A hybrid physics-informed neural network based multiscale solver as a partial differential equation constrained optimization problem
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| Format: | Preprint |
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2023
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| author | Hintermüller, Michael Korolev, Denis |
| author_facet | Hintermüller, Michael Korolev, Denis |
| contents | In this work, we study physics-informed neural networks (PINNs) constrained by partial differential equations (PDEs) and their application in approximating PDEs with two characteristic scales. From a continuous perspective, our formulation corresponds to a non-standard PDE-constrained optimization problem with a PINN-type objective. From a discrete standpoint, the formulation represents a hybrid numerical solver that utilizes both neural networks and finite elements. For the problem analysis, we introduce a proper function space, and we develop a numerical solution algorithm. The latter combines an adjoint-based technique for the efficient gradient computation with automatic differentiation. This new multiscale method is then applied exemplarily to a heat transfer problem with oscillating coefficients. In this context, the neural network approximates a fine-scale problem, and a coarse-scale problem constrains the associated learning process. We demonstrate that incorporating coarse-scale information into the neural network training process via a weak convergence-based regularization term is beneficial. Indeed, while preserving upscaling consistency, this term encourages non-trivial PINN solutions and also acts as a preconditioner for the low-frequency component of the fine-scale PDE, resulting in improved convergence properties of the PINN method. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_04439 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A hybrid physics-informed neural network based multiscale solver as a partial differential equation constrained optimization problem Hintermüller, Michael Korolev, Denis Optimization and Control In this work, we study physics-informed neural networks (PINNs) constrained by partial differential equations (PDEs) and their application in approximating PDEs with two characteristic scales. From a continuous perspective, our formulation corresponds to a non-standard PDE-constrained optimization problem with a PINN-type objective. From a discrete standpoint, the formulation represents a hybrid numerical solver that utilizes both neural networks and finite elements. For the problem analysis, we introduce a proper function space, and we develop a numerical solution algorithm. The latter combines an adjoint-based technique for the efficient gradient computation with automatic differentiation. This new multiscale method is then applied exemplarily to a heat transfer problem with oscillating coefficients. In this context, the neural network approximates a fine-scale problem, and a coarse-scale problem constrains the associated learning process. We demonstrate that incorporating coarse-scale information into the neural network training process via a weak convergence-based regularization term is beneficial. Indeed, while preserving upscaling consistency, this term encourages non-trivial PINN solutions and also acts as a preconditioner for the low-frequency component of the fine-scale PDE, resulting in improved convergence properties of the PINN method. |
| title | A hybrid physics-informed neural network based multiscale solver as a partial differential equation constrained optimization problem |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2309.04439 |