Generative AI in Turbulence: Foundations, Applications, Validations and Benchmarks

Luca Biferale; Michele Buzzicotti

Figure 1: Top left: Artistic reinterpretation of the results presented in ( Mayer et al., 2010 ) where an elephant contour is coded in a set of four complex parameters, the real part of the 5th parameter is the “wiggle” parameter, a demonstration of von Neumann’s statement cited in the main text. Top right: picture obtained from a Generative AI algorithm. Bottom table: comparison of three epistemologies: physical laws (rationalist, theory-driven), animal brains (strong innate priors and efficient learning), and deep neural networks (data-driven, highly empirical). While physics relies on compact theories and biological brains on structured priors, modern ML learns primarily from large datasets with weak priors and high computational costs, representing a distinct “third epistemology” of knowledge generation ( Mitra, 2021 )

Figure 2: The Big-4.

\#1

: Scale separation and effective dimensionality. 3D Homogeneous and Isotropic Turbulence (HIT), is characterized by two properties. First, the development of power-law spectra,

E(k)\sim k^{-5/3+\delta}

, within two cutoffs wavenumber:

k_{f}

, the infrared scale where energy is injected, and

k_{\nu}

, the ultraviolet scale where energy is dissipated ( Frisch, 1995 ; Pope, 2001 ) . In between, the inertial terms, given by self advection

v_{i}\partial_{i}v_{j}

and pressure stress,

\partial_{i}p

, are leading and a power law distribution of energy per wavenumber is observed, with a slope close to the Kolmogorov prediction

5/3

. Statistical stationary imposes a balance between the energy injection,

\epsilon_{in}

and the energy dissipation,

\epsilon_{out}

. Second, the ratio between the IR an UV cutoffs grows with the Reynolds number

k_{\nu}/k_{f}\sim Re^{3/4}

, where the Reynolds number is given by the dimensionless ratio among non-linear and linear terms in the Navier-Stokes equations,

Re\sim U_{0}L_{0}/\nu

. Here,

U_{0}

and

L_{0}

are the characteristic velocity and length scales of the flow and

\nu

is the kinematic viscosity. The presence of a power-law inertial range with an exponent

<3

is the indication of the development of a highly intricate set of hierarchical fluctuations (see Fig. 3 ) which result in a quasi-singular velocity profile, never differentiable (see also Big-4 #3 in Fig. 5 ). Replacing typical Reynolds number observed in real-life turbulent flows (from

Re\sim 10^{6}

for turbo-machinery up to

Re\sim 10^{10}

in the atmosphere, and more in the sky) one easy realizes that 3D HIT turbulence is to all possible meaning an ”infinite”

\#_{dof}

problem, requiring a number of

\sim Re^{9/4}

to fully resolve all scales in between

k_{\nu}

and

k_{f}

. This is a discontinuous difference from 2D flows, where the Energy spectrum goes in the UV as

k^{-y}

, with

y\geq 3

( Boffetta & Ecke, 2012 ) leading to a smooth and differentiable small-scale flow, with much less energy content in the high wavenumber range (see also Fig. 7 for more info about 2D set-ups). Other 3D configurations, e.g. wall bounded flows, are even more complicated than HIT, because of anisotropic and non-homogeneous dissipative, multi-scale statistics, which depends on the distance from the wall, as well as boundary layer physics dominated by the simultaneous presence of production and dissipative terms ( Pope, 2001 ) .

Figure 3: The Big-4

\#1

(continued): Three different zoomed (

\times 10

) rendering of the vortex tube structure observed at scales within the inertial range of 3D HIT turbulence (Fig. 2 ), representing the nested hierarchy of fluctuations characterising 3D turbulence. Frames extracted from ( Johns Hopkins Turbulence Databases, 2026 ) , permission kindly provided by C. Meneveau, see also Bürger et al. (2013) . It is unknown how to fully characterize the distribution of these huge number of dof , and it is an unresolved problem to develop reliable Reduce Order Models (ROM) that fully describe the intricate set of sub-grid-scale fluctuations, see also the other 3 Big-4 properties described in Figs. 4 - 6 . A huge body of applied literature is devoted to the development of Large-Eddy-Simulations (LES) approaches, modeling small-scales fluctuations ( Meneveau & Katz, 2000 ) . Other attempts to compress the

\#_{dof}

in a smaller support have shown that blind decimation might fully destroy important flow features already by removing fraction of percent of data ( Alexakis & Biferale, 2018 ) , a bad news concerning possibility to compress the information in turbulence. One of the main obstacles for applications of ML is given by the huge size of any input data (a minimum of billions of data point are needed to be representative of the minimal flow complexity at moderate Reynolds) and the lack of data (no one has thousands, or hundred-thousands, uncorrelated samples for training). Moreover, the huge set of hierarchically nested scales and frequencies involved, implies a non trivial importance of different multi-scale features for different observable, questioning the possibility to identify a unique set of optimal metrics for training and validation. As a result, none is able to satisfactory generate, using data-driven or other stochastic phenomenological model, a 3D configuration of a turbulent flow with billions or trillions of grid points, as required for moderate Reynolds number flows. For the latter task the only fully reliable way is still to use the Navier-Stokes equations.

Figure 4: The Big-4.

\#2

: Big-tails, intermittency, extreme and rare events, non-perturbative and non-Gaussian fluctuations . Left: 3D Eulerian and Lagrangian turbulence is characterized by strong deviations from Gaussianity and the presence of anomalous scaling Alexakis & Biferale (2018) . The most striking properties being the development of fat tails (fater and fater by increasing Reynolds) as shown by the standardized PDF of the flow acceleration on the left panel, where extreme intense fluctuations up to 50-60 standard deviations can be easily detected. In the figure, red data comes from a DNS of HIT at

1024^{3}

resolution points. The black curve corresponds to the prediction obtained by a global homogeneous

h=1/3

scaling exponent for velocity increments as predicted by the mean field Kolmogorov 1941 theory (K41). The strong discrepancy between K41 theory and the ground truth data is known to be connected to multifractal small-scales properties ( Frisch, 1995 ; Biferale et al., 2004 ) , see also Fig. 5 . Intermittency and fat tails are a key fingerprints of all multi-scale turbulent flows with and without boundaries at small enough scales (in 3D) and must be considered as a primary feature to be reproduced. They emerge only going beyond spectral properties , observing third-order energy or helicity fluxes and high-order moments, as, e.g., Flatness and generalised Flatness, see ( Frisch, 1995 ) and right panel. Due to these strong fluctuations, it is not uncommon to observe extreme events at the gradient scale as large as the global large-scale rms signal, as it happens for cold/warm fronts in atmosphere physics. As a results, small-scales turbulent fluctuations cannot be put under the carpet of the low-energy-content , even if one is interested only to large-scale smooth evolutions, because it might become locally

O(1)

and disruptive for global control/predictability. Any ROM or data-driven approach which limits its benchmarks to spectral properties and/or low order moments might therefore be blatantly wrong concerning these important effects. Right: An example of 4th order Flatness,

F_{4}(\tau)

, defined in terms of the structure functions of single velocity components,

S_{p}(\tau)

measured along Lagrangian particles evolution in 3D turbulence at changing the time lag

\tau

. Showing the relative increase of non-Gaussian (intermittent) fluctuations by decreasing the time lag in the inertial range, and the tendency to go from a Gaussian value

\sim 3

, for low frequency fluctuations, till very high intensities for

\tau\to 0

, close to the inverse of the viscous Kolmogorov time,

\tau_{\eta}

.

Figure 5: The Big-4.

\#3

: Rough velocity profiles, loss of regularity and Multifractal fluctuations . Eulerian velocity profile (one component): pictorical description of one-dimensional cut of the 3D HIT configuration shown in Fig ( 3 ), with two consecutive zooms. The presence of power-law Fourier spectra is connected to fractional Brownian-like profiles in the real space. Velocity increments in the inertial range of scales, between the IR and UV cutoffs,

1/k_{\nu}\ll rll1/k_{f}

, do show anomalous power-law behavior in fully developed 3D turbulence ( Frisch, 1995 ) . Actual understanding supports a multifractal picture, i.e. the existence of a set of quasi-singularities characterizing the local scaling properties (Holder exponent) of generic velocity increments,

\delta_{r}v=v(x+r)-v(x)\sim r^{h(x)}

, with

h_{min}\leq h\leq h_{max}

and the existence of different fractal dimensions characterizing the support of each exponent

D(h)

( Frisch, 1995 ) . Multifractals are connected to anomalous scaling for the velocity structure functions:

S_{p}(r)=\langle|\delta_{r}v|^{p}\rangle\sim r^{\zeta(p)}

, with a non-linear dependency of

\zeta(p)

on

p

. As a result, also the Eulerian generalised Flatness of order

p

,

F_{p}(r)=S_{p}(r)/S_{2}(r)^{p/2}\sim r^{\zeta(p)-p/2\zeta(2)}

developed a scale-dependency. On the contrary, Kolmogorov theory predicts

h=1/3

everywhere, mono-fractal and self-similar scaling properties,

\zeta(p)=p/3

, a constant generalised Flatness, and the spectral slope

5/3

. Nowadays, we have more than 40 year of observations from experiments and direct numerical simulations supporting the more complex MF scenario, which is strictly needed to explain the growth of fat tails shown in Figure 4 . Notice, that the limit

Re\to\infty

corresponds to send the UV cutoffs to zero

r_{\nu}=1/k_{\nu}\to 0

, i.e. leading to a quasi-singular multifractal velocity profile, never differentiable almost everywhere. This is another qualitative discontinuity from 2D turbulence, where one can prove the regularity (smoothness) of solutions for all

Re

.

Figure 6: The Big-4.

\#4

: Lagrangian (multiphase) turbulence. Long-time enhanced diffusivity, small-scales preferential concentration and intrinsic stochastic transport . The quasi-singular velocity profile present in fully developed turbulence (Fig. 5 ) leads to an extremely complex dispersion and spatial distributions of (small) particles ( Toschi & Bodenschatz, 2009 ) . Including the presence of Richardson like diffusivity (left panel), where clouds of particles can explosively disperse at large scales even if initially almost superposing, as soon as they reach the smallest distance where the velocity field exhibit rough-scaling properties (see Fig. 5 ) A phenomenon known since the time of Lorenz, and recently baptized spontaneous stochasticity ( Lorenz, 1969 ; Falkovich et al., 2001 ; Thalabard et al., 2020 ; Bandak et al., 2024 ) . Small-scales mass distribution exhibits clustering and inertia-dependent, multi-scale multi-fractal properties (right panel). As a result, generating and/or inferring particles distributions in 3D turbulence is a challenge even more formidable then the one for the underlying Eulerian fields, not to mention the requirements to work on a mesh-less domain (see section 3.1 ) for more details on AI interesting perspectives.

Figure 7: Not all 2D turbulent flows are equally informative as ML benchmarks. Top row: Representative velocity-field snapshots for four canonical two-dimensional regimes: Kolmogorov flow in a direct-cascade regime, homogeneous isotropic turbulence in the inverse-energy-cascade regime, quasi-geostrophic turbulence, and bulk fluctuations in two-dimensional Rayleigh–Benard convection. The snapshots highlight substantial differences in spatial regularity: Kolmogorov flow is comparatively smooth, whereas the inverse-cascade, convective, and quasi-geostrophic configurations exhibit rougher, multiscale structures. Bottom row: Energy spectra for the different flow realizations, horizontally aligned with respect to their forcing scale to facilitate a direct comparison of their ultraviolet and infrared behaviour. The spectral slopes quantify the qualitative differences visible in the snapshots. Kolmogorov flow displays the steepest high-wavenumber decay very fast,

\sim k^{-5}

, implying strongly suppressed small-scale fluctuations and a smooth, differentiable velocity field. By contrast, inverse-cascade turbulence and quasi-geostrophic turbulence exhibit rough-velocity configurations and spectra close to the 3D K41 scaling

\sim k^{-5/3}

, while Rayleigh–Benard convection shows an approximate Bolgiano scaling

\sim k^{-11/5}

, also leading to quasi-singular velocity profiles with

\delta_{r}v\sim r^{6/5}

in the range with constant energy flux. The spectrum of two-dimensional HIT spanning both inverse-energy and direct-enstrophy cascade ranges is also shown, with an ultraviolet decay close to

k^{-3}

as predicted by Kraichnan ( Alexakis & Biferale, 2018 ) . The superposition of all spectra makes clear that although two-dimensional turbulence can provide physically meaningful and challenging test cases, the commonly used (by AI practitioners) Kolmogorov-flow is the smoothest and least demanding end of these paradigmatic examples and does not provide a strong benchmark for typical challenges of fully developed turbulence. Data were kindly provided by G. Boffetta (2D double cascade), F. Fossella (Convection) and V. Valadao (SQG).

Figure 8: Adapted with permission from ( Frisch, 1995 ) ; original image courtesy of Hélène Frisch. The original illustration, the celebrated “Guide de l’alpiniste turbulent” (Route to the Navier-Stokes peak) was drawn by P. Delache, where the challenge of reaching the infinite-Reynolds-number limit is represented as the ascent of a difficult mountain. The original sketch depicts several classical routes used to study turbulence: the direct mathematical ascent, the renormalization-group path, the experimental route through cascades and intermittent torrents, and the numerical-simulation rock-climbing school, each with its own obstacles, approximations and partial successes. In the present version, we add a new route: the AI helicopter, represented as a black box, collecting turbulent data in the ’cloud’ with a question mark. This “neural helicopter” symbolizes the promise of modern data-driven and generative methods to bypass some traditional bottlenecks. At the same time, its black-box nature and unexplained shortcut emphasize the central caution of this Perspective: AI may offer a powerful new way to approach high-Reynolds-number turbulence, but its ability to “land” on physically meaningful statistics must be assessed through turbulence-specific validation.

Figure 9: Generative modelling across heterogeneous data representations. Schematic examples illustrating the flexibility of modern generative models across markedly different classes of high-dimensional data. From top to bottom: two-dimensional images on regular pixel grids; point-cloud objects represented by irregular sets of spatial coordinates; turbulent Lagrangian trajectories described as time-dependent particle paths; and two- and three-dimensional turbulent flow fields defined on structured spatial domains. In each case, the generative process starts from a simple noise distribution and progressively transforms it into samples with the geometry, correlations, and statistical structure characteristic of the target dataset. The comparison highlights that the underlying generative principle is largely independent of the particular data representation. At the same time, this flexibility does not imply methodological uniformity: both the neural architecture and the benchmarks used to assess generative quality must be adapted to the geometry of the data and to the physically relevant observables of each problem.

Figure 10: Top. Schematic phase-space defined by increasing physical and data complexity. The dashed curve denotes a heuristic frontier of ML applicability at the time of writing. Its non-convex shape reflects the uneven, problem-specific optimization efforts of the ML community, which have advanced selected applications beyond what their nominal complexity alone would suggest. Problems shown in yellow are those for which ML methods have demonstrated established utility and, in some cases, advantages over conventional approaches. Orange-colored problems correspond to regimes in which ML remains comparatively less explored or has not yet been validated and bench-marked sufficiently for quantitative advancements. Canonical low-dimensional systems and simplified 1D and 2D flows, including Lorenz and KS dynamics, 1D and 2D Burgers equations, Kolmogorov flow, and 2D HIT in the direct cascade regime form the principal proof-of-concept region. Fully developed 3D HIT, wall-bounded and convective turbulence, and geophysical flows involving rotation, stratification, or multiphase effects occupy the high-complexity region. 2D HIT in the inverse cascade regime is somehow in between due its reduced complexity wrt 3D case ( Boffetta & Ecke, 2012 ) . 3D problems may nevertheless approach the current ML frontier when restricted to reduced-order descriptions, LES or RANS closures, local super-resolution, data assimilation, or small spatial subdomains. Lagrangian turbulence ( Toschi & Bodenschatz, 2009 ) occupies a distinct position because, for a moderate number of tracked particles, its data complexity can remain relatively low even when the physical complexity is comparable to that of 3D HIT. ROMs span a broad range, from low-dimensional Lorenz-type systems to physically complex Shell Models ( Frisch, 1995 ; Bohr et al., 1998 ; Biferale, 2003 ) . Bottom: qualitative comparison of selected problem classes with respect to the sources of complexity discussed in Figs. 2 - 6 . YES/NO ranking are subjected to specific conditions and do not apply necessarily to all possible flow configurations for each row.

Figure 11: Unified view of generative modelling as probability transport in high-dimensional data space. The target data distribution,

p_{\mathrm{data}}(x)

, shown on the right as a highly structured and multimodal probability landscape, is connected to a simple reference distribution, typically an isotropic Gaussian, shown on the left. Different generative architectures realize this connection in distinct ways. Variational Autoencoders (VAE) and Generative Adversarial Networks (GAN) construct an implicit mapping between latent and data spaces, whereas Normalizing Flows use a sequence of deterministic and invertible transformations, allowing the probability density to be tracked explicitly along the transport path. Diffusion models instead introduce a progressive stochastic deformation of the data distribution: in the forward direction (from right to left), probability mass is gradually dispersed until the structured data distribution approaches Gaussian noise; in the generative, reverse -backward- direction (form left to right), samples initialized from the Gaussian reference are transported back toward the data manifold. In discrete-time diffusion models, this evolution is represented by a finite sequence of noisy states,

x_{t}

, while continuous-time Score-based models describe it through a stochastic differential equation. The coloured surface at the bottom provides a schematic representation of the probability density encountered along the transport process. At each intermediate time, the score field,

s(x,t)

points toward regions of increasing probability and supplies the additional drift required to reverse the diffusive spreading. Learning this time-dependent vector field therefore makes it possible to steer -during the backward process- random trajectories from the simple reference measure toward the multiple high-probability modes of the target distribution without explicitly evaluating the generally intractable normalization constant of the underlying density. Flow-matching and related probability-flow formulations offer a deterministic counterpart to this construction by learning directly the velocity field that transports probability mass between the same endpoints. The trajectories displayed in the figure should thus be interpreted as alternative stochastic or deterministic paths in probability space.

Figure 12: Schematic representation of the forward and backward processes underlying score-based generative models. Top: The forward Ornstein-Uhlenbeck process progressively perturbs samples drawn from the data distribution (here taken from a passive scalar advected by a 2d inverse cascade turbulent flow, courtesy of C. Calascibetta),

p_{\mathrm{data}}(x)

, driving the corresponding marginal probability density

p(x,t)

toward the attracting Gaussian distribution

\mathcal{N}(0,I)

in the long-time limit. The forward transition probability,

p(x,t|x_{0},0)

, is analytically known, while the marginal distribution is obtained by averaging over the unknown data distribution. Bottom: The generative process follows the corresponding reverse-time Langevin dynamics, initialized from the Gaussian prior. If learning succeed, the learned score function,

s_{\theta}(x,t)\approx\nabla_{x}\log p(x,t)

, enters the drift and drives the stochastic dynamics through the sequence of forward marginals in reverse order, ultimately recovering the target data distribution.

Figure 13: Schematic representation of the reverse diffusion process in probability space. The colored surfaces represent the time-dependent marginal probability density

p(x,t)

along the diffusion path, interpolating between the Gaussian prior

\mathcal{N}(0,I)

and the target data distribution

p_{\mathrm{data}}(x)

. Starting from independent realizations sampled from the Gaussian prior, the reverse-time Langevin dynamics generates stochastic trajectories that progressively follow the evolving probability landscape toward regions of high probability of the data distribution.

Figure 14: Schematic representation of the Forward noising and learned Backward denoising processes in a discrete-time diffusion model. The forward process, from right to left, is prescribed (not learned): at every step, a small amount of Gaussian noise is added according to a chosen variance schedule, gradually destroying the fine structure and statistical correlations of the original data. Intermediate states

x_{t}

therefore contain different levels of signal and noise, while the full sequence

x_{0},\dots,x_{T}

provides a discrete approximation to the continuous deformation of probability densities introduced in the score-based formulation (see Fig. 12 ). Generation proceeds through the backward process, from left to right. Starting from an independently sampled Gaussian realization

x_{T}\sim N(0,I)

, a neural network parameterizes the reverse conditional distributions,

q_{\theta}(x_{t-1}|x_{t})

and iteratively removes the estimated noise to reconstruct increasingly structured states. The reverse transition is not obtained by simply undoing the particular noise realization used in the forward process. Instead, the model learns, from the training data, the statistical rule required to map a noisy state

x_{t}

onto a distribution of plausible less-noisy states

x_{t-1}

. Repetition of this stochastic denoising step eventually produces a new realization

x_{0}

sampled from the learned approximation to

p_{\mathrm{data}}

(see Sec. 2.4 for a discussion about the memorization/generalization transitions). The central pair of states illustrates the local learning problem at a generic diffusion step. As discussed in the text, for Gaussian forward kernels, noise prediction and conditional-score estimation are directly related; hence, learning to denoise at all noise levels provides the information needed to approximate the full reverse Markov chain. The intermediate arrows emphasize that practical models employ many intermediate steps, so that the complex transport between the data and Gaussian distributions is decomposed into a sequence of comparatively simple local transformations.

Figure 15: Schematic representation of the generalization-memorization transition in diffusion models and its characterization through maximum cosine similarity, introduced in Eq. ( 60 ). Left: Qualitative phase diagram of the learning dynamics as a function of the number of training backpropagation steps, the number of training samples

N

, for a fixed model size. At early training stages, generated samples are dominated by noise; at intermediate times, the model enters a generalization regime in which it reproduces the statistical features of the data while generating novel realizations; after long training time, assuming a model size large enough, optimization eventually drives the model toward memorization of individual training samples. Representative generated trajectories

x^{\mathrm{gen}}

and their closest training samples

a^{\mu}

illustrate the three regimes along a fixed-

N

section of the diagram. Right: Probability density of the maximum cosine similarity

S_{\max}(x^{\mathrm{gen}})

between each generated sample and the training dataset. In the noise regime, generated and training samples have negligible similarity. In the generalization regime, the distribution of

S_{\max}

approaches the intrinsic nearest-neighbor similarity between distinct samples of the training set (black curve), indicating that generated samples are statistically comparable to, but do not reproduce, individual training examples. In the memorization regime, the distribution instead concentrates close to

S_{\max}=1

, revealing the near-exact reproduction of training samples.

Figure 16: Guided diffusion for reconstruction of missing turbulent data. Top left: example of a Lagrangian velocity trajectory along tracers in which a central time interval is unobserved (gap), while the measurements

Y

are available on the two sides of the gap. Top right: schematic of the guided backward dynamics, where the usual score term driving unconditional generation is supplemented by the gradient of the likelihood,

\nabla_{x}\log p(Y|x,t)

, which enforces consistency with the available observations ( Li et al., 2025 ) . Bottom: evolution of two independent guided samples during the backward process, from two different initial Gaussian statse to the final reconstructed trajectories. Here, the backward process is parametrized in terms of the forward diffusion time

t

, spanning in the reverse direction from

t=T

to

t=0

during generation. The observed regions progressively constrain the generated signal, while the missing interval is filled with statistically plausible fluctuations compatible with both the learned prior and the measurements. The black curve denotes the reference ground truth, while the colored curves show the generated trajectories during the reconstruction process.

Figure 17: Schematic representation of conditional reverse diffusion in probability space. Following the notation adopted in this section, the reverse dynamics is parametrized using the forward diffusion time

t

, which evolves from

T

to

0

during generation, rather than introducing the backward time variable

\tau=T-t

. The gray surfaces represent the unconditional marginal probability densities

p(x,t)

along the diffusion path (as shown in Fig. 13 ), while the colored surfaces show the corresponding conditional densities

p(x,t|Y)

obtained after imposing the linear constraint

Y:=x_{1}+x_{2}=1

. Starting from realizations sampled from the Gaussian prior

\mathcal{N}(0,I)

, the additional likelihood score

\nabla_{x}\log p(Y|x,t)

modifies the reverse-time drift and guides the stochastic trajectories toward regions of probability space compatible with the prescribed constraint, see ( Li et al., 2026a ) for an application to 3D rotating turbulence, imposing incompressibility. As the reverse dynamics proceeds, the conditional probability progressively concentrates around the manifold

x_{1}+x_{2}=1

, indicated by the dashed line, ultimately sampling the conditional data distribution

p_{\mathrm{data}}(x|Y)

. The comparison between the gray and colored probability paths illustrates how likelihood guidance restricts the unconditional generative process to the subset of the learned data distribution consistent with the imposed constraint.

Figure 18: Physics-aware metrics: beyond visual plausibility.

Figure 19: Conditional generation of three-dimensional turbulent channel flow with CoNFiLD. Left: Inpainting of missing flow information. Instantaneous velocity-magnitude fields are shown for the reference DNS (label data), artificially damaged inputs containing a masked region, and two independent CoNFiLD reconstructions at several time instants. The lower panels compare the probability density function of the velocity magnitude and representative wall-normal profiles, showing that the recovered fields reproduce the instantaneous structures and the single-point statistics of the reference flow despite the missing data. Right: Super-resolution of turbulent channel flow from progressively coarsened inputs. CoNFiLD reconstructs high-resolution spatio-temporal fields from low-resolution observations going from

64\times 16

down to

4\times 1

. The corresponding energy spectra show that the generated fields recover a substantial fraction of the unresolved small-scale content and remain closer to the DNS spectra than standard bi-cubic interpolation. The figure illustrates the application of conditional latent diffusion to different inverse problems, including data restoration and super-resolution, within the same generative framework. In perspective, future benchmarking against more specific turbulent metrics (see Fig. 18 ) will unlock the tool’s potential across a broader range of applications, if successful. Reproduced from Fig. 7 and Fig. 8 (panels b and c) of Du, Pan; Parikh, Meet Hemant; Fan, Xiantao; Liu, Xin-Yang; Wang, Jian-Xun. 2024 CoNFiLD: Conditional neural field latent diffusion model generating spatiotemporal turbulence. arXiv:2403.05940v2 under CC BY license (http://creativecommons.org/licenses/by/4.0/), see ( Du et al., 2024b ) .

Figure 20: (Panel a) Long-horizon vorticity predictions obtained with the autoregressive DM. The top row correspond to a sequence from 2D incompressible wake flow at

Re=1000

, while the bottom row correspond to a sequence from the transonic flow at

Re=10k

and Mach number

Ma=0.5

. (Panel b) Spatial-frequency spectra evaluated along a vertical line located one cylinder diameter downstream, for (left) incompressible wake flow and for the transonic flow (right). The predicted spectra are computed and compared against baselines from the mean flow obtained by averaging over a long rollout horizon. View in perspective, successful benchmarking against more specific turbulent metrics (see Fig. 18 ) will potentially unlock the tool for a broader set of empirical applications. Reproduced from Fig. F1 (data for ACDM only) and Fig. F2 of Kohl, Georg; Chen, Li-Wei; Thuerey, Nils. 2026 Benchmarking autoregressive conditional diffusion models for turbulent flow simulation . Neural Networks p. 108641 under CC BY license (http://creativecommons.org/licenses/by/4.0/), see ( Kohl et al., 2026 ) .

Figure 21: Generative modelling of Lagrangian turbulence. Comparison between DNS and diffusion-model (DM) results for single-particle Lagrangian statistics and trajectories. (Left) Representative three-dimensional particle trajectories, with the local acceleration magnitude highlighted along selected portions of the paths. (Right) Probability density functions of normalized Lagrangian velocity increments,

\delta_{\tau}V_{i}(t)=V_{i}(t+\tau)-V_{i}(t)

, for different time lags from

\tau/\tau_{\eta}=100

down to

1

. Reproduced from Fig 1 (panels a and f) of Li, Tianyi; Biferale, Luca; Bonaccorso, Fabio; Scarpolini, Martino Andrea; Buzzicotti, Michele. 2024 Synthetic lagrangian turbulence by generative diffusion models . Nature Machine Intelligence 6 (4), 393–403 under CC BY license (http://creativecommons.org/licenses/by/4.0/), see ( Li et al., 2024a ) .

Figure 22: Generative modelling of mesh-based simulations with diffusion graph networks.. Top: examples illustrating how diffusion graph networks (DGN) and latent diffusion graph networks (LDGN) can generate distributions of states for mesh-based simulations conditioned on the mesh geometry and physical parameters. The benchmarks include (a) the pressure distribution on an ellipse in two-dimensional flow, (b) the corresponding velocity and pressure fields, and (c) the pressure distribution on a wing in three-dimensional turbulent flow. Bottom: representative samples for two configurations from the Wing-Test dataset, comparing the ground truth with predictions generated by the graph-mesh GNN (GM-GNN), DGN, and LDGN models. The comparison illustrates the ability of the generative graph-based approaches to reproduce spatially heterogeneous pressure distributions on complex three-dimensional geometries. In perspective, more extended benchmarking against turbulent specific metrics (see Fig. 18 ) will make these tools available for a broader range of applications, if successful. Reproduced from Fig. 3 (top panels a,b,c) and Fig. 14 (bottom panels a,b) of Lino, Mario; Pfaff, Tobias; Thuerey, Nils. 2025. Learning distributions of complex fluid simulations with diffusion graph networks arXiv: 2504.02843v1, under CC-BY license (http://creativecommons.org/licenses/by/4.0/), see ( Lino et al., 2025a ) .

Figure 23: Representative conditional generation and data-assimilation problems in turbulent flows. The reconstruction tasks can be broadly separated into two classes. Full-state reconstruction. Sparse observations can be used to reconstruct the complete turbulent field, while more general inverse problems aim at inferring unobserved fields or variables from partial information on dynamically coupled quantities, as illustrated here for the reconstruction of the velocity and passive-scalar fields. Small-scale reconstruction. Super-resolution aims at recovering the unresolved high-wavenumber content of a coarse-grained turbulent field, whereas sub-grid closure seeks to infer the effect of unresolved scales on the resolved dynamics through the corresponding sub-grid stresses. These examples illustrate applications where generative models can be conditioned on incomplete, coarse-grained, or indirect observations to reconstruct statistically and dynamically consistent turbulent fields.

Figure 24: Reconstruction of ocean-drifter dynamics with a pretrained diffusion model. Application of the conditional diffusion model (C-DM) discussed in Sec. 2.8 to velocity records from the NOAA Global Drifter Program ( Lagrangian Drifter Laboratory, 2026 ) , following ( Li et al., 2025 ) . The pretrained generative model is conditioned on the observed portions of gappy drifter trajectories to reconstruct the missing dynamics. Top: spatial distribution of the drifter dataset, with representative regions including the Tropics, Western Boundary Currents, and the Antarctic Circumpolar Current. Bottom left: scale-dependent Lagrangian flatness in the three oceanic regions, highlighting their distinct intermittent statistics. Solid symbols correspond to the ground-truth observations, while open symbols report the statistics of the reconstructed data in the corresponding geographic regions, identified by the same colors. The reconstructed statistics are computed using velocity increments for which at least one of the two points lies within the reconstructed gap. Bottom right: examples of trajectories containing unobserved temporal gaps.

Figure 25: Summary of the Web of Science search strategy used to construct the non-AI turbulence baseline corpus. The query is restricted to the last 10 years 2016-2025 and combines turbulence- and fluid-mechanics-related terms searched in titles, abstracts, and author keywords, restricts the results to a selected set of journals representative of fluid mechanics, computational physics and nonlinear dynamics, and excludes records containing machine-learning, artificial-intelligence or generative-AI terminology. This baseline is used for comparison with the AI-related turbulence set obtained by including at least one of the AI key-words listed in the third row.

Figure 26: Country/region distribution of turbulence publications indexed in Web of Science over the last ten years. The left panel shows the AI-related turbulence subset, dominated by China and USA; the right panel shows the broader turbulence subset

Figure 27: Comparison between the yearly number of publications retrieved using AI-related turbulence queries, left, and generic turbulence queries, right. The AI-related subset shows a strongly accelerating trend, increasing from only 22 records in 2016 to 537 records in 2025, with 2024–2025 alone accounting for almost half of the indexed records in this subset. The comparison highlights that AI-based turbulence research remains much smaller in absolute terms, but has expanded at a far larger relative rate.

Figure 28: Distribution of the top 10 publication journals for turbulence-related papers indexed in Web of Science over the last decade, comparing the AI-related subset with the broader turbulence subset. The AI-related literature is more strongly concentrated in a few outlets, with Physics of Fluids alone accounting for nearly

37\%

of the records.

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