Long-time predictive modeling of nonlinear dynamical systems using neural networks
This work addresses the problem of improving long-term predictions in dynamical systems for researchers in computational physics and engineering, though it is incremental with hybrid methods.
The paper tackled long-time predictive modeling of nonlinear dynamical systems using neural networks with limited data, introducing Jacobian regularization to improve accuracy and robustness, showing enhanced performance in numerical examples from ODEs to PDEs.
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.