Neural operators meet conjugate gradients: The FCG-NO method for efficient PDE solving

TL;DR

This paper introduces FCG-NO, a neural operator-based preconditioning method for the conjugate gradient solver, enabling efficient and resolution-invariant PDE solutions with improved accuracy over classical methods.

Contribution

It proposes a novel neural operator preconditioner for FCG, leveraging discretization-invariant architectures and a new training scheme, allowing cross-resolution PDE solving.

Findings

Outperforms classical preconditioners in numerical tests

Preconditioners learned at lower resolutions generalize to higher resolutions

Achieves efficient PDE solving with $O(N)$ complexity

Abstract

Deep learning solvers for partial differential equations typically have limited accuracy. We propose to overcome this problem by using them as preconditioners. More specifically, we apply discretization-invariant neural operators to learn preconditioners for the flexible conjugate gradient method (FCG). Architecture paired with novel loss function and training scheme allows for learning efficient preconditioners that can be used across different resolutions. On the theoretical side, FCG theory allows us to safely use nonlinear preconditioners that can be applied in $O(N)$ operations without constraining the form of the preconditioners matrix. To justify learning scheme components (the loss function and the way training data is collected) we perform several ablation studies. Numerical results indicate that our approach favorably compares with classical preconditioners and allows to reuse of preconditioners learned for lower resolution to the higher resolution data.