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

Abstract

Neural networks have recently emerged as powerful tools for accelerated solving of partial differential equations (PDEs) in both academic and industrial settings. However, their use as standalone surrogate models raises concerns about reliability, as solution accuracy heavily depends on data quality, volume, and training algorithms. This concern is particularly pronounced in tasks that prioritize computational precision and deterministic outcomes. In response, this study introduces “super-fidelity”, a method that employs neural networks for initial warm-starts, significantly speeding up the solution of steady-state PDEs without compromising on accuracy. Drawing from super-resolution in computer vision, super-fidelity maps solutions from low-fidelity computational models to high-fidelity ones using a vector-cloud neural network with equivariance (VCNN-e)—a neural operator that preserves physical symmetries and adapts to different spatial discretizations. We evaluated the proposed method across scenarios with varying degrees of nonlinearity, including (1) two-dimensional laminar flows around elliptical cylinders at low Reynolds numbers, exhibiting monotonic convergence, (2) two-dimensional turbulent flows over airfoils at high Reynolds numbers, characterized by oscillatory convergence, and (3) practical three-dimensional turbulent flows over a wing. The results demonstrate that our neural operator-based initialization can accelerate convergence by at least a factor of two while maintaining the same level of accuracy, outperforming traditional initialization methods using uniform fields or potential flows. The approach's robustness and scalability are confirmed across different linear equation solvers and multi-process computing configurations. Additional investigations highlight its reduced dependence on high quality of training data, and real time savings across multiple simulations, even when including the neural-network model preparation time. Our study presents a promising strategy for accelerated solving of steady-state PDEs using neural operators, ensuring high accuracy in applications where precision is of utmost importance.