Particle clustering in turbulence: Prediction of spatial and statistical properties with deep learning

TL;DR

The paper addresses predicting small-scale particle clustering in turbulent flows by learning a mapping from 3D fluid fields to gridded particle fields using a 3D U-Net. Training data come from Epstein-drag Lagrangian particles in isotropic forced turbulence simulated with ATHENA++, providing $ ho_g$, $oldsymbol{v}_g$ as inputs and $ ho_p$, $oldsymbol{v}_{rel}$ as targets on a $256^3$ grid. The authors show the network reproduces filamentary clustering and accurately recovers statistical measures such as the Radial Distribution Function and relative velocity within a few tens of percent, with typical errors around $<10 ext{\%}$ for many regimes. Generalization tests reveal robustness to some changes in driving scales and energy injection but highlight limitations when key physical parameters (notably the stopping time $ au_s$) differ significantly from the training data, underscoring the need for training on diverse parameter sets. Overall, the study demonstrates that deep learning can complement direct numerical simulations by providing fast, 3D predictions of particle clustering and related statistics in turbulent flows, with clear pathways for extending to broader regimes.

Abstract

We investigate the utility of deep learning for modeling the clustering of particles that are aerodynamically coupled to turbulent fluids. Using a Lagrangian particle module within the Athena++ hydrodynamics code, we simulate the dynamics of particles in the Epstein drag regime within a periodic domain of isotropic forced hydrodynamic turbulence. This setup is an idealized model relevant to the collisional growth of micron to mm-sized dust particles in early stage planet formation. The simulation data are used to train a U-Net deep learning model to predict gridded three-dimensional representations of the particle density and velocity fields, given as input the corresponding fluid fields. The trained model qualitatively captures the filamentary structure of clustered particles in a highly non-linear regime. We assess model fidelity by calculating metrics of the density field (the radial distribution function) and of the velocity field (the relative velocity and the relative radial velocity between particles). Although trained only on the spatial fields, the model predicts these statistical quantities with errors that are typically <10%. Our results suggest that, given appropriately expanded training data, deep learning could complement direct numerical simulations in predicting particle clustering within turbulent flows.

Figures (13)

• Figure 1: The architecture of the U-Net. The thickness of individual layers corresponds roughly to the number of channels in each layer and the width and height corresponds to the spatial size. The color scheme corresponds to: convolution layers of kernel size 3 and stride 1 (orange), Leaky ReLU activation with batch normalization (deep orange), downsampling layers using convolution of kernel size 2 and stride 2 (red) and upsampling layers using transposed convolution of kernel size 2 and stride 2 (purple). Residue connection is implemented for all convolution layers except those used in up and down sampling. Similar to other U-Net architectures, large scale information is incorporated by narrow down and concatenating compression layer outputs with the upsampled outputs.
• Figure 2: Particle density field slice $\rho_p$ from ATHENA++ (left), UNET-V (center, a network trained on the relative velocity between particles and gas) and UNET-R (right, a network trained on the absolute particle velocity), for a typical frame. Qualitatively, there is good agreement between the predicted structures and the ground truth, though the network outputs are visually less sharp than the target simulation data.
• Figure 3: Normalized cross correlation of $\rho_p$ of target field (ATHENA++) with itself, UNET-R and UNET-V. The peak values of the statistics are: 1.0, 0.622 and 0.613 respectively.
• Figure 4: Predicted versus simulated strength of particle clustering as a function of spatial scale, quantified via the radial distribution function statistic (RDF). From left to right, the RDF plots show different simulation setups. The left-most panels show the fiducial setup, while the remaining panels show the results of generalization tests. (See §\ref{['sec:forced_turbulence_simulation']},\ref{['sec:lagrangian_dust_particles']} and §\ref{['sec:generalization']} for details.) ATHENA++ shows the RDF computed directly from the simulation, UNET-R is the U-Net output trained with relative velocity between gas and dust, and UNET-V is the U-Net output trained directly on dust velocity. The first row compares the RDF of the simulation output and the U-Net prediction. The second row shows the ratio between the outputs and the target to better quantify the error. The zoom-in plot in the lower-right panel shows the peak of the error at small scales.
• Figure 5: Prediction of the spatial distribution of particle relative velocity ($v_g-v_p$), in domains of isotropic forced turbulence. The figure compares the ground truth, obtained with ATHENA++, with predictions from two deep learning models: UNET-V trained directly with particle velocity $v_p$, and UNET-R with relative velocity $v_{rel}$. The columns represent different setups (ATHENA++, UNET-V, UNET-R) and the rows show the 3 components of the velocity field. Each panel is a slice through the three-dimensional domain.