A Least-Squares-Based Neural Network (LS-Net) for Solving Linear Parametric PDEs
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arXiv
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| Format: | Preprint |
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2024
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| author | Baharlouei, Shima Taylor, Jamie M. Uriarte, Carlos Pardo, David |
| author_facet | Baharlouei, Shima Taylor, Jamie M. Uriarte, Carlos Pardo, David |
| contents | Developing efficient methods for solving parametric partial differential equations is crucial for addressing inverse problems. This work introduces a Least-Squares-based Neural Network (LS-Net) method for solving linear parametric PDEs. It utilizes a separated representation form for the parametric PDE solution via a deep neural network and a least-squares solver. In this approach, the output of the deep neural network consists of a vector-valued function, interpreted as basis functions for the parametric solution space, and the least-squares solver determines the optimal solution within the constructed solution space for each given parameter. The LS-Net method requires a quadratic loss function for the least-squares solver to find optimal solutions given the set of basis functions. In this study, we consider loss functions derived from the Deep Fourier Residual and Physics-Informed Neural Networks approaches. We also provide theoretical results similar to the Universal Approximation Theorem, stating that there exists a sufficiently large neural network that can theoretically approximate solutions of parametric PDEs with the desired accuracy. We illustrate the LS-net method by solving one- and two-dimensional problems. Numerical results clearly demonstrate the method's ability to approximate parametric solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_15089 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Least-Squares-Based Neural Network (LS-Net) for Solving Linear Parametric PDEs Baharlouei, Shima Taylor, Jamie M. Uriarte, Carlos Pardo, David Numerical Analysis 35A17, 68T07 G.1.2; G.1.8 Developing efficient methods for solving parametric partial differential equations is crucial for addressing inverse problems. This work introduces a Least-Squares-based Neural Network (LS-Net) method for solving linear parametric PDEs. It utilizes a separated representation form for the parametric PDE solution via a deep neural network and a least-squares solver. In this approach, the output of the deep neural network consists of a vector-valued function, interpreted as basis functions for the parametric solution space, and the least-squares solver determines the optimal solution within the constructed solution space for each given parameter. The LS-Net method requires a quadratic loss function for the least-squares solver to find optimal solutions given the set of basis functions. In this study, we consider loss functions derived from the Deep Fourier Residual and Physics-Informed Neural Networks approaches. We also provide theoretical results similar to the Universal Approximation Theorem, stating that there exists a sufficiently large neural network that can theoretically approximate solutions of parametric PDEs with the desired accuracy. We illustrate the LS-net method by solving one- and two-dimensional problems. Numerical results clearly demonstrate the method's ability to approximate parametric solutions. |
| title | A Least-Squares-Based Neural Network (LS-Net) for Solving Linear Parametric PDEs |
| topic | Numerical Analysis 35A17, 68T07 G.1.2; G.1.8 |
| url | https://arxiv.org/abs/2410.15089 |