End-to-End Mesh Optimization of a Hybrid Deep Learning Black-Box PDE Solver
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866917652113391616 |
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| author | Ma, Shaocong Diffenderfer, James Kailkhura, Bhavya Zhou, Yi |
| author_facet | Ma, Shaocong Diffenderfer, James Kailkhura, Bhavya Zhou, Yi |
| contents | Deep learning has been widely applied to solve partial differential equations (PDEs) in computational fluid dynamics. Recent research proposed a PDE correction framework that leverages deep learning to correct the solution obtained by a PDE solver on a coarse mesh. However, end-to-end training of such a PDE correction model over both solver-dependent parameters such as mesh parameters and neural network parameters requires the PDE solver to support automatic differentiation through the iterative numerical process. Such a feature is not readily available in many existing solvers. In this study, we explore the feasibility of end-to-end training of a hybrid model with a black-box PDE solver and a deep learning model for fluid flow prediction. Specifically, we investigate a hybrid model that integrates a black-box PDE solver into a differentiable deep graph neural network. To train this model, we use a zeroth-order gradient estimator to differentiate the PDE solver via forward propagation. Although experiments show that the proposed approach based on zeroth-order gradient estimation underperforms the baseline that computes exact derivatives using automatic differentiation, our proposed method outperforms the baseline trained with a frozen input mesh to the solver. Moreover, with a simple warm-start on the neural network parameters, we show that models trained by these zeroth-order algorithms achieve an accelerated convergence and improved generalization performance. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_11766 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | End-to-End Mesh Optimization of a Hybrid Deep Learning Black-Box PDE Solver Ma, Shaocong Diffenderfer, James Kailkhura, Bhavya Zhou, Yi Machine Learning Numerical Analysis Optimization and Control Deep learning has been widely applied to solve partial differential equations (PDEs) in computational fluid dynamics. Recent research proposed a PDE correction framework that leverages deep learning to correct the solution obtained by a PDE solver on a coarse mesh. However, end-to-end training of such a PDE correction model over both solver-dependent parameters such as mesh parameters and neural network parameters requires the PDE solver to support automatic differentiation through the iterative numerical process. Such a feature is not readily available in many existing solvers. In this study, we explore the feasibility of end-to-end training of a hybrid model with a black-box PDE solver and a deep learning model for fluid flow prediction. Specifically, we investigate a hybrid model that integrates a black-box PDE solver into a differentiable deep graph neural network. To train this model, we use a zeroth-order gradient estimator to differentiate the PDE solver via forward propagation. Although experiments show that the proposed approach based on zeroth-order gradient estimation underperforms the baseline that computes exact derivatives using automatic differentiation, our proposed method outperforms the baseline trained with a frozen input mesh to the solver. Moreover, with a simple warm-start on the neural network parameters, we show that models trained by these zeroth-order algorithms achieve an accelerated convergence and improved generalization performance. |
| title | End-to-End Mesh Optimization of a Hybrid Deep Learning Black-Box PDE Solver |
| topic | Machine Learning Numerical Analysis Optimization and Control |
| url | https://arxiv.org/abs/2404.11766 |