Long-Time Predictive Modeling of Nonlinear Dynamical Systems Using Neural Networks
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...
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| Format: | Article |
| Language: | English |
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Wiley
2018-01-01
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| Series: | Complexity |
| Online Access: | http://dx.doi.org/10.1155/2018/4801012 |
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| _version_ | 1850173031852277760 |
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| author | Shaowu Pan Karthik Duraisamy |
| author_facet | Shaowu Pan Karthik Duraisamy |
| author_sort | Shaowu Pan |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-cbdb8edcae8340cda6ef0e84e7058721 |
| institution | OA Journals |
| issn | 1076-2787 1099-0526 |
| language | English |
| publishDate | 2018-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Complexity |
| spelling | doaj-art-cbdb8edcae8340cda6ef0e84e70587212025-08-20T02:19:57ZengWileyComplexity1076-27871099-05262018-01-01201810.1155/2018/48010124801012Long-Time Predictive Modeling of Nonlinear Dynamical Systems Using Neural NetworksShaowu Pan0Karthik Duraisamy1Department of Aerospace Engineering, University of Michigan, Ann Arbor, MI, USADepartment of Aerospace Engineering, University of Michigan, Ann Arbor, MI, USAWe 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.http://dx.doi.org/10.1155/2018/4801012 |
| spellingShingle | Shaowu Pan Karthik Duraisamy Long-Time Predictive Modeling of Nonlinear Dynamical Systems Using Neural Networks Complexity |
| title | Long-Time Predictive Modeling of Nonlinear Dynamical Systems Using Neural Networks |
| title_full | Long-Time Predictive Modeling of Nonlinear Dynamical Systems Using Neural Networks |
| title_fullStr | Long-Time Predictive Modeling of Nonlinear Dynamical Systems Using Neural Networks |
| title_full_unstemmed | Long-Time Predictive Modeling of Nonlinear Dynamical Systems Using Neural Networks |
| title_short | Long-Time Predictive Modeling of Nonlinear Dynamical Systems Using Neural Networks |
| title_sort | long time predictive modeling of nonlinear dynamical systems using neural networks |
| url | http://dx.doi.org/10.1155/2018/4801012 |
| work_keys_str_mv | AT shaowupan longtimepredictivemodelingofnonlineardynamicalsystemsusingneuralnetworks AT karthikduraisamy longtimepredictivemodelingofnonlineardynamicalsystemsusingneuralnetworks |