  In recent years, using neural networks to speed up the solving of partial
differential equations (PDEs) has gained significant traction in both academic
and industrial settings. However, the use of neural networks as standalone
surrogate models raises concerns about the reliability of solutions due to
their dependence on data volume, quality, and training algorithms, especially
in precision-critical scientific tasks. This study introduces a novel
"super-fidelity" method, which uses neural networks for initial warm-starting
in solving steady-state PDEs, ensuring both speed and accuracy. Drawing from
super-resolution concepts in computer vision, our approach maps low-fidelity
model solutions to high-fidelity targets using a vector-cloud neural network
with equivariance (VCNN-e), maintaining all necessary invariance and
equivariance properties for scalar and vector solutions. This method adapts
well to different spatial resolutions. We tested this approach in two
scientific computing scenarios: one with weak nonlinearity, using low Reynolds
number flows around elliptical cylinders, and another with strong nonlinearity,
using high Reynolds number flows over airfoils. In both cases, our neural
operator-based initialization significantly accelerated convergence by at least
two-fold, without sacrificing accuracy, compared to traditional methods. Its
robustness is confirmed across various iterative algorithms with different
linear equation solvers. The approach also demonstrated time savings in
multiple simulations, even including model development time. Additionally, we
propose an efficient training data generation strategy. Overall, our method
offers an efficient way to accelerate steady-state PDE solutions using neural
operators without loss of accuracy, especially relevant in precision-focused
scientific applications.
