Learning turbulence with machines: data-driven closures and generative models
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general physics↗
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Results for “closure modeling”
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A weak form equation learning approach for identifying closures in optically thin thermal radiation transport
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36 MATERIALS SCIENCE↗
Data for A Hybrid Biophysical-Machine Learning Framework for Diurnal Surface Energy Flux Estimation Using Proximal Sensing
Thermal infrared-based remote sensing of surface energy fluxes has traditionally relied on high spatial resolution satellite data with revisit frequencies on the order of weeks. In this study, we evaluate a biophysics-based analytical surface energy balance model for predicting latent energy (LE) and sensible heat (H) fluxes using proximal sensing observations. The Surface Temperature Initiated Closure (STIC1.2) model has been extensively validated across a wide range of spatial and temporal scales using various satellite-derived thermal infrared data sets. Here we extend this validation by applying STIC at sub-hourly temporal resolution over multiple growing seasons for four distinct agricultural systems. We further develop and evaluate novel STIC variants that incorporate machine learning (ML) techniques to eliminate the need for surface energy balance observations, specifically net radiation and soil heat flux, thereby enhancing model applicability in data-sparse settings. The integration of a ML component to estimate surface available energy is shown to have strong predictive performance for both LE (R2 = 0.81–0.94) and H (R2 = 0.46–0.72) across all agricultural systems examined here, demonstrating the potential of hybrid biophysical-machine learning approaches for surface energy balance modeling with minimal data requirements. This study concludes with a novel application of explainable machine learning (exML) to diagnose sources of model error. This exML framework attributes residual prediction errors to both model input variables and environmental drivers not explicitly included in the simulation experiments. This approach provides a new pathway for improving model design and integrating previously overlooked yet influential variables into future model iterations.
AI/ML↗
Modeling hydraulic fracture opening and closure with proppant transport and settlement
Hydraulic fracturing is a widely used reservoir stimulation technique for improving fluid circulation in rock formations with extremely low permeability, particularly in enhanced geothermal systems (EGS). To better understand the complex processes involved and improve hydraulic stimulation performance, we have developed ELK (ELectrical fracKing), a MOOSE-based 3D finite element application designed to model the behavior of proppant-fluid mixtures in propagating fractures. ELK integrates both the fluid and proppant components, incorporating particle-driven processes such as gravity settling, particle-particle interactions, and strong density and viscosity contracts, in addition to conventional fluid-driven fracture propagation. In this contribution, we extend ELK to model propped fracture closure, which occurs after the injection phase due to a dramatic drop in the effective stress on the fracture plane. During the shut-in, flowback, and production periods, the fracture width decreases, with the closure behavior depending on proppant concentration.. At low concentrations, closure follows a nonlinear joint law linked to the stiffness of asperities in the fracture walls. While at high concentrations, it is controlled by the properties of packed proppant bed. The extended ELK application is validated against several benchmark examples, including the propagation of an inclined frictional crack, fracture opening and sliding in response to fluid injection, and flowback analysis. We believe that ELK’s enhanced capabilities can serve as a valuable tool for the design and optimization of EGS deployment.
15 - GEOTHERMAL ENERGY↗
A Hybrid Biophysical‐Machine Learning Framework for Diurnal Surface Energy Flux Estimation Using Proximal Sensing
Thermal infrared-based remote sensing of surface energy fluxes has traditionally relied on high spatial resolution satellite data with revisit frequencies on the order of weeks. In this study, we evaluate a biophysics-based analytical surface energy balance model for predicting latent energy ( LE ) and sensible heat ( H ) fluxes using proximal sensing observations. The Surface Temperature Initiated Closure (STIC1.2) model has been extensively validated across a wide range of spatial and temporal scales using various satellite-derived thermal infrared data sets. Here we extend this validation by applying STIC at sub-hourly temporal resolution over multiple growing seasons for four distinct agricultural systems. We further develop and evaluate novel STIC variants that incorporate machine learning (ML) techniques to eliminate the need for surface energy balance observations, specifically net radiation and soil heat flux, thereby enhancing model applicability in data-sparse settings. The integration of a ML component to estimate surface available energy is shown to have strong predictive performance for both LE (R 2 = 0.81–0.94) and H (R 2 = 0.46–0.72) across all agricultural systems examined here, demonstrating the potential of hybrid biophysical-machine learning approaches for surface energy balance modeling with minimal data requirements. This study concludes with a novel application of explainable machine learning (exML) to diagnose sources of model error. This exML framework attributes residual prediction errors to both model input variables and environmental drivers not explicitly included in the simulation experiments. This approach provides a new pathway for improving model design and integrating previously overlooked yet influential variables into future model iterations.
evapotranspiration↗
Data for: A hybrid biophysical-machine learning framework for diurnal surface energy flux estimation using proximal sensing
Thermal-based remote sensing of surface energy fluxes has traditionally relied on high spatial resolution satellite data with revisit frequencies on the order of weeks. In this study, we evaluate a biophysics-based analytical surface energy balance model for predicting latent energy (LE) and sensible heat (H) fluxes using proximal sensing observations. The Surface Temperature Initiated Closure (STIC1.2) model has been extensively validated across a wide range of spatial and temporal scales using various satellite-derived thermal datasets. Here we extend this validation by applying STIC at sub-hourly temporal resolution over multiple growing seasons for four distinct agricultural systems. We further develop and evaluate novel STIC variants that incorporate machine learning (ML) techniques to eliminate the need for specific surface energy balance observations, specifically net radiation and soil heat flux, thereby enhancing model applicability in data-sparse settings. The integration of an ML component to estimate surface available energy is shown to have strong predictive performance for both LE (R2 = 0.81-0.94) and H (R2 = 0.46-0.72) across all agricultural systems examined here, demonstrating the potential of hybrid biophysical – machine learning approaches for surface energy balance modeling with minimal data requirements. This study concludes with a novel application of explainable machine learning (exML) to diagnose sources of model error. This exML framework attributes residual prediction errors to both model input variables and environmental drivers not explicitly included in the simulation experiments. This approach provides a new pathway for improving model design and integrating previously overlooked yet influential variables into future model iterations.
Agricultural Sciences↗
A robust spectral element implementation of the $k - τ$ RANS model in Nek5000/NekRS
The $k - ω$ Reynolds Averaged Navier Stokes (RANS) model is one of the industry standard approaches for modeling of turbulent flows. It performs better than the $k - ϵ$ model for low Reynolds number flows and is also more suitable for boundary layers with adverse pressure gradients. Major drawback of the model, however, is that the asymptotic value of $ω$ at the walls is singular, necessitating the use of a contrived “sufficiently” large value for $ω$ as the boundary condition for its transport equation. Here, this invariably leads to the solution being sensitive to near wall grid spacing. While an acceptable solution for low order (finite volume) methods, the excessive near wall gradients lead to persistent numerical stability issues in high order codes. To alleviate the problem, specifically in the context of the high order spectral element code Nek5000, a regularized $k - ω$ approach was formulated in our prior work (Tomboulides et al., 2018). The formulation, however, relies on the use of wall distance and its gradients for modeling the closure terms and can pose problems for simulations in complex geometries. This work presents a novel implementation of the $k - τ$ RANS model in Nek5000, where $τ = 1/ω$, eliminating the need for regularization, owing to the asymptotically bounded behavior of the source terms in the $τ$ transport equation, and also eliminating dependence on wall distance. Robustness and stability of the $k - τ$ model is ensured through implicit treatment of the source terms and their careful numerical implementation and demonstrated through several cases aimed at verification and validation. Studies include both canonical and engineering relevant problems, viz., turbulent channel flow, pipe flow, backward facing step, flow over NACA 0012 airfoil and flow in a T-junction. Results from the $k - τ$ model are shown to be consistent with regularized $k - ω$ model and also with the $k - ω$ SST model in OpenFOAM (for select studies). Comparison with experimental data is also shown, where available, to bolster validation efforts for the $k - τ$ model implementation through prediction of key turbulent quantities of interest.
Nek5000↗
Online learning of quadratic manifolds from streaming data for nonlinear dimensionality reduction and nonlinear model reduction
Here, this work introduces an online greedy method for constructing quadratic manifolds from streaming data, designed to enable in situ analysis of numerical simulation data on the Petabyte scale. Unlike traditional batch methods, which require all data to be available upfront and take multiple passes over the data, the proposed online greedy method incrementally updates quadratic manifolds in one pass as data points are received, eliminating the need for expensive disk input/output operations as well as storing and loading data points once they have been processed. A range of numerical examples demonstrate that the online greedy method learns accurate quadratic manifold embeddings while being capable of processing data that far exceed common disk input/output capabilities and volumes as well as main-memory sizes.
97 MATHEMATICS AND COMPUTING↗
Multiscale and multidimensional modeling of particle acceleration and transport in solar flares
Multi-messenger, multi-viewpoint, and time-resolved observations of solar flares are now providing unprecedented constraints on particle acceleration sites, energy conversion, and energy transport. The interpretation of current observations, including microwave imaging spectroscopy from EOVSA, hard x-ray (HXR) imaging from Solar Orbiter/STIX, gamma-ray diagnostics from Fermi, and in situ measurements from Parker Solar Probe and Solar Orbiter, collectively demands modeling frameworks that go beyond traditional spatially unresolved, one-zone models or single-mechanism descriptions. This review surveys multiscale and multidimensional modeling approaches, including kinetic, magnetohydrodynamic (MHD), and macroscopic particle models, that are being developed to meet the need. Kinetic simulations reveal that three-dimensional (3D) effects, including field-line chaos and self-generated turbulence, are essential for sustained power-law particle acceleration. MHD simulations now capture flux-rope eruptions, plasmoid-unstable current sheets, and turbulent flare regions in realistic magnetic topologies. Macroscopic models coupling MHD with energetic-particle models produce spatially resolved electron distributions and synthetic HXR and microwave emissions for direct comparison with observations. Despite these advances, outstanding challenges remain in bridging kinetic and global scales, improving MHD simulations and macroscopic particle models, and achieving quantitative model-observation closure.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Machine-learned closure of URANS for stably stratified turbulence: connecting physical timescales & data hyperparameters of deep time-series models
Stably stratified turbulence (SST), a model that is representative of the turbulence found in the oceans and atmosphere, is strongly affected by fine balances between forces and becomes more anisotropic in time for decaying scenarios. Moreover, there is a limited understanding of the physical phenomena described by some of the terms in the Unsteady Reynolds-Averaged Navier–Stokes (URANS) equations—used to numerically simulate approximate solutions for such turbulent flows. Rather than attempting to model each term in URANS separately, it is attractive to explore the capability of machine learning (ML) to model groups of terms, i.e. to directly model the force balances. We develop deep time-series ML for closure modeling of the URANS equations applied to SST. We consider decaying SST which are homogeneous and stably stratified by a uniform density gradient, enabling dimensionality reduction. We consider two time-series ML models: long short-term memory and neural ordinary differential equation. Both models perform accurately and are numerically stable in a posteriori (online) tests. Furthermore, we explore the data requirements of the time-series ML models by extracting physically relevant timescales of the complex system. We find that the ratio of the timescales of the minimum information required by the ML models to accurately capture the dynamics of the SST corresponds to the Reynolds number of the flow. The current framework provides the backbone to explore the capability of such models to capture the dynamics of high-dimensional complex dynamical system like SST flows.
97 MATHEMATICS AND COMPUTING↗
Implications of a weakening N = 126 shell closure away from stability for r -process astrophysical conditions
The formation of the third r-process abundance peak near A ∼ 195 is highly sensitive to both nuclear structure far from stability and the astrophysical conditions that produce the heaviest elements. In particular, the N = 126 shell closure plays a crucial role in shaping this peak. Experimental data hints that the shell weakens as proton number departs from Z = 82, a trend largely missed by global mass models. To investigate its impact on r-process nucleosynthesis, we employ both standard global models with strong closures and modified Duflo-Zuker (DZ) models that reproduce the weakening, combined with three sets of β − -decay rates. Strong shell closures generate sharply peaked abundances, whereas weakened closures consistent with the experimental trend produce broader, flatter patterns. Accurately reproducing the solar third peak under weakened shell strength requires sufficiently neutron-rich conditions that significant fission occurs, and slower decay rates. These results demonstrate that a weakening N = 126 shell closure away from stability imposes significant constraints on the astrophysical environments of the r-process and underscores the need for precise mass measurements and improved characterization of β − -decay properties in this region.
N = 126 closed shells↗
Estimation of General Area Dose Rates During Closure Weld Operations on the DOE Standard Canister Pacific Basin Nuclear Conference 2024
The Department of Energy (DOE) Spent Nuclear Fuel (SNF) Road Ready Demonstration Project is an effort to package DOE-managed SNF for road-ready dry storage. A key operation in this demonstration is performing the closure weld on the DOE Standard Canister after it has been loaded with the SNF. Planning the closure weld operations is ongoing and must consider what parts of the operations (if any) can be done manually and what parts must be done remotely. A key parameter to inform the evolving planning of these operations is the dose rate in the vicinity of the canister and the dose hazard posed to the closure weld operators and collocated workers. This study developed a Monte Carlo N-Particle (MCNP) model of the closure weld operation, which was used to estimate general area dose rates as well as dose rates to the welding location, weld operators, and collocated workers. These estimates provide some insight into the potential radiation exposure hazard for workers during the operations and are a factor to be considered in the ongoing development of designs, technology, processes, and administrative controls for packaging DOE-managed SNF for road-ready dry storage.
11 - NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
Atwood effects on nonlocality of the scalar transport closure in Rayleigh-Taylor mixing
The importance of nonlocality is assessed in modeling mean scalar transport for turbulent Rayleigh-Taylor (RT) mixing at different Atwood numbers. Building on the two-dimensional incompressible work of Lavacot et al. [J. Fluid Mech. 985, A47 (2024)], the present work extends the macroscopic forcing method to variable density problems in three-dimensional space to measure moments of the generalized eddy diffusivity kernel in RT mixing for increasing Atwood numbers (𝐴 = 0.05, 0.3, 0.5, 0.8). It is found that as 𝐴 increases, (1) the eddy diffusivity moments become asymmetric and (2) the higher-order eddy diffusivity moments become larger relative to the leading-order diffusivity, indicating that nonlocality becomes more important at higher 𝐴. There is a particularly strong temporal nonlocality at higher 𝐴, suggesting stronger history effects. In conclusion, the implications of these findings for closure modeling for finite-Atwood RT are discussed.
general physics↗
Resolving discrepancies in bang-time predictions for indirect-drive ICF experiments on the NIF: Insights from the Build-A-Hohlraum campaign
This study investigated discrepancies between measured and simulated x-ray drive in Indirect-Drive Inertial Confinement Fusion (ID-ICF) hohlraums at the National Ignition Facility. Despite advances in radiation-hydrodynamic simulations, a consistent “drive deficit” remains. Experimentally measured ID-ICF capsule bang-times are systematically 400–700 ps later than simulations predict. The Build-A-Hohlraum (BAH) campaign explored potential causes for this discrepancy by systematically varying hohlraum features, including laser entrance hole (LEH) windows, capsules, and gas fills. Overall, the agreement between simulated and experimental x-ray drive was found to be largely unaffected by these changes. The data allow us to exclude some hypotheses put forward to potentially explain the discrepancy. Errors in the local thermodynamic equilibrium (LTE) atomic modeling, errors in the modeling of LEH closure, and errors due to a lack of plasma species mix physics in simulations are shown to be inconsistent with our measurements. Instead, the data support the hypothesis that errors in NLTE emission modeling are a significant contributor to the discrepancy. X-ray emission in the 2–4 keV range is found to be approximately 30% lower than in simulations. This is accompanied by higher than predicted electron temperatures in the gold bubble region, pointing to errors in non-LTE modeling. Introducing an opacity multiplier of 0.87 on energy groups above 1.8 keV improves agreement with experimental data, reducing the bang-time discrepancy from 300 to 100 ps. These results underscore the need for refined NLTE opacity models to enhance the predictive power of hohlraum simulations.
Band emission↗
Modeling information flow in a computer processor with a multi-stage queuing model
In this paper, we introduce a nonlinear stochastic model to describe the propagation of information inside a computer processor. In this model, a computational task is divided into stages, and information can flow from one stage to another. The model is formulated as a spatially-extended, continuous-time Markov chain where space represents different stages. This model is equivalent to a spatially-extended version of the M/M/s queue. The main modeling feature is the throttling function which describes the processor slowdown when the amount of information falls below a certain threshold. We derive the stationary distribution for this stochastic model and develop a closure for a deterministic ODE system that approximates the evolution of the mean and variance of the stochastic model. In conclusion, we demonstrate the validity of the closure with numerical simulations.
97 MATHEMATICS AND COMPUTING↗
An Optimized Parameterization of Sub‐Grid Scale Advection for Convection Permitting Models
Convection‐permitting models (CPMs) explicitly resolve deep convection yet under‐resolve the organized lateral exchanges among drafts and their environment that control entrainment/detrainment, precipitation efficiency, and mesoscale structure. In this work, we introduce the Optimized Advection Scheme (OAS), which introduces a small rotation of the Cartesian frame of reference for the horizontal winds relative to other variables used in advection that induces cross‐gradient transport to mimic under‐resolved convective mixing. The rotation angle is selected to minimize the Kullback–Leibler divergence between the simulated and satellite observed precipitation intensity distributions, yielding a physically consistent perturbation that is computationally inexpensive and portable. Optimized Advection Scheme is implemented in WRF and evaluated over Amazon (April 2014). It shifts precipitation–precipitable‐water joint distributions toward lighter rain, reduces overly intense rates, and improves mesoscale convective system (MCS) lifetime and propagation. Mechanistically, the added cross‐gradient transport promotes convective detrainment and environmental mixing, which cools and moistens the mid‐troposphere, weakens downward momentum transport, alleviates excessive downwelling shortwave biases, and warms the surface temperature. The optimized rotation angle yields comparable improvements at 4‐km and 1‐km grid spacing, demonstrating resolution‐independent benefits across the CPM gray zone. By targeting the dynamical root of under‐mixed convective circulations, rather than tuning model microphysics or closures, OAS delivers robust, scale‐aware improvements in precipitation statistics, cloud vertical structure, and characteristics of MCS (MCSs), offering a practical pathway to more reliable CPM simulations for weather and climate applications.
CPM↗
Quantum mechanical closure of partial differential equations with symmetries
We develop a statistical framework for the dynamical closure of spatiotemporal dynamics governed by partial differential equations. Employing the mathematical framework of quantum mechanics to embed the original classical dynamics into a quantum mechanical representation, we use the space of quantum density operators to model the unresolved degrees of freedom of the original dynamics in a statistical sense, and the framework of quantum measurement to predict their contributions to the resolved dynamics. The embedded dynamics is discretized by a positivity preserving process, leading to a compressed representation that is invariant under the dynamical symmetries of the resolved dynamics. We present a data based formulation of the closure scheme and apply it to a closure problem for the shallow water equations. The numerical results demonstrate that our closure model can accurately predict the main features of the true dynamics, including for out of sample initial conditions.
Delay embedding↗
Scalar flux transport models for self-similar turbulent mixing
A common approach to closing turbulent species flux in multicomponent Reynolds-averaged Navier-Stokes models is to use the standard gradient diffusion approximation. While such an approach has been shown to work well when applied to many canonical turbulent mixing configurations, a gradient diffusion approach is fundamentally limited in its ability to capture complex phenomena such as countergradient transport. For this reason, complicated mixing applications may benefit by treating the turbulent diffusivity with a model transport equation in a manner analogous to second-moment momentum closure in Reynolds-stress transport models. Here, the present work explores the development and application of two different scalar flux transport (SFT) models. Self-similarity constraints are derived for these models, and they are evaluated against gradient-diffusion-based models in several one- and two-dimensional problems of turbulent mixing. It is found that the new SFT models out-perform gradient diffusion models in problems involving rapid acceleration reversal and in problems involving anisotropic transport of materials. In addition, it is found that even a hybrid-SFT approach, in which an SFT equation is utilized along with a gradient diffusion closure, provides some measure of improvement over models that transport the mass flux rather than the scalar flux.
Reynolds-averaged Navier Stokes↗