Automatic discovery of optimal meta-solvers for time-dependent nonlinear PDEs

TL;DR

This paper introduces a scalable framework that combines classical numerical methods with deep learning to automatically discover optimal solvers for complex time-dependent nonlinear PDEs, improving efficiency and adaptability.

Contribution

It presents a novel multi-objective optimization approach for automated solver discovery, integrating neural operators with traditional methods for tailored, high-performance PDE solutions.

Findings

Discovered meta-solvers outperform conventional methods in various PDE problems.

The framework effectively balances accuracy, speed, and memory usage.

Applicable across reaction-diffusion, fluid dynamics, and solid mechanics.

Abstract

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.