Conservative physics-informed neural networks on discrete domains for conservation laws: Applications to forward and inverse problems
We propose a conservative physics-informed neural network (cPINN) on discrete domains for nonlinear conservation laws. Here, the term discrete domain represents the discrete sub-domains obtained after division of the computational domain, where PINN is applied and the conservation property of cPINN...
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| Published in: | Computer methods in applied mechanics and engineering Vol. 365; no. C; p. 113028 |
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| Main Authors: | , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Amsterdam
Elsevier B.V
15.06.2020
Elsevier BV Elsevier |
| Subjects: | |
| ISSN: | 0045-7825, 1879-2138 |
| Online Access: | Get full text |
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| Abstract | We propose a conservative physics-informed neural network (cPINN) on discrete domains for nonlinear conservation laws. Here, the term discrete domain represents the discrete sub-domains obtained after division of the computational domain, where PINN is applied and the conservation property of cPINN is obtained by enforcing the flux continuity in the strong form along the sub-domain interfaces. In case of hyperbolic conservation laws, the convective flux contributes at the interfaces, whereas in case of viscous conservation laws, both convective and diffusive fluxes contribute. Apart from the flux continuity condition, an average solution (given by two different neural networks) is also enforced at the common interface between two sub-domains. One can also employ a deep neural network in the domain, where the solution may have complex structure, whereas a shallow neural network can be used in the sub-domains with relatively simple and smooth solutions. Another advantage of the proposed method is the additional freedom it gives in terms of the choice of optimization algorithm and the various training parameters like residual points, activation function, width and depth of the network etc. Various forms of errors involved in cPINN such as optimization, generalization and approximation errors and their sources are discussed briefly. In cPINN, locally adaptive activation functions are used, hence training the model faster compared to its fixed counterparts. Both, forward and inverse problems are solved using the proposed method. Various test cases ranging from scalar nonlinear conservation laws like Burgers, Korteweg–de Vries (KdV) equations to systems of conservation laws, like compressible Euler equations are solved. The lid-driven cavity test case governed by incompressible Navier–Stokes equation is also solved and the results are compared against a benchmark solution. The proposed method enjoys the property of domain decomposition with separate neural networks in each sub-domain, and it efficiently lends itself to parallelized computation, where each sub-domain can be assigned to a different computational node.
•Domain decomposition technique is proposed for PINNs with tailored neural network in each sub-domain for solving conservation laws.•Due to presence of multiple neural networks, the representation capacity of the proposed cPINN method increases.•Based on prior knowledge of the solution regularity in each sub-domain, the hyper-parameter set of corresponding PINN can be properly adjusted.•The partial independence of individual PINNs in decomposed domains can be further employed to implement cPINN in a parallelized algorithm. |
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| AbstractList | We propose a conservative physics-informed neural network (cPINN) on discrete domains for nonlinear conservation laws. Here, the term discrete domain represents the discrete sub-domains obtained after division of the computational domain, where PINN is applied and the conservation property of cPINN is obtained by enforcing the flux continuity in the strong form along the sub-domain interfaces. In case of hyperbolic conservation laws, the convective flux contributes at the interfaces, whereas in case of viscous conservation laws, both convective and diffusive fluxes contribute. Apart from the flux continuity condition, an average solution (given by two different neural networks) is also enforced at the common interface between two sub-domains. One can also employ a deep neural network in the domain, where the solution may have complex structure, whereas a shallow neural network can be used in the sub-domains with relatively simple and smooth solutions. Another advantage of the proposed method is the additional freedom it gives in terms of the choice of optimization algorithm and the various training parameters like residual points, activation function, width and depth of the network etc. Various forms of errors involved in cPINN such as optimization, generalization and approximation errors and their sources are discussed briefly. In cPINN, locally adaptive activation functions are used, hence training the model faster compared to its fixed counterparts. Both, forward and inverse problems are solved using the proposed method. Various test cases ranging from scalar nonlinear conservation laws like Burgers, Korteweg–de Vries (KdV) equations to systems of conservation laws, like compressible Euler equations are solved. The lid-driven cavity test case governed by incompressible Navier–Stokes equation is also solved and the results are compared against a benchmark solution. The proposed method enjoys the property of domain decomposition with separate neural networks in each sub-domain, and it efficiently lends itself to parallelized computation, where each sub-domain can be assigned to a different computational node.
•Domain decomposition technique is proposed for PINNs with tailored neural network in each sub-domain for solving conservation laws.•Due to presence of multiple neural networks, the representation capacity of the proposed cPINN method increases.•Based on prior knowledge of the solution regularity in each sub-domain, the hyper-parameter set of corresponding PINN can be properly adjusted.•The partial independence of individual PINNs in decomposed domains can be further employed to implement cPINN in a parallelized algorithm. We propose a conservative physics-informed neural network (cPINN) on discrete domains for nonlinear conservation laws. Here, the term discrete domain represents the discrete sub-domains obtained after division of the computational domain, where PINN is applied and the conservation property of cPINN is obtained by enforcing the flux continuity in the strong form along the sub-domain interfaces. In case of hyperbolic conservation laws, the convective flux contributes at the interfaces, whereas in case of viscous conservation laws, both convective and diffusive fluxes contribute. Apart from the flux continuity condition, an average solution (given by two different neural networks) is also enforced at the common interface between two sub-domains. One can also employ a deep neural network in the domain, where the solution may have complex structure, whereas a shallow neural network can be used in the sub-domains with relatively simple and smooth solutions. Another advantage of the proposed method is the additional freedom it gives in terms of the choice of optimization algorithm and the various training parameters like residual points, activation function, width and depth of the network etc. Various forms of errors involved in cPINN such as optimization, generalization and approximation errors and their sources are discussed briefly. In cPINN, locally adaptive activation functions are used, hence training the model faster compared to its fixed counterparts. Both, forward and inverse problems are solved using the proposed method. Various test cases ranging from scalar nonlinear conservation laws like Burgers, Korteweg–de Vries (KdV) equations to systems of conservation laws, like compressible Euler equations are solved. The lid-driven cavity test case governed by incompressible Navier–Stokes equation is also solved and the results are compared against a benchmark solution. The proposed method enjoys the property of domain decomposition with separate neural networks in each sub-domain, and it efficiently lends itself to parallelized computation, where each sub-domain can be assigned to a different computational node. |
| ArticleNumber | 113028 |
| Author | Kharazmi, Ehsan Jagtap, Ameya D. Karniadakis, George Em |
| Author_xml | – sequence: 1 givenname: Ameya D. surname: Jagtap fullname: Jagtap, Ameya D. email: ameyadjagtap@gmail.com, ameya_jagtap@brown.edu organization: Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912, USA – sequence: 2 givenname: Ehsan orcidid: 0000-0002-3680-5500 surname: Kharazmi fullname: Kharazmi, Ehsan email: ehsan_kharazmi@brown.edu organization: Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912, USA – sequence: 3 givenname: George Em surname: Karniadakis fullname: Karniadakis, George Em email: george_karniadakis@brown.edu organization: Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912, USA |
| BackLink | https://www.osti.gov/biblio/1616479$$D View this record in Osti.gov |
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| Snippet | We propose a conservative physics-informed neural network (cPINN) on discrete domains for nonlinear conservation laws. Here, the term discrete domain... |
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| SubjectTerms | Activation Algorithms Artificial neural networks Compressibility Computational fluid dynamics Conservation laws cPINN Domain decomposition Euler-Lagrange equation Fluxes Inverse problems Machine learning Mortar PINN Neural networks Optimization Parallel processing Training |
| Title | Conservative physics-informed neural networks on discrete domains for conservation laws: Applications to forward and inverse problems |
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