An Implicit Adaptive Fourier Neural Operator for Long-term Predictions of Three-dimensional Turbulence

TL;DR

The paper tackles the challenge of long-term prediction of 3D turbulence with data-driven surrogates. It introduces the implicit adaptive Fourier neural operator (IAFNO), built on AFNO with self-attention in Fourier space and enhanced by an implicit iterative scheme to stabilize deep recursions. Across forced HIT, a temporally evolving mixing layer, and turbulent channel flow, IAFNO outperforms the implicit U-Net enhanced FNO (IUFNO) and dynamic Smagorinsky (DSM) in accuracy and stability, while achieving substantial reductions in parameters ($\sim 1/80$), memory ($\sim 1/3$), and training time (4× faster). These results demonstrate that IAFNO is a scalable, efficient surrogate for LES-like turbulence predictions, with potential for physics-informed enhancements and applicability to complex geometries.

Abstract

Long-term prediction of three-dimensional (3D) turbulent flows is one of the most challenging problems for machine learning approaches. Although some existing machine learning approaches such as implicit U-net enhanced Fourier neural operator (IUFNO) have been proven to be capable of achieving stable long-term predictions for turbulent flows, their computational costs are usually high. In this paper, we use the adaptive Fourier neural operator (AFNO) as the backbone to construct a model that can predict 3D turbulence. Furthermore, we employ the implicit iteration to our constructed AFNO and propose the implicit adaptive Fourier neural operator (IAFNO). IAFNO is systematically tested in three types of 3D turbulence, including forced homogeneous isotropic turbulence (HIT), temporally evolving turbulent mixing layer and turbulent channel flow. The numerical results demonstrate that IAFNO is more accurate than IUFNO and the traditional large-eddy simulation using dynamic Smagorinsky model (DSM), while exhibiting greater stability compared to IUFNO. Meanwhile, the AFNO model exhibits instability in numerical simulations. Moreover, the training efficiency of IAFNO is 4 times higher than that of IUFNO, and the number of parameters and GPU memory occupation of IAFNO are only 1/80 and 1/3 of IUFNO, respectively in HIT. In other tests, the improvements are slightly lower but still considerable. These improvements mainly come from patching and self-attention in 3D space. Besides, the well-trained IAFNO is significantly faster than the DSM. The code (and datasets) of the current work can be accessed at https://github.com/yuchi-richard-jiang/IAFNO.

Figures (20)

• Figure 1: (a) The model constructed with AFNO as the backbone; (b) the architecture of AFNO proposed by Guibas et al. guibas2021adaptive.
• Figure 2: The architecture of IAFNO.
• Figure 3: The velocity spectra in the forced HIT at different time instants: (a) $t/\tau\approx 4.0$; (b) $t/\tau\approx 8.0$; (c) $t/\tau\approx 20.0$; (d) $t/\tau\approx 50.0$.
• Figure 4: The PDFs of the normalized velocity increments $\delta_{r} \bar{u} / \bar{u}^{\textrm{rms}}$ in the forced HIT at at (a) $t/\tau\approx 20.0$; (b) $t/\tau\approx 50.0$. The PDFs of the normalized vorticity $\bar{\omega} / \bar{\omega}^{\textrm{rms}}_{\textrm{fDNS}}$ at (c) $t/\tau\approx 20.0$; (d) $t/\tau\approx 50.0$.
• Figure 5: Temporal evolutions of the velocity rms value and vorticity rms value in the forced HIT.