Principal Component Flow Map Learning of PDEs from Incomplete, Limited, and Noisy Data
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866912167879507968 |
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| author | Churchill, Victor |
| author_facet | Churchill, Victor |
| contents | We present a computational technique for modeling the evolution of dynamical systems in a reduced basis, with a focus on the challenging problem of modeling partially-observed partial differential equations (PDEs) on high-dimensional non-uniform grids. We address limitations of previous work on data-driven flow map learning in the sense that we focus on noisy and limited data to move toward data collection scenarios in real-world applications. Leveraging recent work on modeling PDEs in modal and nodal spaces, we present a neural network structure that is suitable for PDE modeling with noisy and limited data available only on a subset of the state variables or computational domain. In particular, spatial grid-point measurements are reduced using a learned linear transformation, after which the dynamics are learned in this reduced basis before being transformed back out to the nodal space. This approach yields a drastically reduced parameterization of the neural network compared with previous flow map models for nodal space learning. This allows for rapid high-resolution simulations, enabled by smaller training data sets and reduced training times. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_10854 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Principal Component Flow Map Learning of PDEs from Incomplete, Limited, and Noisy Data Churchill, Victor Machine Learning Dynamical Systems We present a computational technique for modeling the evolution of dynamical systems in a reduced basis, with a focus on the challenging problem of modeling partially-observed partial differential equations (PDEs) on high-dimensional non-uniform grids. We address limitations of previous work on data-driven flow map learning in the sense that we focus on noisy and limited data to move toward data collection scenarios in real-world applications. Leveraging recent work on modeling PDEs in modal and nodal spaces, we present a neural network structure that is suitable for PDE modeling with noisy and limited data available only on a subset of the state variables or computational domain. In particular, spatial grid-point measurements are reduced using a learned linear transformation, after which the dynamics are learned in this reduced basis before being transformed back out to the nodal space. This approach yields a drastically reduced parameterization of the neural network compared with previous flow map models for nodal space learning. This allows for rapid high-resolution simulations, enabled by smaller training data sets and reduced training times. |
| title | Principal Component Flow Map Learning of PDEs from Incomplete, Limited, and Noisy Data |
| topic | Machine Learning Dynamical Systems |
| url | https://arxiv.org/abs/2407.10854 |