Neural operator-based super-fidelity: A warm-start approach for accelerating steady-state simulations

TL;DR

This paper presents a neural operator-based super-fidelity method that efficiently accelerates steady-state PDE simulations by providing accurate warm-starts, demonstrating significant convergence speedups across various fluid flow scenarios without compromising solution accuracy.

Contribution

The study introduces a novel neural operator approach using VCNN-e for reliable, high-fidelity initializations in steady-state PDE solvers, enhancing efficiency and scalability.

Findings

At least two-fold acceleration in convergence times.

Maintains high accuracy across different flow regimes.

Effective across various linear solvers and computing setups.

Abstract

Recently, the use of neural networks to accelerate the solving of partial differential equations (PDEs) has gained significant traction in both academia and industry. However, employing neural networks as standalone surrogate models raises concerns about solution reliability, especially in precision-critical scientific tasks. This study introduces a novel "super-fidelity" method that leverages neural networks for warm-starting steady-state PDE solvers, ensuring both efficiency and accuracy. Inspired by super-resolution techniques in computer vision, this method maps low-fidelity solutions to high-fidelity targets using a vector-cloud neural network with equivariance (VCNN-e), a neural operator that preserves all necessary invariance and equivariance properties for scalar and vector predictions while seamlessly adapting to different spatial discretizations. We evaluated this approach in three scenarios: (1) a weakly nonlinear case involving low Reynolds number flows around elliptical cylinders, (2) a strongly nonlinear case with high Reynolds number flows over airfoils, and (3) a practical case with high Reynolds number flows over a wing. In all cases, the neural operator-based initialization accelerated convergence by at least two-fold compared to traditional methods, without sacrificing accuracy. The method's robustness and scalability are further demonstrated across different linear equation solvers and multi-process computing configurations. It also achieves overall time savings in scenarios with multiple simulations, even when accounting for model development time. Overall, our approach provides an effective means to accelerate steady-state PDE solutions using neural operators, maintaining high accuracy while significantly improving computational efficiency, particularly in precision-driven scientific applications.