Neural ordinary differential equations with irregular and noisy data
Measurement noise is an integral part of collecting data of a physical process. Thus, noise removal is necessary to draw conclusions from these data, and it often becomes essential to construct dynamical models using these data. We discuss a methodology to learn differential equation(s) using noisy...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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The Royal Society
2023-07-01
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| Series: | Royal Society Open Science |
| Subjects: | |
| Online Access: | https://royalsocietypublishing.org/doi/10.1098/rsos.221475 |
| _version_ | 1797310790112378880 |
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| author | Pawan Goyal Peter Benner |
| author_facet | Pawan Goyal Peter Benner |
| author_sort | Pawan Goyal |
| collection | DOAJ |
| description | Measurement noise is an integral part of collecting data of a physical process. Thus, noise removal is necessary to draw conclusions from these data, and it often becomes essential to construct dynamical models using these data. We discuss a methodology to learn differential equation(s) using noisy and irregularly sampled measurements. In our methodology, the main innovation can be seen in the integration of deep neural networks with the neural ordinary differential equations (ODEs) approach. Precisely, we aim at learning a neural network that provides (approximately) an implicit representation of the data and an additional neural network that models the vector fields of the dependent variables. We combine these two networks by constraints using neural ODEs. The proposed framework to learn a model describing the vector field is highly effective under noisy measurements. The approach can handle scenarios where dependent variables are unavailable at the same temporal grid. Moreover, a particular structure, e.g. second order with respect to time, can easily be incorporated. We demonstrate the effectiveness of the proposed method for learning models using data obtained from various differential equations and present a comparison with the neural ODE method that does not make any special treatment to noise. Additionally, we discuss an ensemble approach to improve the performance of the proposed approach further. |
| first_indexed | 2024-03-08T01:49:05Z |
| format | Article |
| id | doaj.art-844daf775a1549b0b67fca02cd23bcb4 |
| institution | Directory Open Access Journal |
| issn | 2054-5703 |
| language | English |
| last_indexed | 2024-03-08T01:49:05Z |
| publishDate | 2023-07-01 |
| publisher | The Royal Society |
| record_format | Article |
| series | Royal Society Open Science |
| spelling | doaj.art-844daf775a1549b0b67fca02cd23bcb42024-02-14T11:18:27ZengThe Royal SocietyRoyal Society Open Science2054-57032023-07-0110710.1098/rsos.221475Neural ordinary differential equations with irregular and noisy dataPawan Goyal0Peter Benner1Max Planck Institute for Dynamics of Complex Technical Systems, Standtorstrasse 1, 39106 Magdeburg, GermanyMax Planck Institute for Dynamics of Complex Technical Systems, Standtorstrasse 1, 39106 Magdeburg, GermanyMeasurement noise is an integral part of collecting data of a physical process. Thus, noise removal is necessary to draw conclusions from these data, and it often becomes essential to construct dynamical models using these data. We discuss a methodology to learn differential equation(s) using noisy and irregularly sampled measurements. In our methodology, the main innovation can be seen in the integration of deep neural networks with the neural ordinary differential equations (ODEs) approach. Precisely, we aim at learning a neural network that provides (approximately) an implicit representation of the data and an additional neural network that models the vector fields of the dependent variables. We combine these two networks by constraints using neural ODEs. The proposed framework to learn a model describing the vector field is highly effective under noisy measurements. The approach can handle scenarios where dependent variables are unavailable at the same temporal grid. Moreover, a particular structure, e.g. second order with respect to time, can easily be incorporated. We demonstrate the effectiveness of the proposed method for learning models using data obtained from various differential equations and present a comparison with the neural ODE method that does not make any special treatment to noise. Additionally, we discuss an ensemble approach to improve the performance of the proposed approach further.https://royalsocietypublishing.org/doi/10.1098/rsos.221475machine learningdynamical systemsneural networksnoisy dataneural ordinary differential equations |
| spellingShingle | Pawan Goyal Peter Benner Neural ordinary differential equations with irregular and noisy data Royal Society Open Science machine learning dynamical systems neural networks noisy data neural ordinary differential equations |
| title | Neural ordinary differential equations with irregular and noisy data |
| title_full | Neural ordinary differential equations with irregular and noisy data |
| title_fullStr | Neural ordinary differential equations with irregular and noisy data |
| title_full_unstemmed | Neural ordinary differential equations with irregular and noisy data |
| title_short | Neural ordinary differential equations with irregular and noisy data |
| title_sort | neural ordinary differential equations with irregular and noisy data |
| topic | machine learning dynamical systems neural networks noisy data neural ordinary differential equations |
| url | https://royalsocietypublishing.org/doi/10.1098/rsos.221475 |
| work_keys_str_mv | AT pawangoyal neuralordinarydifferentialequationswithirregularandnoisydata AT peterbenner neuralordinarydifferentialequationswithirregularandnoisydata |