Predicting Time-Dependent Flow Over Complex Geometries Using Operator Networks

TL;DR

The paper tackles the challenge of fast, geometry-generalizing surrogates for unsteady incompressible flow by introducing a time-dependent, geometry-aware DeepONet that encodes geometry via a signed distance field and recent flow history via a CNN branch. Trained on 1,103 FlowBench Flow Past Object simulations, the approach achieves ~5% relative L2 error in single-step predictions and delivers up to 1000x CFD speedups, with physics-centric diagnostics validating short-term fidelity. Rollouts reveal accurate near-term transients but show systematic error accumulation over long horizons, especially for geometries with sharp corners, highlighting stability limits and areas for improvement. The work provides detailed ablations, diagnostic metrics, and public code/data to support reproducibility and benchmarking in geometry-rich unsteady CFD surrogates.

Abstract

Fast, geometry-generalizing surrogates for unsteady flow remain challenging. We present a time-dependent, geometry-aware Deep Operator Network that predicts velocity fields for moderate-Re flows around parametric and non-parametric shapes. The model encodes geometry via a signed distance field (SDF) trunk and flow history via a CNN branch, trained on 841 high-fidelity simulations. On held-out shapes, it attains $\sim 5\%$ relative L2 single-step error and up to 1000X speedups over CFD. We provide physics-centric rollout diagnostics, including phase error at probes and divergence norms, to quantify long-horizon fidelity. These reveal accurate near-term transients but error accumulation in fine-scale wakes, most pronounced for sharp-cornered geometries. We analyze failure modes and outline practical mitigations. Code, splits, and scripts are openly released at: https://github.com/baskargroup/TimeDependent-DeepONet to support reproducibility and benchmarking.

Figures (14)

• Figure 1: (a) Representative snapshots from the FlowBench FPO dataset, illustrating vortex‐shedding behind four representative shapes. (b) Time-dependent Geometric DeepONet surrogate model: the branch network in Stage-1 process $N_t$ velocity frames through parallel convolutional streams (Inception‐style CNN) fed into an MLP, while the trunk network encodes spatial $(x,y,\mathrm{SDF})$ via another MLP. These are fused element‐wise, passed through a Stage‐2 branch MLP (ReLU) and trunk MLP (sine), and finally contracted to predict the next‐step velocity field.
• Figure 2: Time‐evolution of single‐step versus rollout prediction errors for varying input sequence lengths. Panels (a) and (b) plot the relative $L_2$ and $L_\infty$ errors over time using an input sequence of length $s=1$. Panels (c) and (d) show the same metrics for $s=4$; panels (e) and (f) for $s=8$; and panels (g) and (h) for $s=16$.
• Figure 3: Single‐step prediction of flow velocity for an example geometry. (a) Relative $L_2$ error over time. (b) RMSE of $u$ and $v$ over time. (c–e) Ground‐truth $u$ at $t = 30, 45, 59$. (f–h) Predicted $u$ at $t = 30, 45, 59$. (i–k) Ground‐truth $v$ at $t = 30, 45, 59$. (l–n) Predicted $v$ at $t = 30, 45, 59$. Colorbars for $u$ are shown in (e) and (h), and for $v$ in (k) and (n).
• Figure 4: Single‐step predictions for four example geometries at $t = 30$. Each pair of rows corresponds to one shape: the top row shows the $u$‐component and the bottom row the $v$‐component.
• Figure 5: Single step time‐series of $u$ and $v$ at two points at downstream distance from geometry $x=1D$ and $x=2D$, where D is the geometry diameter. We show a collection of 4 shapes where each row corresponds to a single geometry.