Controlling Chaos in Van Der Pol Dynamics Using Signal-Encoded Deep Learning
Controlling nonlinear dynamics is a long-standing problem in engineering. Harnessing known physical information to accelerate or constrain stochastic learning pursues a new paradigm of scientific machine learning. By linearizing nonlinear systems, traditional control methods cannot learn nonlinear f...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-01-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/3/453 |
| _version_ | 1797486356187840512 |
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| author | Hanfeng Zhai Timothy Sands |
| author_facet | Hanfeng Zhai Timothy Sands |
| author_sort | Hanfeng Zhai |
| collection | DOAJ |
| description | Controlling nonlinear dynamics is a long-standing problem in engineering. Harnessing known physical information to accelerate or constrain stochastic learning pursues a new paradigm of scientific machine learning. By linearizing nonlinear systems, traditional control methods cannot learn nonlinear features from chaotic data for use in control. Here, we introduce Physics-Informed Deep Operator Control (PIDOC), and by encoding the control signal and initial position into the losses of a physics-informed neural network (PINN), the nonlinear system is forced to exhibit the desired trajectory given the control signal. PIDOC receives signals as physics commands and learns from the chaotic data output from the nonlinear van der Pol system, where the output of the PINN is the control. Applied to a benchmark problem, PIDOC successfully implements control with a higher stochasticity for higher-order terms. PIDOC has also been proven to be capable of converging to different desired trajectories based on case studies. Initial positions slightly affect the control accuracy at the beginning stage yet do not change the overall control quality. For highly nonlinear systems, PIDOC is not able to execute control with a high accuracy compared with the benchmark problem. The depth and width of the neural network structure do not greatly change the convergence of PIDOC based on case studies of van der Pol systems with low and high nonlinearities. Surprisingly, enlarging the control signal does not help to improve the control quality. The proposed framework can potentially be applied to many nonlinear systems for nonlinear controls. |
| first_indexed | 2024-03-09T23:32:01Z |
| format | Article |
| id | doaj.art-4a3fad95da834094a6c5c768e5dc9774 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T23:32:01Z |
| publishDate | 2022-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-4a3fad95da834094a6c5c768e5dc97742023-11-23T17:07:45ZengMDPI AGMathematics2227-73902022-01-0110345310.3390/math10030453Controlling Chaos in Van Der Pol Dynamics Using Signal-Encoded Deep LearningHanfeng Zhai0Timothy Sands1Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853, USASibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853, USAControlling nonlinear dynamics is a long-standing problem in engineering. Harnessing known physical information to accelerate or constrain stochastic learning pursues a new paradigm of scientific machine learning. By linearizing nonlinear systems, traditional control methods cannot learn nonlinear features from chaotic data for use in control. Here, we introduce Physics-Informed Deep Operator Control (PIDOC), and by encoding the control signal and initial position into the losses of a physics-informed neural network (PINN), the nonlinear system is forced to exhibit the desired trajectory given the control signal. PIDOC receives signals as physics commands and learns from the chaotic data output from the nonlinear van der Pol system, where the output of the PINN is the control. Applied to a benchmark problem, PIDOC successfully implements control with a higher stochasticity for higher-order terms. PIDOC has also been proven to be capable of converging to different desired trajectories based on case studies. Initial positions slightly affect the control accuracy at the beginning stage yet do not change the overall control quality. For highly nonlinear systems, PIDOC is not able to execute control with a high accuracy compared with the benchmark problem. The depth and width of the neural network structure do not greatly change the convergence of PIDOC based on case studies of van der Pol systems with low and high nonlinearities. Surprisingly, enlarging the control signal does not help to improve the control quality. The proposed framework can potentially be applied to many nonlinear systems for nonlinear controls.https://www.mdpi.com/2227-7390/10/3/453physics-informed neural networksvan der Pol dynamicsnonlinear controlmachine learning |
| spellingShingle | Hanfeng Zhai Timothy Sands Controlling Chaos in Van Der Pol Dynamics Using Signal-Encoded Deep Learning Mathematics physics-informed neural networks van der Pol dynamics nonlinear control machine learning |
| title | Controlling Chaos in Van Der Pol Dynamics Using Signal-Encoded Deep Learning |
| title_full | Controlling Chaos in Van Der Pol Dynamics Using Signal-Encoded Deep Learning |
| title_fullStr | Controlling Chaos in Van Der Pol Dynamics Using Signal-Encoded Deep Learning |
| title_full_unstemmed | Controlling Chaos in Van Der Pol Dynamics Using Signal-Encoded Deep Learning |
| title_short | Controlling Chaos in Van Der Pol Dynamics Using Signal-Encoded Deep Learning |
| title_sort | controlling chaos in van der pol dynamics using signal encoded deep learning |
| topic | physics-informed neural networks van der Pol dynamics nonlinear control machine learning |
| url | https://www.mdpi.com/2227-7390/10/3/453 |
| work_keys_str_mv | AT hanfengzhai controllingchaosinvanderpoldynamicsusingsignalencodeddeeplearning AT timothysands controllingchaosinvanderpoldynamicsusingsignalencodeddeeplearning |