Long-time predictive modeling of nonlinear dynamical systems using neural networks
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Abstract (arXiv)
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.
Key facts. Cited by 14 papers in the map (data through 7 Oct 2026). First posted on arXiv 31 May 2018.
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Citing papers per year: 2021: 5; 2022: 6; 2023: 0; 2024: 3.
14 papers.
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- 2022Neural Implicit Flow: a mesh-agnostic dimensionality reduction paradigm of spatio-temporal datacited by 37
- 2021Model Order Reduction based on Runge-Kutta Neural Networkcited by 9
- 2022On Fast Simulation of Dynamical System with Neural Vector Enhanced Numerical Solvercited by 9
- 2021Non-intrusive Nonlinear Model Reduction via Machine Learning Approximations to Low-dimensional Operatorscited by 8
- 2021Initializing LSTM internal states via manifold learningcited by 7
- 2021Learning Dynamics from Noisy Measurements using Deep Learning with a Runge-Kutta Constraintcited by 7
- 2022Comparison of neural closure models for discretised PDEscited by 6
- 2022Neural ODEs with Irregular and Noisy Datacited by 5
- 2022A Model-Constrained Tangent Slope Learning Approach for Dynamical Systemscited by 3
- 2024TAEN: A Model-Constrained Tikhonov Autoencoder Network for Forward and Inverse Problemscited by 3
- 2022Active-learning-based non-intrusive Model Order Reductioncited by 2
- 2021Accelerating Dynamical System Simulations with Contracting and Physics-Projected Neural-Newton Solverscited by 1
- 2024TAEN: A Model-Constrained Tikhonov Autoencoder Network for Forward and Inverse Problems9 Dec 2024
- 2024A Novel Neural Filter to Improve Accuracy of Neural Network Models of Dynamic Systems20 Sep 2024
- 2024Fast Deep Predictive Coding Networks for Videos Feature Extraction without Labels8 Sep 2024
- 2022Comparison of neural closure models for discretised PDEs26 Oct 2022
- 2022A Model-Constrained Tangent Slope Learning Approach for Dynamical Systems9 Aug 2022
- 2022On Fast Simulation of Dynamical System with Neural Vector Enhanced Numerical Solver7 Aug 2022
- 2022Neural ODEs with Irregular and Noisy Data19 May 2022
- 2022Active-learning-based non-intrusive Model Order Reduction8 Apr 2022
- 2022Neural Implicit Flow: a mesh-agnostic dimensionality reduction paradigm of spatio-temporal data7 Apr 2022
- 2021Learning Dynamics from Noisy Measurements using Deep Learning with a Runge-Kutta Constraint23 Sep 2021
- 2021Non-intrusive Nonlinear Model Reduction via Machine Learning Approximations to Low-dimensional Operators17 Jun 2021
- 2021Accelerating Dynamical System Simulations with Contracting and Physics-Projected Neural-Newton Solvers4 Jun 2021
2 more
- 2021Initializing LSTM internal states via manifold learning27 Apr 2021
- 2021Model Order Reduction based on Runge-Kutta Neural Network25 Mar 2021
- 2022Neural Implicit Flow: a mesh-agnostic dimensionality reduction paradigm of spatio-temporal datacited by 37
- 2021Model Order Reduction based on Runge-Kutta Neural Networkcited by 9
- 2022On Fast Simulation of Dynamical System with Neural Vector Enhanced Numerical Solvercited by 9
- 2021Non-intrusive Nonlinear Model Reduction via Machine Learning Approximations to Low-dimensional Operatorscited by 8
- 2021Initializing LSTM internal states via manifold learningcited by 7
- 2021Learning Dynamics from Noisy Measurements using Deep Learning with a Runge-Kutta Constraintcited by 7
- 2022Comparison of neural closure models for discretised PDEscited by 6
- 2022Neural ODEs with Irregular and Noisy Datacited by 5
- 2022A Model-Constrained Tangent Slope Learning Approach for Dynamical Systemscited by 3
- 2024TAEN: A Model-Constrained Tikhonov Autoencoder Network for Forward and Inverse Problemscited by 3
- 2022Active-learning-based non-intrusive Model Order Reductioncited by 2
- 2021Accelerating Dynamical System Simulations with Contracting and Physics-Projected Neural-Newton Solverscited by 1
12 most cited of 14 All 14 (JSON)
- 2024TAEN: A Model-Constrained Tikhonov Autoencoder Network for Forward and Inverse Problems9 Dec 2024
- 2024A Novel Neural Filter to Improve Accuracy of Neural Network Models of Dynamic Systems20 Sep 2024
- 2024Fast Deep Predictive Coding Networks for Videos Feature Extraction without Labels8 Sep 2024
- 2022Comparison of neural closure models for discretised PDEs26 Oct 2022
- 2022A Model-Constrained Tangent Slope Learning Approach for Dynamical Systems9 Aug 2022
- 2022On Fast Simulation of Dynamical System with Neural Vector Enhanced Numerical Solver7 Aug 2022
- 2022Neural ODEs with Irregular and Noisy Data19 May 2022
- 2022Active-learning-based non-intrusive Model Order Reduction8 Apr 2022
- 2022Neural Implicit Flow: a mesh-agnostic dimensionality reduction paradigm of spatio-temporal data7 Apr 2022
- 2021Learning Dynamics from Noisy Measurements using Deep Learning with a Runge-Kutta Constraint23 Sep 2021
- 2021Non-intrusive Nonlinear Model Reduction via Machine Learning Approximations to Low-dimensional Operators17 Jun 2021
- 2021Accelerating Dynamical System Simulations with Contracting and Physics-Projected Neural-Newton Solvers4 Jun 2021
12 newest of 14 All 14 (JSON)