Automatic discovery of optimal meta-solvers for time-dependent nonlinear PDEs
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866918076709076992 |
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| author | Lee, Youngkyu Liu, Shanqing Darbon, Jerome Karniadakis, George Em |
| author_facet | Lee, Youngkyu Liu, Shanqing Darbon, Jerome Karniadakis, George Em |
| contents | We present a general and scalable framework for the automated discovery of optimal meta-solvers for the solution of time-dependent nonlinear partial differential equations after appropriate discretization. By integrating classical numerical methods (e.g., Krylov-based methods) with modern deep learning components, such as neural operators, our approach enables flexible, on-demand solver design tailored to specific problem classes and objectives. The fast solvers tackle the large linear system resulting from the Newton--Raphson iteration or by using an implicit-explicit (IMEX) time integration scheme. Specifically, we formulate solver discovery as a multi-objective optimization problem, balancing various performance criteria such as accuracy, speed, and memory usage. The resulting Pareto optimal set provides a principled foundation for solver selection based on user-defined preference functions. When applied to problems in reaction--diffusion, fluid dynamics, and solid mechanics, the discovered meta-solvers consistently outperform conventional iterative methods, demonstrating both practical efficiency and broad applicability. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_00278 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Automatic discovery of optimal meta-solvers for time-dependent nonlinear PDEs Lee, Youngkyu Liu, Shanqing Darbon, Jerome Karniadakis, George Em Numerical Analysis We present a general and scalable framework for the automated discovery of optimal meta-solvers for the solution of time-dependent nonlinear partial differential equations after appropriate discretization. By integrating classical numerical methods (e.g., Krylov-based methods) with modern deep learning components, such as neural operators, our approach enables flexible, on-demand solver design tailored to specific problem classes and objectives. The fast solvers tackle the large linear system resulting from the Newton--Raphson iteration or by using an implicit-explicit (IMEX) time integration scheme. Specifically, we formulate solver discovery as a multi-objective optimization problem, balancing various performance criteria such as accuracy, speed, and memory usage. The resulting Pareto optimal set provides a principled foundation for solver selection based on user-defined preference functions. When applied to problems in reaction--diffusion, fluid dynamics, and solid mechanics, the discovered meta-solvers consistently outperform conventional iterative methods, demonstrating both practical efficiency and broad applicability. |
| title | Automatic discovery of optimal meta-solvers for time-dependent nonlinear PDEs |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2507.00278 |