MultiAdam: Parameter-wise Scale-invariant Optimizer for Multiscale Training of Physics-informed Neural Networks

TL;DR

This paper introduces MultiAdam, a scale-invariant optimizer designed for multiscale training of Physics-informed Neural Networks, addressing training imbalance issues and improving accuracy significantly.

Contribution

The paper provides a theoretical analysis of second-order PDEs and proposes MultiAdam, a novel optimizer that balances loss terms and enhances PINN training performance.

Findings

MultiAdam improves predictive accuracy by 1-2 orders of magnitude.

Theoretical guarantees of convergence for the proposed method.

Effective across multiple physical domain problems.

Abstract

Physics-informed Neural Networks (PINNs) have recently achieved remarkable progress in solving Partial Differential Equations (PDEs) in various fields by minimizing a weighted sum of PDE loss and boundary loss. However, there are several critical challenges in the training of PINNs, including the lack of theoretical frameworks and the imbalance between PDE loss and boundary loss. In this paper, we present an analysis of second-order non-homogeneous PDEs, which are classified into three categories and applicable to various common problems. We also characterize the connections between the training loss and actual error, guaranteeing convergence under mild conditions. The theoretical analysis inspires us to further propose MultiAdam, a scale-invariant optimizer that leverages gradient momentum to parameter-wisely balance the loss terms. Extensive experiment results on multiple problems from different physical domains demonstrate that our MultiAdam solver can improve the predictive accuracy by 1-2 orders of magnitude compared with strong baselines.