Approximation of Solution Operators for High-dimensional PDEs
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910302701879296 |
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| author | Gaby, Nathan Ye, Xiaojing |
| author_facet | Gaby, Nathan Ye, Xiaojing |
| contents | We propose a finite-dimensional control-based method to approximate solution operators for evolutional partial differential equations (PDEs), particularly in high-dimensions. By employing a general reduced-order model, such as a deep neural network, we connect the evolution of the model parameters with trajectories in a corresponding function space. Using the computational technique of neural ordinary differential equation, we learn the control over the parameter space such that from any initial starting point, the controlled trajectories closely approximate the solutions to the PDE. Approximation accuracy is justified for a general class of second-order nonlinear PDEs. Numerical results are presented for several high-dimensional PDEs, including real-world applications to solving Hamilton-Jacobi-Bellman equations. These are demonstrated to show the accuracy and efficiency of the proposed method. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_10385 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Approximation of Solution Operators for High-dimensional PDEs Gaby, Nathan Ye, Xiaojing Numerical Analysis Machine Learning Optimization and Control 65M99, 49M05 We propose a finite-dimensional control-based method to approximate solution operators for evolutional partial differential equations (PDEs), particularly in high-dimensions. By employing a general reduced-order model, such as a deep neural network, we connect the evolution of the model parameters with trajectories in a corresponding function space. Using the computational technique of neural ordinary differential equation, we learn the control over the parameter space such that from any initial starting point, the controlled trajectories closely approximate the solutions to the PDE. Approximation accuracy is justified for a general class of second-order nonlinear PDEs. Numerical results are presented for several high-dimensional PDEs, including real-world applications to solving Hamilton-Jacobi-Bellman equations. These are demonstrated to show the accuracy and efficiency of the proposed method. |
| title | Approximation of Solution Operators for High-dimensional PDEs |
| topic | Numerical Analysis Machine Learning Optimization and Control 65M99, 49M05 |
| url | https://arxiv.org/abs/2401.10385 |