Learning Turbulent Flows with Generative Models: From operator-enhanced super-resolution and forecasting to sparse flow reconstructions

Vivek Oommen, Siavash Khodakarami, Aniruddha Bora, Zhicheng Wang, George Em Karniadakis
Brown University
Paper Code

Abstract

Neural operators are promising surrogates for dynamical systems but when trained with standard L2 losses they tend to oversmooth fine-scale turbulent structures. Here, we show that combining operator learning with generative modeling overcomes this limitation. We consider three practical turbulent-flow challenges where conventional neural operators fail: spatio-temporal super-resolution, forecasting, and sparse flow reconstruction. For Schlieren jet super-resolution, an adversarially trained neural operator (adv-NO) reduces the energy-spectrum error by 15x while preserving sharp gradients at neural operator-like inference cost. For 3D homogeneous isotropic turbulence, adv-NO trained on only 160 timesteps from a single trajectory forecasts accurately for five eddy-turnover times and offers 114x wall-clock speed-up at inference than the baseline diffusion-based forecasters, enabling near-real-time rollouts. For reconstructing cylinder wake flows from highly sparse Particle Tracking Velocimetry-like inputs, a conditional generative model infers full 3D velocity and pressure fields with correct phase alignment and statistics. These advances enable accurate reconstruction and forecasting at low compute cost, bringing near-real-time analysis and control within reach in experimental and computational fluid mechanics.

Outline of our Study

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We address three turbulence modeling challenges in this work: (1) super-resolution of low-resolution flow fields in both space and time, (2) forecasting turbulent flow evolution from limited training data, and (3) zero-shot full-field reconstruction from partial observations. We compare vanilla MSE-trained neural operators (NO) against generative models and physics-informed variants, highlighting mitigation of spectral bias. For 3D tasks, we train with limited data, acknowledging the difficulty of obtaining high-fidelity DNS or experimental datasets.

Task 1: Spatio-temporal Super-resolution

Case: Schlieren Visualizations of an Impinging Jet (M=1)

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Spatio-temporal super-resolution. a) Our objective: learn the mapping from low resolution low frame-rate (LRLF) input to high resolution high frame-rate (HRHF) output. b) Schematic of learning setups for conventionally trained neural operator (NO), adversarially trained NO (adv-NO), and NO combined with generative (Gen) models - variational autoencoder (VAE), generative adversarial network (GAN), and diffusion model (DM). Gen is used to super-resolve NO-predicted states. c) Comparing the energy spectrum. d) Spectrum error versus per-sample inference cost. Adv-NO attains low error at orders-of-magnitude lower compute than NO+GAN and NO+DM while outperforming NO and NO+VAE that have similar costs.

Task 2: Forecasting

Case: Homogeneous Isotropic Turbulence (\(Re_{\lambda} = 90\))

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We forecast homogeneous isotropic turbulence for 5 eddy-turnover time scales (\(t_E\)). Only a single simulation trajectory with 160 snapshots was available for training. Physics-informed training reduces the field error, but does not help improve the energy spectrum. Adv-NO achieves a good spectral fidelity till 5\(t_E\).

Task 3: Flow Reconstruction (from PTV-like meassurements)

Case: Turbulent Cylinder Wake (\(Re = 11,000\))

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Here we address a practical problem in experimental fluid mechanics - Reconstructing the full 3D velocity and pressure fields from sparse PIV-like velocity measurements. We compare a diffusion model (DM) and a generative adversarial network (GAN) in this study. First we train on the DNS of the system. The training dataset consist of a single simulation trajectory of 150 snapshots. Test Setup 1 - mimics volumetric PIV. Test Setup 2 - mimics planar PIV. Test Setup 3 - represents volumetric PIV in a sub-domain. Test Setup 4 - pressure reconstruction. Under extremely sparse velocity measurements, when GAN fails, the diffusion model achieves phase-alignement near the observation region and obeys the global statistics.

BibTeX

@article{oommen2025learning,
      author={Oommen, Vivek and Khodakarami, Siavash and Bora, Aniruddha and Wang, Zhicheng and Karniadakis, George Em},
      title={Learning Turbulent Flows with Generative Models: Super-resolution, Forecasting, and Sparse Flow Reconstruction},
      journal={arXiv preprint arXiv:2509.08752},
      year={2025}
    }