Loss Jump During Loss Switch in Solving PDEs with Neural Networks

TL;DR

This paper investigates the loss-jump phenomenon in neural network training for PDEs, revealing how switching loss functions causes significant deviations due to frequency preference differences, and provides theoretical insights into this behavior.

Contribution

It uncovers and analyzes the loss-jump phenomenon during loss function switching in PDE-solving neural networks, offering theoretical understanding of frequency preferences.

Findings

Loss jump occurs when switching from data loss to model loss.

Neural networks exhibit different frequency preferences under different loss functions.

Theoretical analysis explains the frequency preference behavior.

Abstract

Using neural networks to solve partial differential equations (PDEs) is gaining popularity as an alternative approach in the scientific computing community. Neural networks can integrate different types of information into the loss function. These include observation data, governing equations, and variational forms, etc. These loss functions can be broadly categorized into two types: observation data loss directly constrains and measures the model output, while other loss functions indirectly model the performance of the network, which can be classified as model loss. However, this alternative approach lacks a thorough understanding of its underlying mechanisms, including theoretical foundations and rigorous characterization of various phenomena. This work focuses on investigating how different loss functions impact the training of neural networks for solving PDEs. We discover a stable loss-jump phenomenon: when switching the loss function from the data loss to the model loss, which includes different orders of derivative information, the neural network solution significantly deviates from the exact solution immediately. Further experiments reveal that this phenomenon arises from the different frequency preferences of neural networks under different loss functions. We theoretically analyze the frequency preference of neural networks under model loss. This loss-jump phenomenon provides a valuable perspective for examining the underlying mechanisms of neural networks in solving PDEs.