Long-time predictive modeling of nonlinear dynamical systems using neural networks

TL;DR

This paper explores long-term predictive modeling of nonlinear dynamical systems using neural networks, introducing Jacobian regularization to enhance stability and robustness, and discusses data augmentation strategies for improved performance.

Contribution

It proposes Jacobian regularization in neural network training for better long-term predictions of nonlinear systems, addressing stability and robustness issues.

Findings

Jacobian regularization improves prediction accuracy

Regularization enhances robustness against local errors

Data augmentation helps when training data lacks low-dimensional attractors

Abstract

We study the use of feedforward neural networks (FNN) to develop models of nonlinear dynamical systems from data. Emphasis is placed on predictions at long times, with limited data availability. Inspired by global stability analysis, and the observation of the strong correlation between the local error and the maximum singular value of the Jacobian of the ANN, we introduce Jacobian regularization in the loss function. This regularization suppresses the sensitivity of the prediction to the local error and is shown to improve accuracy and robustness. Comparison between the proposed approach and sparse polynomial regression is presented in numerical examples ranging from simple ODE systems to nonlinear PDE systems including vortex shedding behind a cylinder, and instability-driven buoyant mixing flow. Furthermore, limitations of feedforward neural networks are highlighted, especially when the training data does not include a low dimensional attractor. Strategies of data augmentation are presented as remedies to address these issues to a certain extent.