Automatic discovery of optimal meta-solvers via multi-objective optimization
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We present a general and scalable framework for the automated discovery of optimal meta-solvers for the solution of 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.
