Physics-constrained convolutional neural networks for inverse problems in spatiotemporal partial differential equations
We propose a physics-constrained convolutional neural network (PC-CNN) to solve two types of inverse problems in partial differential equations (PDEs), which are nonlinear and vary both in space and time. In the first inverse problem, we are given data that is offset by spatially varying systematic...
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Cambridge University Press
2024-01-01
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| author | Daniel Kelshaw Luca Magri |
| author_facet | Daniel Kelshaw Luca Magri |
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| description | We propose a physics-constrained convolutional neural network (PC-CNN) to solve two types of inverse problems in partial differential equations (PDEs), which are nonlinear and vary both in space and time. In the first inverse problem, we are given data that is offset by spatially varying systematic error (i.e., the bias, also known as the epistemic uncertainty). The task is to uncover the true state, which is the solution of the PDE, from the biased data. In the second inverse problem, we are given sparse information on the solution of a PDE. The task is to reconstruct the solution in space with high resolution. First, we present the PC-CNN, which constrains the PDE with a time-windowing scheme to handle sequential data. Second, we analyze the performance of the PC-CNN to uncover solutions from biased data. We analyze both linear and nonlinear convection-diffusion equations, and the Navier–Stokes equations, which govern the spatiotemporally chaotic dynamics of turbulent flows. We find that the PC-CNN correctly recovers the true solution for a variety of biases, which are parameterized as non-convex functions. Third, we analyze the performance of the PC-CNN for reconstructing solutions from sparse information for the turbulent flow. We reconstruct the spatiotemporal chaotic solution on a high-resolution grid from only 1% of the information contained in it. For both tasks, we further analyze the Navier–Stokes solutions. We find that the inferred solutions have a physical spectral energy content, whereas traditional methods, such as interpolation, do not. This work opens opportunities for solving inverse problems with partial differential equations. |
| format | Article |
| id | doaj-art-d9aeb6698717486eb056fc485355a96e |
| institution | OA Journals |
| issn | 2632-6736 |
| language | English |
| publishDate | 2024-01-01 |
| publisher | Cambridge University Press |
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| series | Data-Centric Engineering |
| spelling | doaj-art-d9aeb6698717486eb056fc485355a96e2025-08-20T02:32:18ZengCambridge University PressData-Centric Engineering2632-67362024-01-01510.1017/dce.2024.46Physics-constrained convolutional neural networks for inverse problems in spatiotemporal partial differential equationsDaniel Kelshaw0Luca Magri1https://orcid.org/0000-0002-0657-2611Department of Aeronautics, Imperial College London, London, UKDepartment of Aeronautics, Imperial College London, London, UK The Alan Turing Institute, The British Library, London, UK DIMEAS, Politecnico di Torino, Torino, Italy.We propose a physics-constrained convolutional neural network (PC-CNN) to solve two types of inverse problems in partial differential equations (PDEs), which are nonlinear and vary both in space and time. In the first inverse problem, we are given data that is offset by spatially varying systematic error (i.e., the bias, also known as the epistemic uncertainty). The task is to uncover the true state, which is the solution of the PDE, from the biased data. In the second inverse problem, we are given sparse information on the solution of a PDE. The task is to reconstruct the solution in space with high resolution. First, we present the PC-CNN, which constrains the PDE with a time-windowing scheme to handle sequential data. Second, we analyze the performance of the PC-CNN to uncover solutions from biased data. We analyze both linear and nonlinear convection-diffusion equations, and the Navier–Stokes equations, which govern the spatiotemporally chaotic dynamics of turbulent flows. We find that the PC-CNN correctly recovers the true solution for a variety of biases, which are parameterized as non-convex functions. Third, we analyze the performance of the PC-CNN for reconstructing solutions from sparse information for the turbulent flow. We reconstruct the spatiotemporal chaotic solution on a high-resolution grid from only 1% of the information contained in it. For both tasks, we further analyze the Navier–Stokes solutions. We find that the inferred solutions have a physical spectral energy content, whereas traditional methods, such as interpolation, do not. This work opens opportunities for solving inverse problems with partial differential equations.https://www.cambridge.org/core/product/identifier/S2632673624000467/type/journal_articlescientific machine learningconvolutional neural networksinverse problems |
| spellingShingle | Daniel Kelshaw Luca Magri Physics-constrained convolutional neural networks for inverse problems in spatiotemporal partial differential equations Data-Centric Engineering scientific machine learning convolutional neural networks inverse problems |
| title | Physics-constrained convolutional neural networks for inverse problems in spatiotemporal partial differential equations |
| title_full | Physics-constrained convolutional neural networks for inverse problems in spatiotemporal partial differential equations |
| title_fullStr | Physics-constrained convolutional neural networks for inverse problems in spatiotemporal partial differential equations |
| title_full_unstemmed | Physics-constrained convolutional neural networks for inverse problems in spatiotemporal partial differential equations |
| title_short | Physics-constrained convolutional neural networks for inverse problems in spatiotemporal partial differential equations |
| title_sort | physics constrained convolutional neural networks for inverse problems in spatiotemporal partial differential equations |
| topic | scientific machine learning convolutional neural networks inverse problems |
| url | https://www.cambridge.org/core/product/identifier/S2632673624000467/type/journal_article |
| work_keys_str_mv | AT danielkelshaw physicsconstrainedconvolutionalneuralnetworksforinverseproblemsinspatiotemporalpartialdifferentialequations AT lucamagri physicsconstrainedconvolutionalneuralnetworksforinverseproblemsinspatiotemporalpartialdifferentialequations |