Meta-learning Loss Functions of Parametric Partial Differential Equations Using Physics-Informed Neural Networks

The chapter delves into the application of meta-learning to optimize loss functions for parametric partial differential equations (PDEs) using Physics-Informed Neural Networks (PINNs). It highlights the limitations of traditional methods like Finite Element Methods (FEMs) and introduces PINNs as a more efficient alternative. The focus is on meta-learning strategies that enable models to quickly adapt to new tasks by learning from a distribution of related tasks. A key innovation is the use of Generalized Additive Models (GAMs) to model the residuals of each meta-learning task, providing a flexible and expressive approach to learning the loss function. The proposed method demonstrates improved convergence and robustness to noisy data, as shown through experiments on the viscous Burgers equation and the 2D heat equation. The chapter concludes by discussing the potential of this approach for discovering analytical PDEs from experimental data and outlines future research directions.