Automatic discovery of optimal meta-solvers via multi-objective optimization

TL;DR

The paper develops a Pareto-optimal framework to automatically discover optimal meta-solvers that hybridize neural operators with classical solvers for linear systems from PDE discretizations. It parameterizes relaxation-based and Krylov-based meta-solvers, defines vector-valued performance maps, and uses multi-objective optimization to construct Pareto fronts, then applies preference functions and LP-based rediscovery to select tailored solvers. Numerical experiments on Poisson equations in 1D–3D demonstrate that DeepONet–SSOR and related combinations frequently emerge as Pareto-optimal, while the approach supports extension to nonlinear and space-time PDEs. This automation provides a scalable pathway to tailor fast, accurate solvers to specific application needs and constraints.

Abstract

We design two classes of ultra-fast meta-solvers for linear systems arising after discretizing PDEs by combining neural operators with either simple iterative solvers, e.g., Jacobi and Gauss-Seidel, or with Krylov methods, e.g., GMRES and BiCGStab, using the trunk basis of DeepONet as a coarse preconditioner. The idea is to leverage the spectral bias of neural networks to account for the lower part of the spectrum in the error distribution while the upper part is handled easily and inexpensively using relaxation methods or fine-scale preconditioners. We create a pareto front of optimal meta-solvers using a plurarilty of metrics, and we introduce a preference function to select the best solver most suitable for a specific scenario. This automation for finding optimal solvers can be extended to nonlinear systems and other setups, e.g. finding the best meta-solver for space-time in time-dependent PDEs.

Figures (12)

• Figure 1: Construction of meta-solvers. A relaxation-based meta-solver combines a neural operator with an iterative solver, instantiated using a fixed proportion. Additionally, we apply a multi-grid technique on top of this combination. For the Krylov-based meta-solver, we first select a relaxation method as a smoother and combine it with a neural operator using the strategy: $N$ steps of smoother, followed by 1 step of the neural operator, and then $N$ more steps of the smoother. Additionally, we apply a multi-grid technique. Finally, a Krylov method is applied on top of this framework.
• Figure 1: Projection of Pareto front intro three dimensional criteria for solving 3-d Poisson equation, using ralxation-based methods. All solvers are depicted in blue while Pareto optimal solvers are highlighted in red. The "gap" due to the adaption thus the improvement of performance by multi-grid techniques.
• Figure 1: Projection of Pareto front onto three dimensional criteria, for relaxation-based methods solving 1-d Poisson equation.
• Figure 2: Sketch of Pareto Front and discovering optimal solvers by preference functions, with computational time and relative error performance criteria. All solvers are depicted in blue while Pareto optimal solvers are highlighted in red. The discovery of optimal solvers by preference function geometrically corresponds to slicing the set of Pareto optimal meta-solvers with a straight line.
• Figure 2: The top-3 optimal solvers discovered by preference function $p^1$ and $p^2$, in the three-dimensional projection: Computational Time -- Relative Error -- Memory allocation of Pareto front, for relaxation-based methods, solving 3-d Poisson equation.

Theorems & Definitions (8)

• Definition 1
• Definition 2
• Definition 3
• Proposition 4
• Remark 1
• Definition 5
• Proposition 6
• Lemma 7: Corollary of \ref{['propo_prefer']}