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

TL;DR

The paper addresses efficient learning of parametric PDEs by meta-learning a task-specific loss via Generalized Additive Models within Physics-Informed Neural Networks. By replacing or augmenting the data loss with a GAM-derived residual term, the method enhances fast adaptation across related PDE tasks, demonstrated on the 1D Burgers and 2D Heat equations. Empirical results show GAM_PINN achieves faster convergence and lower mean-squared errors than random initialization and standard MAML-based PINNs, and can even de-noise PDEs under noisy observations. The work points to significant practical impact in rapidly solving parameterized PDEs and suggests future directions in symbolic PDE discovery and broader experimental data applications.

Abstract

This paper proposes a new way to learn Physics-Informed Neural Network loss functions using Generalized Additive Models. We apply our method by meta-learning parametric partial differential equations, PDEs, on Burger's and 2D Heat Equations. The goal is to learn a new loss function for each parametric PDE using meta-learning. The derived loss function replaces the traditional data loss, allowing us to learn each parametric PDE more efficiently, improving the meta-learner's performance and convergence.

Figures (6)

• Figure 1: Burgers’ equation: The convergence of mean loss overall parametric values for the number of training iterations. Confidence intervals (95%) assuming a normal distribution indicated by error bars illustrate the uncertainty in the loss measurements.
• Figure 2: Burgers’ equation: Solution of $\hat{u}_{xt}$ with periodic initial conditions ($\theta$ = 0). The solution approximates the ground truth of the equation with an error less than $5e^{-4}$.
• Figure 3: 2D Heat equation: The convergence of mean loss overall parametric values concerning the number of training iterations using initial conditions defined in Eq. 13.
• Figure 4: 2D Heat equation: The convergence of mean loss overall parametric values concerning the number of training iterations using initial conditions defined in Eq. 14. Confidence intervals (95%) assuming a normal distribution that are indicated by error bars illustrate the uncertainty in the loss measurements.
• Figure 5: Burgers' equation: comparison of the noisy solution u(x,t) (top) with de-noised solution (bottom) for initial conditions (Eq. 15) with noise $p = 5\%$.