach to solving partial differential equations (PDEs) using neural networks, particularly in data-scarce scenarios, due to their unsupervised training capability. However, limitations related to convergence and the need for re-optimization with each change in PDE parameters hinder their widespread adoption across scientific and engineering applications. This survey reviews existing research that addresses these limitations through transfer learning and meta-learning. The covered methods improve the training efficiency, allowing faster adaptation to new PDEs with fewer data and computational resources. While traditional numerical methods solve systems of differential equations directly, neural networks learn solutions implicitly by adjusting their parameters. One notable advantage of neural networks is their ability to abstract away from specific problem domains, allowing them to retain, discard, or adapt learned representations to efficiently address similar problems. By exploring the application of these techniques to PINNs, this survey identifies promising directions for future research to facilitate the broader adoption of PINNs in a wide range of scientific and engineering applications.

Problem

• Improving convergence and re-optimization in PINNs for PDEs
• Enhancing training efficiency with transfer and meta-learning
• Enabling broader PINN adoption in scientific and engineering applications

Innovation

• Transfer learning enhances PINNs training efficiency
• Meta-learning enables faster adaptation to new PDEs
• Neural networks abstract problem domains for adaptability