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Multiplicative diffusion models
Multiplicative diffusion models
We introduce a new class of generative models based on multiplicative score-driven diffusion. In contrast to classical diffusion models that rely on additive Gaussian noise, our construction is driven by skew-symmetric multiplicative noise. It yields conservative forward-backward dynamics inspired by the principles of physics.
We prove that the forward process converges exponentially fast to a tractable non-Gaussian latent distribution, and we characterize this limit explicitly. A key property of our diffusion is that it preserves the distribution of data norms, resulting in a latent space that is inherently data-aware. Unlike the standard Gaussian prior, this structure better adapts to heavy-tailed and anisotropic data, providing a closer match between latent and observed distributions.
On the algorithmic side, we derive the reverse-time stochastic differential equation and associated probability flow, and show that sliced score matching furnishes a consistent estimator for the backward dynamics. This estimation procedure is equivalent to maximizing an evidence lower bound (ELBO), bridging our framework with established variational principles.
Empirically, we demonstrate the advantages of our model in challenging settings, including correlated Cauchy distributions and experimental fluid dynamics images (d=1024). Across these tasks, our approach more accurately captures extreme events and tail behavior than classical diffusion models, particularly in the low-data regime. Our results suggest that multiplicative conservative diffusions open a principled alternative to current score-based generative models, with strong potential for domains where rare but critical events dominate.
Some related publications & communications
Gruhlke, R., Resseguier, V., & Talla, M.. "Multiplicative Diffusion Models: Beyond Gaussian Latents". ICLR 2026 [HAL][Preprint] [github repo]
Open-source code
MSGM [github repo]
Stochastic reduced order models
for real-time data assimilation
Stochastic reduced order models
for real-time data assimilation
Results : Impressive numerical results have been obtained for a 3-dimensional wake flows at moderate Reynolds for up to 14 vortex shedding cycles after the learning window, using a single measurement point.
Some related publications & communications
Resseguier, V., Ladvig, M., Heitz, D. (2022). "Real-time estimation and prediction of unsteady flows using reduced-order models coupled with few measurements". Journal of Computational Physics. [HAL][Preprint] [github repo]
Resseguier, V., Picard, A. M., Mémin, E. & Chapron, B. (2021). Quantifying truncation-related uncertainties in unsteady fluid dynamics reduced order models. SIAM/ASA Journal on Uncertainty Quantification. [HAL][file]
Valentin Resseguier, Matheus Ladvig, Augustin Picard, Etienne Mémin, Bertrand Chapron (2020)
"Toward real-time embedded observer of unsteady fluid flow environment",
10th European Congress on Embedded Real Time Software and Systems (ERTS 2020). Toulouse.
[Conference poster]Matheus Ladvig, Agustin Martin Picard, Valentin Resseguier, Dominique Heitz, Etienne Mémin, Chapron Bertrand (2020)
"Real-time observer from stochastic reduced order model",
[Slides]
Open-source code
pyReDA [github repo]
The python code performs the data assimilation using precomputed ROM coefficients and ROM noise statistics.
Wave-turbulence interaction
Wave-turbulence interaction
Project goal : Simulate more precisely and at lower cost the interactions of surface currents on swells and better characterize these effects and their impacts (bias and variability) on satellite radar measurements.
Methodology : The small-scale currents being generally badly resolved by models and unobserved by altimetric measurements, we have developed stochastic multiscale self-similar and self-adaptive models to represent them and simulate their effects on swell spectra. From there, wave fields are emulated and radar measurements can be simulated.
Results : Self-similarity in both time and space ensures very good skills to our stochastic turbulence models in every simulated situations. In contrast, the usual time-decorrelation assumption is valid only for wavenumber larger than a threshold.
Heterogeneous oceanic currents like jets locally enhances wave energy, and this strengthens several biases on altimetric satellite measurements due to wave vertical asymmetries.
Related publications & communications
Valentin Resseguier, Erwan Hascoet, Bertrand Chapron (2024). "Wave propagation in random 2D turbulence: a multi-scale approach". Journal of Fluid Mechanics. [HAL][file]
Valentin Resseguier, Erwan Hascoet, Fabrice Collard, Bertrand Chapron (2024). Stochastic closures for linear wave propagation in turbulence. 5th Stochastic Transport in Upper Ocean Dynamics (STUOD) Annual Workshop [HAL] [slides]
Valentin Resseguier, Erwan Hascoet, Bertrand Chapron, Baylor Fox-Kemper (2021) "Validity domains and parametrizations for white-noise and multiscale models in turbulence and wave-turbulence interactions". EGU [HAL][PICO slide]
Models under location uncertainty
Models under location uncertainty
Project goal : Model the effect of unresolved turbulence and above all quantify the induced simulation model errors in order to improve ensemble-based data assimilation algorithms.
Methodology : Physically, the two central assumptions are the temporal decorrelation of small-scale turbulence and the transport of physical quantities by the partially random velocity of the fluid. Compared to classical fluid mechanics, three new terms appear in the equations: a multiplicative noise, a diffusion term, and a large-scale advecting velocity correction. For ensemble forecasts, we propose several parametrization choice (i.e. possibly heterogeneous spatial covariance) for the random turbulence.
Results : This framework enables new fluid dynamics model derivations and faster attractor visit. But more importantly, the dynamics under location uncertainty accurately quantifies model errors unlike traditional methods (e.g. random initial conditions).
Poster-Houches-2024.pdfRelated publications
Valentin Resseguier (2024). "Maximum likelihood estimation of subgrid flows from tracer image sequences". in Stochastic Transport in Upper Ocean Dynamics 2023 [HAL][file]
Yicun Zhen, Valentin Resseguier, Bertrand Chapron (2023). "Physically Constrained Covariance Inflation from Location Uncertainty and Optimal Transportation". [HAL][file]
Resseguier, Long Li, Gabriel Jouan, Pierre Dérian, Etienne Mémin, Chapron Bertrand (2020)"New trends in ensemble forecast strategy: uncertainty quantification for coarse-grid computational fluid dynamics",Archives of Computational Methods in Engineering.[HAL][bibtex][file]
Valentin Resseguier, Wei Pan, Baylor Fox-Kemper (2020)"Data-driven versus self-similar parameterizations for Stochastic Advection by Lie Transport and Location Uncertainty",Nonlinear Processes in Geophysics.[HAL][bibtex][file] [github repo]
Bertrand Chapron, Pierre Dérian, Etienne Mémin, Valentin Resseguier (2018)"Large scale flows under location uncertainty: a consistent stochastic framework",Quarterly Journal of the Royal Meteorological Society.[HAL][bibtex][file]
Valentin Resseguier, Etienne Mémin, Bertrand Chapron (2017)"Geophysical flows under location uncertainty, Part III SQG and frontal dynamics under strong turbulence conditions",Geophysical and Astrophysical Fluid Dynamics.[HAL][bibtex][file]
Valentin Resseguier, Etienne Mémin, Bertrand Chapron (2017)"Geophysical flows under location uncertainty, Part II Quasi-geostrophy and efficient ensemble spreading",Geophysical and Astrophysical Fluid Dynamics.[HAL][bibtex][file] [github repo]
Valentin Resseguier, Etienne Mémin, Bertrand Chapron (2017)"Geophysical flows under location uncertainty, Part I Random transport and general models",Geophysical and Astrophysical Fluid Dynamics.[HAL][bibtex][file]
Some related communications
Baylor Fox-Kemper, Darryl Holm, Wei Pan, Valentin Resseguier (2019)
"Stochastic transport to quantify errors in geophysical fluid dynamic simulations",
EGU 2019 (European Geophysical Union). Vienne, AT.[Slides]Valentin Resseguier, Long Li, Gabriel Jouan, Pierre Dérian, Etienne Mémin, Bertrand Chapron (2019)
"Dynamics under location uncertainty and other energy-related stochastic subgrid schemes",
2019 - Workshop Conservation Principles, Data and Uncertainty in Atmosphere-Ocean Modelling. Potsdam, DE.[Slides]
Valentin Resseguier, Etienne Mémin, Bertrand Chapron (2016)
"Likely chaotic transitions of large-scale fluid flows using a stochastic transport model",
9th Chaotic Modeling and Simulation International Conference (CHAOS2016). Londres, GB.[Slides]
Valentin Resseguier, Etienne Mémin, Bertrand Chapron (2016)
"Stochastic subgrid tensor for geophysical flow modeling",
8th European Postgraduate Fluid Dynamics Conference. Varsovie, PL.[Poster]
Open-source codes
SQGMU [github repo]
This MATLAB codes simulates deterministic or randomized version of the Surface Quasi-Geostrophic (SQG) model. The random dynamics is based on the transport under location uncertainty. It is associated to : "Geophysical flows under location uncertainty, Part II", V. Resseguier et al., 2017
LU_SALT_Selfsim [github repo]
This MATLAB codes simulates deterministic or randomized version of 2D Euler and Surface Quasi-Geostrophic (SQG) models. The random dynamics is based on LU and SALT frameworks. Several parameterisations are available. It is associated to : "Data-driven versus self-similar parameterizations for Stochastic Advection by Lie Transport and Location Uncertainty", V. Resseguier et al., 2020
Fluid mixing
Fluid mixing
Project goal : Identify the Lagrangian coherent structures (i.e. the fluid subdomains which stay isolated from each other without mixing) but also understand and provide parametrization proxy for Lagragian-advection-based downscaling methods (e.g. Sutton et al., 1994, for the atmosphere or Desprès et al., 2011a,b for the ocean).
Methodology :
To identify Lagrangian coherent structures we simulate (or observe) streaklines from many source points, and then we compute the streaklines' tangents norm kurtosis (4th-order moment). This method detects streaklines breakings which occur at Lagrangian coherent structures boundaries.
Besides, oceanographic Lagragian-advection-based downscaling methods focus on the tracer “submesoscale” structures (≤10 km), created by mesoscale currents (∼100 km) indirectly measured by satellite. And in this case, we can assume these (Eulerian) currents as constant in time. Thus, the key mixing processes are only locally uniform shears and foldings around stationary convective cells. From there, we propose a pseudo-analytical deterministic model, inexpensive in computing power, to predict the deformations generated by the currents and the evolution of the tracer 2nd-order statistics after a finite time.
Results :
Our streaklines tangents kurtosis method successfully identify Lagrangian coherent structures. But, contrary to the state of the art (finite-time Lyapunov exponents, abbreviated FLTE), our method is not restricted by the weak precision of the spatial derivatives involved (finite differences).
For oceanographic Lagragian-advection-based downscaling methods, we successfully predict the evolution of spectral energy density of the tracer after a finite time, the effective diffusion coefficient and the optimal time to stop a downscaling, without needing any time integration.
Related publications & communications
Valentin Resseguier, Bertrand Chapron, Etienne Mémin (2022). "Effects of smooth divergence-free flows on tracer gradients and spectra: Eulerian prognosis description". Journal of Physical Oceanography. [HAL][file][github repo]
Open-source codes
Mixing [github repo]
Compute Lagrangian advection and several mixing diagnoses, from synthetic and real oceanographic data. For real satellite image tests, globcurrent data can be used. It is associated to : "Effects of smooth divergence-free flows on tracer gradients and spectra: Eulerian prognosis description", V. Resseguier et al., 2021
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