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1. PFGM++: Unlocking the Potential of Physics-Inspired Generative Models

   Yilun Xu, Ziming Liu (December 2023, Proceedings of Machine Learning Research)

   We introduce a new family of physics-inspired generative models termed PFGM++ that unifies diffusion models and Poisson Flow Generative Models (PFGM). These models realize generative trajectories for N dimensional data by embedding paths in N+D dimensional space while still controlling the progression with a simple scalar norm of the D additional variables. The new models reduce to PFGM when D=1 and to diffusion models when D→∞. The flexibility of choosing D allows us to trade off robustness against rigidity as increasing D results in more concentrated coupling between the data and the additional variable norms. We dispense with the biased large batch field targets used in PFGM and instead provide an unbiased perturbation-based objective similar to diffusion models. To explore different choices of D, we provide a direct alignment method for transferring well-tuned hyperparameters from diffusion models (D→∞) to any finite D values. Our experiments show that models with finite D can be superior to previous state-of-the-art diffusion models on CIFAR-10/FFHQ 64×64 datasets/LSUN Churches 256×256, with median Ds. In class-conditional setting, D=2048 yields current state-of-the-art FID of 1.74 on CIFAR-10 without additional training. Furthermore, we demonstrate that models with smaller D exhibit improved robustness against modeling errors. Code is available at https://github.com/Newbeeer/pfgmpp 

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2. Estimating Spectral Fluxes in Quasi-Two-Dimensional Flows with Advective Structure Functions and Bessel Functions

   https://doi.org/10.1175/JPO-D-24-0211.1  

   Pearson, Brodie; Wagner, Cassidy; Fox-Kemper, Baylor; Samelson, Roger (September 2025, Journal of Physical Oceanography)

   Abstract Several new methods are proposed that can diagnose the interscale transfer (or spectral flux) of kinetic energy (KE) and other properties in oceanic and broader geophysical systems, using integrals of advective structure functions and Bessel functions (herein “Bessel methods”). The utility of the Bessel methods is evaluated using simulations of anisotropic flow within two-dimensional (2D), surface quasigeostrophic (SQG), and two-layer QG systems. The Bessel methods diagnose various spectral fluxes within all of these systems, even under strong anisotropy and complex dynamics (e.g., multiple cascaded variables, coincident and opposing spectral fluxes, and nonstationary systems). In 2D turbulence, the Bessel methods capture the inverse KE cascade at large scales and the downscale enstrophy cascade (and associated downscale energy flux) at small scales. In SQG turbulence, the Bessel methods capture the downscale buoyancy variance cascade and the coincident upscale wavenumber-dependent KE flux. In QG turbulence, the Bessel methods capture the upscale kinetic energy flux. It is shown that these Bessel methods can be applied to data with limited extent or resolution, provided the scales of interest are captured by the range of separation distances. The Bessel methods are shown to have several advantages over other flux-estimation methods, including the ability to diagnose downscale energy cascades and to identify sharp transition scales. Analogous Bessel methods are also discussed for third-order structure functions, along with some caveats due to boundary terms. Significance StatementBig ocean eddies play an important role in Earth’s energy cycle by moving energy to both larger and smaller scales, but it is difficult to measure these “eddy energy fluxes” from oceanic observations. We develop a new method to estimate eddy energy fluxes that utilizes spatial differences between pairs of points and can be applied to various ocean data. This new method accurately diagnoses key eddy energy flux properties, as we demonstrate using idealized numerical simulations of various large-scale ocean systems. 

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3. A Generative Super‐Resolution Model for Enhancing Tropical Cyclone Wind Field Intensity and Resolution

   https://doi.org/10.1029/2024JH000375  

   Lockwood, Joseph_W; Gori, Avantika; Gentine, Pierre (November 2024, Journal of Geophysical Research: Machine Learning and Computation)

   Abstract Extreme winds associated with tropical cyclones (TCs) can cause significant loss of life and economic damage globally, highlighting the need for accurate, high‐resolution modeling and forecasting for wind. However, due to their coarse horizontal resolution, most global climate and weather models suffer from chronic underprediction of TC wind speeds, limiting their use for impact analysis and energy modeling. In this study, we introduce a cascading deep learning framework designed to downscale high‐resolution TC wind fields given low‐resolution data. Our approach maps 85 TC events from ERA5 data (0.25° resolution) to high‐resolution (0.05° resolution) observations at 6‐hr intervals. The initial component is a debiasing neural network designed to model accurate wind speed observations using ERA5 data. The second component employs a generative super‐resolution strategy based on a conditional denoising diffusion probabilistic model (DDPM) to enhance the spatial resolution and to produce ensemble estimates. The model is able to accurately model intensity and produce realistic radial profiles and fine‐scale spatial structures of wind fields, with a percentage mean bias of −3.74% compared to the high‐resolution observations. Our downscaling framework enables the prediction of high‐resolution wind fields using widely available low‐resolution and intensity wind data, allowing for the modeling of past events and the assessment of future TC risks. 

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4. Signed distance function–biased flow importance sampling for implicit neural compression of flow fields

   https://doi.org/10.1111/mice.13526  

   Mures, Omar A; Cid_Montoya, Miguel (July 2025, Computer-Aided Civil and Infrastructure Engineering)

   Abstract The rise of exascale supercomputing has motivated an increase in high‐fidelity computational fluid dynamics (CFD) simulations. The detail in these simulations, often involving shape‐dependent, time‐variant flow domains and low‐speed, complex, turbulent flows, is essential for fueling innovations in fields like wind, civil, automotive, or aerospace engineering. However, the massive amount of data these simulations produce can overwhelm storage systems and negatively affect conventional data management and postprocessing workflows, including iterative procedures such as design space exploration, optimization, and uncertainty quantification. This study proposes a novel sampling method harnessing the signed distance function (SDF) concept: SDF‐biased flow importance sampling (BiFIS) and implicit compression based on implicit neural network representations for transforming large‐size, shape‐dependent flow fields into reduced‐size shape‐agnostic images. Designed to alleviate the above‐mentioned problems, our approach achieves near‐lossless compression ratios of approximately :, reducing the size of a bridge aerodynamics forced‐vibration simulation from roughly to about while maintaining low reproduction errors, in most cases below , which is unachievable with other sampling approaches. Our approach also allows for real‐time analysis and visualization of these massive simulations and does not involve decompression preprocessing steps that yield full simulation data again. Given that image sampling is a fundamental step for any image‐based flow field prediction model, the proposed BiFIS method can significantly improve the accuracy and efficiency of such models, helping any application that relies on precise flow field predictions. The BiFIS code is available onGitHub. 

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5. Stacked Deep Learning Models for Fast Approximations of Steady-State Navier–Stokes Equations for Low Re Flow

   https://doi.org/10.34133/icomputing.0093  

   Wang, Shen; Nikfar, Mehdi; Agar, Joshua C; Liu, Yaling (January 2024, Intelligent Computing)

   Computational fluid dynamics (CFD) simulations are broadly used in many engineering and physics fields. CFD requires the solution of the Navier–Stokes (N-S) equations under complex flow and boundary conditions. However, applications of CFD simulations are computationally limited by the availability, speed, and parallelism of high-performance computing. To address this, machine learning techniques have been employed to create data-driven approximations for CFD to accelerate computational efficiency. Unfortunately, these methods predominantly depend on large labeled CFD datasets, which are costly to procure at the scale required for robust model development. In response, we introduce a weakly supervised approach that, through a multichannel input capturing boundary and geometric conditions, solves steady-state N-S equations. Our method achieves state-of-the-art results without relying on labeled simulation data, instead using a custom data-driven and physics-informed loss function and small-scale solutions to prime the model for solving the N-S equations. By training stacked models, we enhance resolution and predictability, yielding high-quality numerical solutions to N-S equations without hefty computational demands. Remarkably, our model, being highly adaptable, produces solutions on a 512 × 512 domain in a swift 7 ms, outpacing traditional CFD solvers by a factor of 1,000. This paves the way for real-time predictions on consumer hardware and Internet of Things devices, thereby boosting the scope, speed, and cost-efficiency of solving boundary-value fluid problems. 

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