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In recent years, using neural networks to speed up the solving of partialdifferential equations (PDEs) has gained significant traction in both academicand industrial settings. However, the use of neural networks as standalonesurrogate models raises concerns about the reliability of solutions due totheir dependence on data volume, quality, and training algorithms, especiallyin precision-critical scientific tasks. This study introduces a novel"super-fidelity" method, which uses neural networks for initial warm-startingin solving steady-state PDEs, ensuring both speed and accuracy. Drawing fromsuper-resolution concepts in computer vision, our approach maps low-fidelitymodel solutions to high-fidelity targets using a vector-cloud neural networkwith equivariance (VCNN-e), maintaining all necessary invariance andequivariance properties for scalar and vector solutions. This method adaptswell to different spatial resolutions. We tested this approach in twoscientific computing scenarios: one with weak nonlinearity, using low Reynoldsnumber flows around elliptical cylinders, and another with strong nonlinearity,using high Reynolds number flows over airfoils. In both cases, our neuraloperator-based initialization significantly accelerated convergence by at leasttwo-fold, without sacrificing accuracy, compared to traditional methods. Itsrobustness is confirmed across various iterative algorithms with differentlinear equation solvers. The approach also demonstrated time savings inmultiple simulations, even including model development time. Additionally, wepropose an efficient training data generation strategy. Overall, our methodoffers an efficient way to accelerate steady-state PDE solutions using neuraloperators without loss of accuracy, especially relevant in precision-focusedscientific applications.

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