Local Constraint Neural Operator for Discretization-Invariant Super-Resolution Turbulent Flow Prediction

Resolving small-scale turbulence through fine-grid Direct Numerical Simulation (DNS) remains computationally expensive, making coarse-grid models such as LES and RANS the practical alternative. However, in highly turbulent regimes, these model approaches exhibit significantly large errors from DNS results. Recent studies have explored machine learning methods to overcome such limitations. Among them, Fourier Neural Operator (FNO) offers the discretization invariant framework that can, in theory, extrapolate from coarse grid training data to finer grid test predictions within bounded error. However, in practice, FNO suffers from spectral bias, over-fitting large scales while under-predicting small scales, and its computational cost grows rapidly with increasing resolution.

In this study, we propose a constraint‑based local operator layer that embeds convolutional kernels within the FNO architecture, enforcing locality and retaining the spectral generalization mapping. Our method enables more accurate reconstruction of unseen high-frequency components during super-resolution, despite training on coarse grids. We conducted intensive turbulence statistics evaluation, including energy spectra on isotropic turbulence (periodic boundaries) and Rayleigh–Bénard convection (non-periodic boundaries) datasets. Our study shows that the discretization invariant property of neural operators can be practically applied for turbulence super-resolution, enabling high-resolution reconstructions without additional training.