DeepOnet Based Preconditioning Strategies For Solving Parametric Linear Systems of Equations

TL;DR

This work presents a novel class of preconditioners for parametric linear systems by hybridizing DeepONet-based operator learning with standard Krylov solvers. It introduces two hybrid strategies: Direct Preconditioning (DP), which uses DeepONet to approximate the inverse action at each step, and Trunk Basis (TB), which extracts deep trunk basis functions to form a coarse space for subspace corrections, enabling effective low-frequency error reduction. The TB approach consistently outperforms DP across diffusion and Helmholtz benchmarks and supports CG preconditioning, yielding scalable, level-independent convergence when integrated into multigrid-like frameworks. The results demonstrate robust performance across parameter variations and mesh refinements, with TB offering a practical, mesh-insensitive coarse space construction that leverages spectral bias and operator learning for accelerated Krylov convergence.

Abstract

We introduce a new class of hybrid preconditioners for solving parametric linear systems of equations. The proposed preconditioners are constructed by hybridizing the deep operator network, namely DeepONet, with standard iterative methods. Exploiting the spectral bias, DeepONet-based components are harnessed to address low-frequency error components, while conventional iterative methods are employed to mitigate high-frequency error components. Our preconditioning framework comprises two distinct hybridization approaches: direct preconditioning (DP) and trunk basis (TB) approaches. In the DP approach, DeepONet is used to approximate an action of an inverse operator to a vector during each preconditioning step. In contrast, the TB approach extracts basis functions from the trained DeepONet to construct a map to a smaller subspace, in which the low-frequency component of the error can be effectively eliminated. Our numerical results demonstrate that utilizing the TB approach enhances the convergence of Krylov methods by a large margin compared to standard non-hybrid preconditioning strategies. Moreover, the proposed hybrid preconditioners exhibit robustness across a wide range of model parameters and problem resolutions.

Figures (13)

• Figure 1: An example of single/multi-input (left/right) DeepONet, see lu2022comprehensive.
• Figure 1: An illustration of a hybrid preconditioning framework. The DeepONet components are illustrated in blue, while components related to FE discretization and standard iterative methods are illustrated in brown. The red color depicts the part of DeepONet that must be evaluated during each preconditioning step.
• Figure 1: Left: An illustration of the computational domain with two different channel patterns used for the diffusion equation test with jumping coefficients. Right: Example of samples used for testing \ref{['sec:jump_diff']} example. The examples are selected such that the value of ${K \in [1, 10^{5}]}$ increases from left to right.
• Figure 1: Eigenvalues of the iteration matrix associated with Jacobi smoother $\boldsymbol{{M}}_1$ with $\gamma_1\in \{1, 2/3, \gamma_{k_{\text{H}}}\}$ and the TB-based operator $\boldsymbol{{C}}$ with $k \in \{3, 6, 12\}$ plotted against eigenvalues of the associated 1D Helmholtz operator. The gray and brown overlays relate to low and high-frequencies parts of the spectrum, respectively.
• Figure 1: An illustration of randomly selected $15$ trunk basis functions from the DeepONet ($p=128$) trained for \ref{['sec:jump_diff']} test problem.