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All sourcesNASA NTRSDOE OSTI
NASA NTRS1975

NASA/MSFC multilayer diffusion models and computer programs, version 5

The transport and diffusion models and algorithms developed for use by NASA in predicting concentrations and dosages downwind from normal and abnormal launches of rocket vehicles are described along with the associated computer programs for use in performing the calculations. Topics discussed include: the mathematical specifications and procedures used in the Preprocessor Program to calculate rocket exhaust cloud rise, cloud dimensions, and other input parameters to the transport and diffusion models; the revised mathematical specifications for the Multilayer Diffusion Models; users' instructions for implementing the Preprocessor and Multilayer Diffusion Models Programs; and worked example problems illustrating the use of the models and computer programs.

Dumbauld, R. K.↗
DOE OSTI2024

Diffusion-Model-Assisted Supervised Learning of Generative Models for Density Estimation

Here, we present a supervised learning framework of training generative models for density estimation. Generative models, including generative adversarial networks (GANs), normalizing flows, and variational auto-encoders (VAEs), are usually considered as unsupervised learning models, because labeled data are usually unavailable for training. Despite the success of the generative models, there are several issues with the unsupervised training, e.g., requirement of reversible architectures, vanishing gradients, and training instability. To enable supervised learning in generative models, we utilize the score-based diffusion model to generate labeled data. Unlike existing diffusion models that train neural networks to learn the score function, we develop a training-free score estimation method. This approach uses mini-batch-based Monte Carlo estimators to directly approximate the score function at any spatial-temporal location in solving an ordinary differential equation (ODE), corresponding to the reverse-time stochastic differential equation (SDE). This approach can offer both high accuracy and substantial time savings in neural network training. Once the labeled data are generated, we can train a simple, fully connected neural network to learn the generative model in the supervised manner. Compared with existing normalizing flow models, our method does not require the use of reversible neural networks and avoids the computation of the Jacobian matrix. Compared with existing diffusion models, our method does not need to solve the reverse-time SDE to generate new samples. As a result, the sampling efficiency is significantly improved. We demonstrate the performance of our method by applying it to a set of 2D datasets as well as real data from the University of California Irvine (UCI) repository.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Random Walks With Tweedie: A Unified View of Score-Based Diffusion Models [In the Spotlight]

We present a concise derivation for several influential score-based diffusion models that relies on only a few textbook results. Diffusion models have recently emerged as powerful tools for generating realistic, synthetic signals—particularly natural images—and often play a role in state-of-the-art algorithms for inverse problems in image processing. While these algorithms are often surprisingly simple, the theory behind them is not, and multiple complex theoretical justifications exist in the literature. Here, in this study, we provide a simple and largely self-contained theoretical justification for score-based diffusion models that is targeted towards the signal processing community. This approach leads to generic algorithmic templates for training and generating samples with diffusion models. We show that several influential diffusion models correspond to particular choices within these templates and demonstrate that alternative, more straightforward algorithmic choices can provide comparable results. This approach has the added benefit of enabling conditional sampling without any likelihood approximation.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Experimental observation of nonlinear relation between pressure and water flux is consistent with the solution-diffusion model

In several recent studies, it has been proposed that the fundamental understanding of penetrant transport in dense polymer membranes occurring via the solution-diffusion model, which has been the generally accepted theoretical framework for describing penetrant transport in such materials for the past several decades, is flawed. An alternate mechanistic framework based on the idea of two-phase flow in a porous medium (i.e., pore-flow) has been broadly advanced instead, with proponents of this approach claiming that the pore-flow theoretical framework provides the necessary mechanistic insight to design novel polymeric membrane materials for emerging applications. In this study, we show experimental results for hydraulic permeation of water that are entirely consistent with the solution-diffusion theory, without modification, for three dense polymeric membranes: crosslinked poly(ethylene glycol diacrylate) (XLPEGDA), Nafion 117 ionomer in the sodium counterion form (Nafion 117-Na), and cellulose acetate (CA). By measuring water flux at transmembrane pressures up to 240 bar, we observe a nonlinear relationship between the transmembrane pressure (TMP) and water flux, J w , for XLPEGDA and Nafion 117-Na, while this relationship is linear for CA. We demonstrate that the behavior of these three materials is described via the solution-diffusion model. According to the solution-diffusion model, flux is, to a good approximation, proportional to the transmembrane concentration difference induced by the pressure difference across the membrane, rather than to TMP itself. Water sorption isotherms are reported for all three materials. They further justify the nonlinear relationship between TMP and J w observed in XLPEGDA and Nafion 117-Na, emphasizing that the nonlinearity in the flux/TMP relationship stems from nonlinearities in the sorption isotherm with pressure. Additionally, the relationship between water flux and TMP can be predicted, a priori, with no adjustable parameters when a predictive model for the diffusion coefficient of water is employed in conjunction with the experimental water sorption isotherms in the solution-diffusion model. Furthermore, our results demonstrate the validity of the solution-diffusion model to describe transport of penetrants in dense polymer membranes, while highlighting the sensitivity of the solution-diffusion model to the many physical and mathematical simplifications commonly applied to the theory in literature.

materials↗
DOE OSTI2017

Validation of a mixture-averaged thermal diffusion model for premixed lean hydrogen flames

The mixture-averaged thermal diffusion model originally proposed by Chapman and Cowling is validated using multiple flame configurations. Simulations using detailed hydrogen chemistry are done on one-, two-, and three-dimensional flames. The analysis spans flat and stretched, steady and unsteady, and laminar and turbulent flames. Quantitative and qualitative results using the thermal diffusion model compare very well with the more complex multicomponent diffusion model. Comparisons are made using flame speeds, surface areas, species profiles, and chemical source terms. Once validated, this model is applied to three-dimensional laminar and turbulent flames. For these cases, thermal diffusion causes an increase in the propagation speed of the flames as well as increased product chemical source terms in regions of high positive curvature. The results illustrate the necessity for including thermal diffusion, and the accuracy and computational efficiency of the mixture-averaged thermal diffusion model.

Schlup, Jason↗
DOE OSTI2015

Extending the diffusion approximation to the boundary using an integrated diffusion model

The widely used diffusion approximation is inaccurate to describe the transport behaviors near surfaces and interfaces. To solve such stochastic processes, an integro-differential equation, such as the Boltzmann transport equation (BTE), is typically required. In this work, we show that it is possible to keep the simplicity of the diffusion approximation by introducing a nonlocal source term and a spatially varying diffusion coefficient. We apply the proposed integrated diffusion model (IDM) to a benchmark problem of heat conduction across a thin film to demonstrate its feasibility. We also validate the model when boundary reflections and uniform internal heat generation are present.

36 MATERIALS SCIENCE↗
DOE OSTI2025

Modeling inter‐reader variability in clinical target volume delineation for soft tissue sarcomas using diffusion model

Abstract Background Accurate delineation of the clinical target volume (CTV) is essential in the radiotherapy treatment of soft tissue sarcomas. However, this process is subject to inter‐reader variability due to the need for clinical assessment of risk and extent of potential microscopic spread. This can lead to inconsistencies in treatment planning, potentially impacting treatment outcomes. Most existing automatic CTV delineation methods do not account for this variability and can only generate a single CTV for each case. Purpose This study aims to develop a deep learning‐based technique to generate multiple CTV contours for each case, simulating the inter‐reader variability in the clinical practice. Methods We employed a publicly available dataset consisting of fluorodeoxyglucose positron emission tomography (FDG‐PET), x‐ray computed tomography (CT), and pre‐contrast T1‐weighted magnetic resonance imaging (MRI) scans from 51 patients with soft tissue sarcoma, along with an independent validation set containing five additional patients. An experienced reader drew a contour of the gross tumor volume (GTV) for each patient based on multi‐modality images. Subsequently, two additional readers, together with the first one, were responsible for contouring three CTVs in total based on the GTV. We developed a diffusion model‐based deep learning method that is capable of generating arbitrary number of different and plausible CTVs to mimic the inter‐reader variability in CTV delineation. The proposed model incorporates a separate encoder to extract features from the GTV masks, leveraging the critical role of GTV information in accurate CTV delineation. Results The proposed diffusion model demonstrated superior performance with the highest Dice Index (0.902 compared to values below 0.881 for state‐of‐the‐art models) and the best generalized energy distance (GED) (0.209 compared to values exceeding 0.221 for state‐of‐the‐art models). It also achieved the second‐highest recall and precision metrics among the compared ambiguous image segmentation models. Results from both datasets exhibited consistent trends, reinforcing the reliability of our findings. Additionally, ablation studies exploring different model structures and input configurations highlighted the significance of incorporating prior GTV information for accurate CTV delineation. Conclusions The proposed diffusion model successfully generates multiple plausible CTV contours for soft tissue sarcomas, effectively capturing inter‐reader variability in CTV delineation.

Dong, Yafei [Yale Biomedical Imaging Institute Yal↗
DOE OSTI2023

Ensemble flow reconstruction in the atmospheric boundary layer from spatially limited measurements through latent diffusion models

Due to costs and practical constraints, field campaigns in the atmospheric boundary layer typically only measure a fraction of the atmospheric volume of interest. Machine learning techniques have previously successfully reconstructed unobserved regions of flow in canonical fluid mechanics problems and two-dimensional geophysical flows, but these techniques have not yet been demonstrated in the three-dimensional atmospheric boundary layer. Here, we conduct a numerical analogue of a field campaign with spatially limited measurements using large-eddy simulation. We pose flow reconstruction as an inpainting problem, and reconstruct realistic samples of turbulent, three-dimensional flow with the use of a latent diffusion model. The diffusion model generates physically plausible turbulent structures on larger spatial scales, even when input observations cover less than 1% of the volume. Through a combination of qualitative visualization and quantitative assessment, we demonstrate that the diffusion model generates meaningfully diverse samples when conditioned on just one observation. These samples successfully serve as initial conditions for a large-eddy simulation code. We find that diffusion models show promise and potential for other applications for other turbulent flow reconstruction problems.

17 WIND ENERGY↗
NASA NTRS1985

Exhaust Effluent Diffusion Model

Rocket Exhaust Effluent Diffusion Model (REEDM) predicts concentrations, dosages, and depositions downwind from normal and abnormal launches of rocket vehicles at NASA's Kennedy Space Center. REEDM written in FORTRAN IV for interactive execution.

Bjorklund, J. R.↗
NASA NTRSDate not supplied

Thermodynamic and Diffusion Model Estimates on Metamorphic Temperatures and Timescales for Basaltic Eucrite GRA 98098

Introduction: HED meteorites are thought to rep-resent igneous rocks from Vesta’s basaltic crust and preserve evidence of early crustal metamorphism. Determining the temperatures and timescales of thermal metamorphism is important for reconstructing crustal evolution in the early solar system. Here, we study basaltic eucrite Graves Nunataks (GRA) 98098, which has been identified as a highly metamorphosed eucrite [1]. We present new estimates on metamorphic temperatures determined via thermodynamic modeling as well as the initial results from diffusion models constraining timescales of thermal metamorphism. Sample Description: GRA 98098 is an unbrecciated eucrite with a granoblastic plagioclase and pyroxene mineralogy. Millimeter to cm-long lathes of tridymite cross-cut and poikilitically enclose plagioclase and pyroxene [1,this work]. Pyroxene grains have exsolved into Ca-rich (~Wo38En29Fs33) and Ca-poor (~Wo4.5En36Fs59.5) lamellae. Both unzoned and zoned plagioclase grains are observed. Unzoned plagioclase grains are found solely with tridymite laths. These grains have ~An92 compositions. The cores of the zoned plagioclase grains have the same composition and thin, relatively sodic rims (~An67), (Fig. 1). The bulk sample is unusually enriched in highly in-compatible elements and has one of the most fractionated REE patterns reported [1]. Maximum metamorphic temperatures of 985±78°C have been estimated using two-pyroxene thermometry [2]. Methods: Thermodynamic modeling. Thermodynamic models were constructed using the software Perple_X, which employs a Gibbs free energy minimization in order to determine the most stable phase assemblage for a given bulk rock composition [3]. Bulk composition was calculated using mineral com-positions acquired via EMPA (this study) and the observed abundancies present in the thin section. Two bulk compositions were estimated; 1) includes all phases present in the thin section, (assumes that all phases are present during metamorphism), 2) excludes tridymite from the bulk calculation (assumes that tridymite was not present during metamorphism). In order to determine whether metamorphic equilibria was achieved and estimate temperatures of metamorphism, we compared measured pyroxene compositions with thermodynamically predicted compositions [4, Fig. 2]. Diffusion Modeling. Several time-temperature de-pendent diffusion profiles were calculated in order to determine the best match for XAn chemical profiles observed at the edges of the zoned plagioclase (Fig. 3). We assumed that the start condition was a stepwise gradient at the plagioclase/pyroxene interface. We also assumed an average diffusion coefficient (D) and a constant temperature using the equation in [5]. D was determined for two temperatures (T = 1060ºC; near eucrite solidus [6] and T = 985ºC; metamorphism reported in [4]) and then XAn was calculated as a function of distance from plagioclase core to rim using an error function solution to Fick’s second law. Results: Thermodynamic model results are summarized in Fig. 2. For a bulk composition that includes all phases in the thin section, pyroxene endmember compositions plot in the following temperature ranges: Fs ~660-860ºC, En~1000ºC & 1150ºC, and Wo~760-900ºC (Fig. 2a). For a bulk composition that excludes tridymite from the peak metamorphic assemblage (i.e., the bulk composition minus the contribution from tridymite), a temperature range could not be determined for the Fs component of pyroxene. For Wo, T~760-900ºC and En, T~1000ºC & 1150ºC (Fig. 2b). Fig. 3 summarizes the diffusion model results. For T = 1060°C & 985°C, the most appropriate time interval was estimated based on which diffusion curve most matched (solid lines, Fig. 3) the EMPA data. For T = 1060°C, the best looking match was t = 500 ka. For T = 985°C, the best match was t = 7 Ma. Discussion and future work: Temperature estimates from thermodynamic models are not conclusive because the temperature ranges determined for pyroxene endmember stability do not overlap (colored fields in Fig. 2), thus implying that there is disequilibrium between pyroxene crystals and the bulk composition considered [4]. Thus, additional exploration is needed to define a metamorphically equilibrated do-main that accurately records peak temperature. The utility of defining metamorphically equilibrated do-mains to improve the accuracy and level of detail elucidated regarding the petrogenetic history of metamorphose samples has been demonstrated previously [4,7]. We suggest that in the case of Fig. 2a, the thin section composition is not representative of the length scales over which metamorphic equilibrium was achieve and in the case of Fig. 2b, the assumption that tridymite was not present during metamorphism was incorrect. However, results from thermodynamic models can provide insight into the relative timing of mineral and compositional textures. For example from texture alone, it is unclear whether tridymite was igneous in origin and represents the last bits of melt in a crystallizing magma chamber, or if it formed during (and possibly initiated) open system thermal metamorphism. The latter could be consistent with a partial melt hypothesis [8,9] while the former implies that simple fractional crystallization can yield the textures present in GRA 98098. The lack of coincidence be-tween pyroxene endmember compositions in Fig. 2b suggest that the bulk composition minus tridymite was not the assemblage in equilibrium with the pyroxene, suggesting that tridymite was present during metamorphism and formed during igneous crystallization. We conclude that the development of the Na-rich plagioclase rims likely occurred during or immediately after peak thermal metamorphism, because eucrites of similar metamorphic grade and texture have unzoned plagioclase (~An92) [2,4,8], and Na zoning is only observed in the plagioclase not included in the tridymite. This suggests that the zoning formed after tridymite formation, and therefore after igneous crystallization. Thus, the timescales calculated via diffusion modeling possibly represent the time interval over which thermal metamorphism occurred. Cooling rates approximated for the Vestan crust predict that the crust cooled below 300°C around 35-40 Ma after formation[10]. This is consistent with our modeling results that predict formation of the Na rich plagioclase rims occurring at higher temperatures over a period of 0.5 to 7 Ma years. Future work. Additional thermodynamic modeling work will focus on selecting an equilibrated bulk rock domain in which to elucidate metamorphic conditions. Diffusion models currently provide a minimum time-scale, since diffusion slows down as the system cools. Future work with will focus on integrating cooling into the diffusion models and constraining the depth at which thermal metamorphism occurs because it could be used to determine whether the range of time-scales calculated for thermal metamorphism are consistent with the geologic environment.

J S Gorce↗
DOE OSTI2018

A reduced thermal diffusion model for H and H 2

Here this work details the development of a new, reduced thermal diffusion model. The proposed model derives from the thermal diffusion model of Chapman and Cowling (1970). In its derivation, a set of mixture-averaged like approximations are made, which results in the number of operations being reduced from $\mathscr{O}$($\mathcal{n}^2$) to sub-linear, where n is the number of species in the chemical model. With these approximations, the new, reduced model thermal diffusion coefficients can be calculated independently for each species. The model is validated against multicomponent thermal diffusion cases using multiple fuel and diluent mixtures at various pressures, temperatures, and equivalence ratios. The resulting reduced model thermal diffusion fluxes agree well with the multicomponent values, with a multiplicative scaling factor identified using a least squares regression. Unstretched laminar flame speeds are compared using the multicomponent and newly developed models. Finally, an a posteriori comparison in a turbulent configuration shows excellent agreement of both the mean and fluctuations of the thermal diffusion coefficients.

42 ENGINEERING↗
DOE OSTI2023

Blackout Diffusion: Generative Diffusion Models in Discrete-State Spaces

Typical generative diffusion models rely on a Gaussian diffusion process for training the backward transformations, which can then be used to generate samples from Gaussian noise. However, real world data often takes place in discrete-state spaces, including many scientific applications. Here, we develop a theoretical formulation for arbitrary discrete-state Markov processes in the forward diffusion process using exact (as opposed to variational) analysis. We relate the theory to the existing continuous-state Gaussian diffusion as well as other approaches to discrete diffusion, and identify the corresponding reverse-time stochastic process and score function in the continuous-time setting, and the reverse-time mapping in the discrete-time setting. As an example of this framework, we introduce “Blackout Diffusion”, which learns to produce samples from an empty image instead of from noise. Numerical experiments on the CIFAR-10, Binarized MNIST, and CelebA datasets confirm the feasibility of our approach. Generalizing from specific (Gaussian) forward processes to discrete-state processes without a variational approximation sheds light on how to interpret diffusion models, which we discuss.

Santos, Javier E.↗
NASA NTRS1988

Analytic solution of the box diffusion model for a global ocean

The paper presents an analytic solution for the temperature of the mixed layer for a box diffusion model of a global ocean. It is shown that the exact solution approaches the appropriate limits of a mixed layer ocean model and of a diffusive ocean without a mixed layer on top. It is also shown that, over a large range of parameters of the box diffusion model, the difference between the temperature calculated using a box diffusion model and the temperature calculated using a purely diffusive ocean is buried in the noise.

Lebedeff, Sergej A.↗
DOE OSTI2023

Code for the manuscript titled "Blackout Diffusion: Generative Diffusion Models in Discrete-State Spaces"

We would like to disclose two scripts, written in Jupyter notebook, in which we implement the "Blackout Diffusion Process" described in the manuscript "Blackout Diffusion: Generative Diffusion Models in Discrete-State Spaces" (LA-UR-23-20509), to be submitted to the International Conference of Machine Learning (ICML). The abstract of the manuscript is append below. == Typical generative diffusion models rely on a Gaussian diffusion process for training the backward transformations, which can then be used generate samples from Gaussian noise. However, real world data often takes place in discrete-state spaces, which occur in many scientific applications. Here, we develop a theoretical formulation for arbitrary discrete-state Markov processes in the forward diffusion process. We relate the theory to the existing continuous-state Gaussian diffusion and identify the corresponding reverse-time stochastic process and score function in the continuous-time setting, and the reverse-time mapping in the discrete-time setting. As an example of this framework, we introduce "Blackout Diffusion", which learns to produce samples from an empty image instead of from noise. Numerical experiments on the CIFAR-10 dataset confirm the feasibility of generative diffusion modeling in a discrete space. Generalizing from specific (Gaussian) forward processes to a more general framework also sheds light on how to interpret generative diffusion models and their mathematical structure, which we comment on.

Lin, Yen Ting↗
DOE OSTI2026

Multi-Objective design of interlocking metasurfaces using conditional diffusion models

Unit cell design remains a major challenge for interlocking metasurfaces, a promising joining technology for dissimilar materials, due to the complex, competing, multivariate design space and the need for rapid adaptation to varying performance requirements. This study explores Conditional Diffusion Models as a design optimization tool for interlocking metasurfaces. Given the complex, competing, multivariate design space for interlocking metasurfaces, unit cell design remains a major challenge for this joining technology. We trained a conditional diffusion model on 25,000 finite element analysis-simulated interlocking metasurface unit cells to generate designs with tailored thermo-mechanical properties (tensile strength, shear strength, and thermal conductivity) based on specified performance criteria. The model demonstrated a success rate of approximately 72 % in producing designs that met specified property bounds. The conditional diffusion model generated both thermally resistive and conductive designs, revealing clear trends in design characteristics: taller, dendritic structures were advantageous for tensile loads, while shorter, robust designs excelled in shear applications. Our findings indicate that the model's performance is more influenced by the breadth of the design space than by the quantity of training data, highlighting the importance of expansive design domains for generating innovative solutions. This work establishes conditional diffusion models as a highly efficient and adaptable tool for rapid interlocking metasurface unit cell design, paving the way for advancements in multi-material joining technologies, as well as highlighting the justification to leverage conditional diffusion models as design tools across complex design domains.

Conditional diffusion models↗