scholarly journals Identifying nonlinear dynamical systems via generative recurrent neural networks with applications to fMRI

2019 ◽  
Vol 15 (8) ◽  
pp. e1007263 ◽  
Author(s):  
Georgia Koppe ◽  
Hazem Toutounji ◽  
Peter Kirsch ◽  
Stefanie Lis ◽  
Daniel Durstewitz
Keyword(s):  
Neural Networks ◽  
Dynamical Systems ◽  
Recurrent Neural Networks ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical

scholarly journals Long-Time Predictive Modeling of Nonlinear Dynamical Systems Using Neural Networks

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-26 ◽  
Author(s):  
Shaowu Pan ◽  
Karthik Duraisamy
Keyword(s):  
Neural Networks ◽  
Dynamical Systems ◽  
Polynomial Regression ◽  
Nonlinear Dynamical Systems ◽  
Feedforward Neural Networks ◽  
Data Availability ◽  
Training Data ◽  
Local Error ◽  
Nonlinear Dynamical ◽  
Sparse Polynomial

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 strong correlation between the local error and the maximal 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.

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