POD-Galerkin reduced order models and physics-informed neural networks for solving inverse problems for the Navier–Stokes equations
Abstract We present a Reduced Order Model (ROM) which exploits recent developments in Physics Informed Neural Networks (PINNs) for solving inverse problems for the Navier–Stokes equations (NSE). In the proposed approach, the presence of simulated data for the fluid dynamics fields is assumed. A POD-...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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SpringerOpen
2023-03-01
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| Series: | Advanced Modeling and Simulation in Engineering Sciences |
| Subjects: | |
| Online Access: | https://doi.org/10.1186/s40323-023-00242-2 |
| _version_ | 1797864054915596288 |
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| author | Saddam Hijazi Melina Freitag Niels Landwehr |
| author_facet | Saddam Hijazi Melina Freitag Niels Landwehr |
| author_sort | Saddam Hijazi |
| collection | DOAJ |
| description | Abstract We present a Reduced Order Model (ROM) which exploits recent developments in Physics Informed Neural Networks (PINNs) for solving inverse problems for the Navier–Stokes equations (NSE). In the proposed approach, the presence of simulated data for the fluid dynamics fields is assumed. A POD-Galerkin ROM is then constructed by applying POD on the snapshots matrices of the fluid fields and performing a Galerkin projection of the NSE (or the modified equations in case of turbulence modeling) onto the POD reduced basis. A POD-Galerkin PINN ROM is then derived by introducing deep neural networks which approximate the reduced outputs with the input being time and/or parameters of the model. The neural networks incorporate the physical equations (the POD-Galerkin reduced equations) into their structure as part of the loss function. Using this approach, the reduced model is able to approximate unknown parameters such as physical constants or the boundary conditions. A demonstration of the applicability of the proposed ROM is illustrated by three cases which are the steady flow around a backward step, the flow around a circular cylinder and the unsteady turbulent flow around a surface mounted cubic obstacle. |
| first_indexed | 2024-04-09T22:46:29Z |
| format | Article |
| id | doaj.art-f3c02454dcd54e2c81ed3145c95fb1f5 |
| institution | Directory Open Access Journal |
| issn | 2213-7467 |
| language | English |
| last_indexed | 2024-04-09T22:46:29Z |
| publishDate | 2023-03-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Advanced Modeling and Simulation in Engineering Sciences |
| spelling | doaj.art-f3c02454dcd54e2c81ed3145c95fb1f52023-03-22T11:52:52ZengSpringerOpenAdvanced Modeling and Simulation in Engineering Sciences2213-74672023-03-0110113810.1186/s40323-023-00242-2POD-Galerkin reduced order models and physics-informed neural networks for solving inverse problems for the Navier–Stokes equationsSaddam Hijazi0Melina Freitag1Niels Landwehr2Institute of Mathematics, University of PotsdamInstitute of Mathematics, University of PotsdamInstitute of Computer Science, University of HildesheimAbstract We present a Reduced Order Model (ROM) which exploits recent developments in Physics Informed Neural Networks (PINNs) for solving inverse problems for the Navier–Stokes equations (NSE). In the proposed approach, the presence of simulated data for the fluid dynamics fields is assumed. A POD-Galerkin ROM is then constructed by applying POD on the snapshots matrices of the fluid fields and performing a Galerkin projection of the NSE (or the modified equations in case of turbulence modeling) onto the POD reduced basis. A POD-Galerkin PINN ROM is then derived by introducing deep neural networks which approximate the reduced outputs with the input being time and/or parameters of the model. The neural networks incorporate the physical equations (the POD-Galerkin reduced equations) into their structure as part of the loss function. Using this approach, the reduced model is able to approximate unknown parameters such as physical constants or the boundary conditions. A demonstration of the applicability of the proposed ROM is illustrated by three cases which are the steady flow around a backward step, the flow around a circular cylinder and the unsteady turbulent flow around a surface mounted cubic obstacle.https://doi.org/10.1186/s40323-023-00242-2Proper orthogonal decompositionInverse problemsPhysics-based machine learningNavier–Stokes equations |
| spellingShingle | Saddam Hijazi Melina Freitag Niels Landwehr POD-Galerkin reduced order models and physics-informed neural networks for solving inverse problems for the Navier–Stokes equations Advanced Modeling and Simulation in Engineering Sciences Proper orthogonal decomposition Inverse problems Physics-based machine learning Navier–Stokes equations |
| title | POD-Galerkin reduced order models and physics-informed neural networks for solving inverse problems for the Navier–Stokes equations |
| title_full | POD-Galerkin reduced order models and physics-informed neural networks for solving inverse problems for the Navier–Stokes equations |
| title_fullStr | POD-Galerkin reduced order models and physics-informed neural networks for solving inverse problems for the Navier–Stokes equations |
| title_full_unstemmed | POD-Galerkin reduced order models and physics-informed neural networks for solving inverse problems for the Navier–Stokes equations |
| title_short | POD-Galerkin reduced order models and physics-informed neural networks for solving inverse problems for the Navier–Stokes equations |
| title_sort | pod galerkin reduced order models and physics informed neural networks for solving inverse problems for the navier stokes equations |
| topic | Proper orthogonal decomposition Inverse problems Physics-based machine learning Navier–Stokes equations |
| url | https://doi.org/10.1186/s40323-023-00242-2 |
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