Fourier neural operators for spatiotemporal dynamics in two-dimensional turbulence

TL;DR

This work tackles the computational burden of high-fidelity turbulence simulations by proposing a physics-guided hybrid emulator that couples Fourier neural operators with a PDE solver to preserve physicality during long-time predictions. It systematically compares 2D FNO with temporal channels, 3D FNO, and a Hybrid FNO-PDE approach, using a dataset of 5000 2D decaying-turbulence simulations generated by lattice Boltzmann methods. The key finding is that hybrid FNO-PDE yields stable long-term predictions, while pure ML approaches diverge, and a 2D FNO with temporal channels offers the best speed–accuracy balance for the studied setting. The study highlights the importance of incorporating governing physics and temporal structure to enable scalable, accurate turbulence surrogates with potential impact on climate and atmospheric modeling where long-time forecasts are essential.

Abstract

High-fidelity direct numerical simulation of turbulent flows for most real-world applications remains an outstanding computational challenge. Several machine learning approaches have recently been proposed to alleviate the computational cost even though they become unstable or unphysical for long time predictions. We identify that the Fourier neural operator (FNO) based models combined with a partial differential equation (PDE) solver can accelerate fluid dynamic simulations and thus address computational expense of large-scale turbulence simulations. We treat the FNO model on the same footing as a PDE solver and answer important questions about the volume and temporal resolution of data required to build pre-trained models for turbulence. We also discuss the pitfalls of purely data-driven approaches that need to be avoided by the machine learning models to become viable and competitive tools for long time simulations of turbulence.

Figures (9)

• Figure 1: Mean [top row], standard deviation [middle row], and Frobenius norm [bottom row] of raw vorticity ($\Omega$) [left column] and normalized vorticity ($\tilde{\Omega}$) [right column]. The normalization is with respect to mean and standard deviation from $t=0$. Each curve represents a different sample from the data set of 5000 samples.
• Figure 2: $L_2$ norm of difference of vorticity fields of ten samples with their respective initial values.
• Figure 3: Normalized projection of vorticity fields of the sample data sets on their respective initial values.
• Figure 4: Lyapunov exponents calculated for the two components of the velocity vector.
• Figure 5: Different number of channels with width 8 (top) and width 40 (bottom) while the other hyperparameters are: layers (4), modes (32), scheduler gamma (0.5), scheduler step (100), and learning rate (0.001).