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DOE OSTI2024

A Finite Difference informed Random Walk solver for simulating radiation defect evolution in polycrystalline structures with strongly inhomogeneous diffusivity

Diffusivity of species and defects on grain boundaries is usually several orders of magnitude larger than that inside grains. Such strongly inhomogeneous diffusivity requires prohibitively high computational demands for modeling microstructural evolution. Here, this paper presents a highly-efficient numerical solver, combining the Finite Difference method and Random Walk model, designed for accurately modeling strongly inhomogeneous diffusion within polycrystalline structures. The proposed solver, termed Finite Difference informed Random Walk (FDiRW), integrates a customized Finite Difference (cFD) scheme tailored for fast diffusion along thin grain boundaries represented by a single-layer of nodes. Numerical experiments demonstrate that the FDiRW solver achieves an impressive efficiency gain of 1560x compared to traditional Finite Difference methods while maintaining accuracy, making it feasible for personal computer machines to handle diffusional systems with strongly inhomogeneous diffusivity across static polycrystalline microstructures. The model has been successfully applied to simulate radiation defect evolution, showcasing its scalability to engineering scales in both length and time dimensions.

36 MATERIALS SCIENCE↗

DOE OSTI2025

Effect of lithium diffusion into Ga 2 O 3 thin films

The integration of lithium based compounds (e.g., Li:NiO/Ga 2 O 3 , LiGa 5 O 8 /Ga 2 O 3 ) in Ga 2 O 3 based pn-heterojunctions raises concerns about interface stability, i.e., Li diffusion effects on Ga 2 O 3 properties. In this work the ex-situ diffusion of Li is investigated in three different Ga 2 O 3 epilayers systems [(001) κ-Ga 2 O 3 and (−201) β-Ga2O3 heteroepitaxy on (001) α-Al 2 O 3 , and (010) β-Ga 2 O 3 homoepitaxy] at relevant temperatures for the synthesis / processing of Li-based epilayers. It is here experimentally demonstrated and quantified the Li diffusion in all the investigated Ga 2 O 3 epilayers systems and Li bulk (D Li,bulk ) and 2D defects (D Li,2D ) diffusion coefficients are provided. In the case of the (010) β-Ga 2 O 3 homoepitaxial layer (nominally free of structural defects), hybrid functional theory calculations foresee a diffusion mechanism mediated by Ga vacancies (VGa). Moreover, in the (010) β-Ga 2 O 3 homo-layer a significant effect on its functional properties (e.g., additional Raman vibrational modes, induced conductivity in an otherwise insulating sample) upon the Li-diffusion process is experimentally highlighted and tentatively related to the passivation of acceptor defects (i.e., formation of V Ga -nLi complexes).

Defects↗

DOE OSTI2025

Effect of Mg and Ni impurities on tritium diffusion in lithium ceramics through cluster dynamics simulations

This study investigates the impact of Mg and Ni doping on tritium diffusion in LiAlO 2 and LiAl 5 O 8 ceramics, that are used in tritium-producing burnable absorber rods (TPBARs). Utilizing Centipede simulations across a broad temperature range (500 K to 1250 K), we explore the interplay between defect dynamics, cluster formation, and tritium mobility. In LiAlO 2 , Mg doping significantly enhances tritium diffusivity by increasing tritium interstitial concentrations and diffusion coefficients of key species, thereby doubling the overall tritium diffusivity. Ni doping, while shifting the dominant defect to Li vacancies, maintains high tritium mobility due to the low binding energy of Li vacancy-tritium complexes, which ensures effective tritium migration. In LiAl 5 O 8 , Mg and Ni doping results in a slight reduction in the diffusion coefficients of key species, yet the dramatic increase in tritium interstitial concentrations compensates, leading to a net small increase in tritium diffusivity. In conclusion, the findings highlight the critical role of defects in tritium transport and the effect of Mg and Ni defects on the performance of these ceramics in demanding nuclear environments.

Cluster dynamics↗

DOE OSTI2025

First-principles investigation of cerium and neodymium diffusion in BCC chromium and vanadium via vacancy-mediated transport

Lanthanide transport plays a crucial role in the performance and longevity of metallic nuclear fuels. This study examines the diffusion behavior of Ce and Nd—two major fission products—in body-centered cubic (BCC) Cr and V, which are potential liner or coating materials for mitigating fuel-cladding chemical interactions (FCCI). Using density functional theory (DFT) calculations and self-consistent mean-field (SCMF) analysis, the vacancy-mediated diffusion coefficients are evaluated. Our findings reveal that Ce and Nd act as oversized solutes and are strongly bound to vacancies in BCC Cr and V, with diffusivities in Cr significantly lower than in V and in hexagonal closed-packed (HCP) Zr, as investigated in our previous work. The activation energies for Ce and Nd diffusion are 3.39 and 3.32 eV, respectively, in BCC Cr, and 2.56 and 2.33 eV, respectively, in BCC V. Analysis of vacancy drag and partial diffusion coefficient ratios indicates a strong tendency for lanthanide enrichment at vacancy sinks in BCC Cr, and to a lesser extent in BCC V, with this effect persisting up to the melting point in Cr and remaining substantial for Nd in V at high temperatures. Under irradiation, the increase in vacancy concentration is expected to enhance lanthanide transport, potentially accelerating interactions at liner-cladding interfaces. Although BCC Cr exhibits relatively low lanthanide diffusivities under equilibrium conditions, the expected segregation tendencies under irradiation suggest that Zr liners may be a more favorable option. Further investigations using rate theory, cluster dynamics, and phase-field modeling are required to quantitatively assess the performance of these materials in reactor environments.

36 - MATERIALS SCIENCE↗

DOE OSTI2024

Adsorption and Solvation Modulate Rhodamine B Diffusion in Ethanol/Water-Filled Anodic Alumina Nanopores

Confinement of solvents and solutes within nanoporous materials frequently leads to the emergence of unique mass transport behaviors that, once fully understood, may lead to improved chemical separations. Here, the diffusion of Rhodamine B (RhB) dye within 10 and 20 nm diameter anodic aluminum oxide (AAO) nanopores filled with binary ethanol/water mixtures is investigated. Mixture compositions spanning from pure ethanol to pure water are employed. The results of confocal fluorescence correlation spectroscopy studies reveal that RhB diffusion occurs by a two-component mechanism comprising composition-dependent fast and slow motions, characterized by diffusion coefficients D f and D s . The results are consistent with those of previous studies performed under more limited conditions [J. Phys. Chem. C, 2023, 127, 411-420]. The fast component scales with mixture viscosity and is assigned to hindered bulk-like diffusion in central pore regions. Slow diffusion likely involves adsorption of RhB to the pore surface and may be described by a desorption mediated mechanism. The occurrence of RhB adsorption to the AAO surface is verified at the single-molecule level by wide-field fluorescence imaging of membrane cross-sectional surfaces. Unique composition dependent trends in the autocorrelation amplitude and in D s that mimic bulk RhB solubility are revealed. D s is found to be smallest in pure ethanol and pure water and largest in intermediate mixtures. These results suggest that RhB surface adsorption is strongest in the pure liquids and weakest in mixtures of intermediate composition, where the dye is least soluble, and most soluble, respectively. As a result, molecular dynamics simulations reveal that a water layer appears on the pore surface under most conditions, while RhB is solvated primarily by ethanol. The composition dependence of RhB diffusion is concluded to reflect its solvation dependent interactions with the pore walls.

36 MATERIALS SCIENCE↗

DOE OSTI2025

Experimental Characterization of Hydrogen Diffusion in Shale Rocks for Geologic Storage Applications

As global energy systems undergo a transition to cleaner alternatives, geologic hydrogen storage has emerged as a promising solution for large-scale energy storage. A critical factor in determining the feasibility of this approach is the effectiveness of caprock formations, such as shale, in preventing hydrogen migration. This study investigates the diffusion behavior of hydrogen through shale to assess its suitability as a caprock for geologic hydrogen storage. Using a novel double-seal core holder design and a through-diffusion apparatus, hydrogen diffusion was measured through shale rock from the Eagle Ford and Wolfcamp Formations under dry conditions. These measurements were complemented by microstructural and mineralogical analyses using low-pressure nitrogen adsorption and X-ray diffraction. The effective diffusion coefficient of hydrogen in these shale caprocks ranged from 2.51 × 10 –8 to 9.85 × 10 –8 m 2 /s. Notably, we observed that the diffusion behavior was more related to the pore network structure and could not be attributed to differences in the total pore volume between shale types alone. Here, to further understand the role of pore network complexity, a fractal pore model was developed to correlate tortuosity with the fractal dimension of the pore structure (a measure of pore network complexity). The proposed model closely matched tortuosity values obtained from diffusion experiments, outperforming existing theoretical tortuosity–porosity correlations. These findings provide key quantitative parameters needed to assess the feasibility of geologic hydrogen storage as well as insights that can be applied to hydrogen storage in a range of geologic formations.

08 HYDROGEN↗

DOE OSTI2025

Defect-Mediated Diffusion Pathways in Spodumene Accelerate Lithium Transport

Lithium extraction from naturally occurring α-spodumene is hindered by poor lithium diffusivity, necessitating high-temperature phase transformation to a low-density β polymorph. Although β spodumene exhibits up to 5 orders of magnitude higher lithium-ion diffusivity, both phases have diffusion activation energies between 0.8 and 1 eV, indicating that polymorph density is not the controlling factor over diffusivity. We show that aluminum vacancies facilitate lithium-ion diffusion in α-spodumene by reducing the migration barrier from 2.4 to 0.9 eV. Bond valence site energy and nudged elastic band calculations show a new lithium local minimum site which promotes a one-dimensional percolation network by reducing the lithium intersite distance from 4.5 Å to 2.9 Å. However, aluminum vacancies are energetically unfavorable to percolate through the whole structure, resulting in very low net lithium diffusivity and highlighting the critical role of nonstoichiometric defects in facilitating lithium transport in rigid aluminosilicate structures.

Chemical structure↗

DOE OSTI2024

Direct Visualization of Defect‐Controlled Diffusion in van der Waals Gaps

Abstract Diffusion processes govern fundamental phenomena such as phase transformations, doping, and intercalation in van der Waals (vdW) bonded materials. Here, the diffusion dynamics of W atoms by visualizing the motion of individual atoms at three different vdW interfaces: hexagonal boron nitride (BN)/vacuum, BN/BN, and BN/WSe 2 , by recording scanning transmission electron microscopy movies is quantified. Supported by density functional theory (DFT) calculations, it is inferred that in all cases diffusion is governed by intermittent trapping at electron beam‐generated defect sites. This leads to diffusion properties that depend strongly on the number of defects. These results suggest that diffusion and intercalation processes in vdW materials are highly tunable and sensitive to crystal quality. The demonstration of imaging, with high spatial and temporal resolution, of layers and individual atoms inside vdW heterostructures offers possibilities for direct visualization of diffusion and atomic interactions, as well as for experiments exploring atomic structures, their in situ modification, and electrical property measurements of active devices combined with atomic resolution imaging.

Chemistry↗

DOE OSTI2026

Nanodomain Formation and Temperature-Dependent Diffusion in Deep Eutectic Solvents Revealed by Single-Molecule Tracking

Deep eutectic solvents (DESs) are typically regarded as homogeneous liquids; however, recent work shows that many exhibit nanoscale structural heterogeneity. Most studies attribute these nanoscale features to short-range chemical interactions. It is still unclear whether a long-range physical mechanism also plays a role. Here, in this study, we examined the nanoscale structure in two hydrophobic DESs, 1:3 tetrabutylammonium bromide: l-menthol (DES-butyl) and 1:3 tetraoctylammonium bromide: l-menthol (DES-octyl). The notation 1:3 represents the molar ratio of the hydrogen bond acceptors to hydrogen bond donors used in the synthesis of the DESs. Single-molecule tracking (SMT) coupled with maximum entropy method (MEM) analysis was used to measure the number of diffusion populations of a dilute concentration of an added fluorescent probe. The presence of more than one population of diffusion coefficients indicates the existence of multiple local environments for the fluorescent probe (i.e., nanoscale structures in the DES). DES-butyl showed a relatively narrow diffusion coefficient distribution centered at 0.55 μm 2 /s, whereas DES-octyl displayed two distinct diffusing populations at 20 °C, with diffusion coefficients of 0.12 μm 2 /s and 0.53 μm 2 /s for the slow and fast populations, respectively. As DES-octyl was heated, the slow-diffusing population steadily diminished and disappeared above ∼30 °C, indicating that the nanodomains present at lower temperatures collapse as the liquid becomes more thermodynamically mixed. This temperature-dependent homogenization is consistent with a physical mechanism of nanostructure formation, for example, liquid–liquid phase separation (LLPS), wherein the structure is not driven solely by specific chemical interactions. The SMT-MEM results suggest that a long-range physical mechanism is the most plausible origin of the measured nanoscale structure in DES-octyl.

Opare-Addo, Jemima [Ames Laboratory (AMES), Ames, ↗

DOE OSTI2025

Translational diffusion in supercooled water at and near the glass transition temperature—136 K

The properties of amorphous solid water at and near the calorimetric glass transition temperature, T g , of 136 K have been debated for years. One hypothesis is that water turns into a “true” liquid at T g (i.e., it becomes ergodic) and exhibits all the characteristics of an ergodic liquid, including translational diffusion. A competing hypothesis is that only rotational motion becomes active at T g , while the “real” glass transition in water is at a considerably higher temperature. To address this dispute, we have investigated the diffusive mixing in nanoscale water films, with thicknesses up to ∼100 nm, using infrared (IR) spectroscopy. The experiments used films that were composed of at least 90% H 2 O with D 2 O making up the balance and were conducted under conditions where H/D exchange was essentially eliminated. Because the IR spectra of multilayer D 2 O films (e.g., thicknesses of ∼3–6 nm) embedded within thick H 2 O films are distinct from the spectrum of isolated D 2 O molecules within H 2 O, the diffusive mixing of (initially) isotopically layered water films could be followed as a function of annealing time and temperature. The results show that water films with total thicknesses ranging from ∼20 to 100 nm diffusively mixed prior to crystallization for temperatures between 120 and 144 K. The translational diffusion had an Arrhenius temperature dependence with an activation energy of 40.8 ± 3.5 kJ/mol, which indicates that water at and near T g is a strong liquid. The measured diffusion coefficient at 136 K is 6.25 ± 1.4 × 10 −21 m 2 /s.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2026

Molecular dynamics based study on the effects of cation size on the local structure and diffusion in polymerized ionic liquids

We have used coarse-grained molecular dynamics simulations to understand the effect of cation size on ion diffusion in polymerized ionic liquids at temperatures well above the glass transition temperature of the polymers. We investigated dependencies of the diffusion constant on the cation radius and static dielectric constant and interpreted these results in terms of underlying structural changes, and decoupling phenomenon. We have found non-monotonic effects of the cation radius on the diffusion constant, with a maximum at an intermediate radius resulting from two different size dependent effects. Changes in the radial distribution functions characterizing spatial distribution of cations with respect to anions result in a monotonic increase in the coordination number. This yields an increase in the effective interaction energy as a function of cation radius, which can explain the decrease in the diffusion constant. This increase is counteracted at small cation radii by a corresponding increase in decoupling of cation and anion dynamics. In addition, the diffusion constant is found to increase on increasing the static dielectric constant for all radii of the cations. These results highlight the significant role of decoupling in designing polymerized ionic liquids with an enhanced diffusion constant of cations.

Gillespie, Colin [ORNL] (ORCID:0000000320287080)↗

DOE OSTI2024

Simulating hindered grain boundary diffusion using the smoothed boundary method

Abstract Grain boundaries can greatly affect the transport properties of polycrystalline materials, particularly when the grain size approaches the nanoscale. While grain boundaries often enhance diffusion by providing a fast pathway for chemical transport, some material systems, such as those of solid oxide fuel cells and battery cathode particles, exhibit the opposite behavior, where grain boundaries act to hinder diffusion. To facilitate the study of systems with hindered grain boundary diffusion, we propose a model that utilizes the smoothed boundary method to simulate the dynamic concentration evolution in polycrystalline systems. The model employs domain parameters with diffuse interfaces to describe the grains, thereby enabling solutions with explicit consideration of their complex geometries. The intrinsic error arising from the diffuse interface approach employed in our proposed model is explored by comparing the results against a sharp interface model for a variety of parameter sets. Finally, two case studies are considered to demonstrate potential applications of the model. First, a nanocrystalline yttria-stabilized zirconia solid oxide fuel cell system is investigated, and the effective diffusivities are extracted from the simulation results and are compared to the values obtained through mean-field approximations. Second, the concentration evolution during lithiation of a polycrystalline battery cathode particle is simulated to demonstrate the method’s capability.

Materials Science↗

DOE OSTI2024

In situ counter-diffusion crystallization and long-term crystal preservation in microfluidic fixed targets for serial crystallography

Compared with batch and vapor diffusion methods, counter diffusion can generate larger and higher-quality protein crystals yielding improved diffraction data and higher-resolution structures. Typically, counter-diffusion experiments are conducted in elongated chambers, such as glass capillaries, and the crystals are either directly measured in the capillary or extracted and mounted at the X-ray beamline. Despite the advantages of counter-diffusion protein crystallization, there are few fixed-target devices that utilize counter diffusion for crystallization. In this article, different designs of user-friendly counter-diffusion chambers are presented which can be used to grow large protein crystals in a 2D polymer microfluidic fixed-target chip. Methods for rapid chip fabrication using commercially available thin-film materials such as Mylar, propylene and Kapton are also detailed. Rules of thumb are provided to tune the nucleation and crystal growth to meet users' needs while minimizing sample consumption. These designs provide a reliable approach to forming large crystals and maintaining their hydration for weeks and even months. This allows ample time to grow, select and preserve the best crystal batches before X-ray beam time. Importantly, the fixed-target microfluidic chip has a low background scatter and can be directly used at beamlines without any crystal handling, enabling crystal quality to be preserved. The approach is demonstrated with serial diffraction of photoactive yellow protein, yielding 1.32 Å resolution at room temperature. Fabrication of this standard microfluidic chip with commercially available thin films greatly simplifies fabrication and provides enhanced stability under vacuum. These advances will further broaden microfluidic fixed-target utilization by crystallographers.

Liu, Zhongrui↗

DOE OSTI2026

Solar-Tracker Diffuse-Response Algorithm for Balancing Energy Gain and Mechanical Wear in Photovoltaic Systems

Single-axis solar tracking maximizes photovoltaic energy production under clear-sky conditions; however, its effectiveness decreases under cloudy and overcast skies, where diffuse irradiance dominates and the optimal module orientation changes. Conventional tracking algorithms either neglect sky conditions or rely on simplified diffuse-response strategies that may trigger frequent tracker repositioning under variable cloud cover, leading to increased mechanical wear with marginal energy gains. This work proposes an enhanced diffuse-response tracking algorithm that explicitly accounts for both the intensity and temporal persistence of cloudiness. By requiring overcast conditions to persist for a minimum duration before reorienting the tracker to a diffuse-stow position, the proposed approach reduces unnecessary movements while preserving the benefits of diffuse-response operation. The algorithm is evaluated through numerical simulations based on historical meteorological data and validated using field measurements on monofacial and bifacial photovoltaic strings. The results show that the proposed strategy reduces excess tracker movement from 114% to 0.16% while maintaining nearly the same energy yield. Compared to a conventional diffuse-response algorithm, the associated energy reduction is minimal (≈0.17%) relative to the ≈0.37% yield gain observed at the studied location. These findings demonstrate that incorporating cloudiness duration enables a practical compromise between energy performance and tracker durability, particularly for monofacial photovoltaic systems.

14 SOLAR ENERGY↗

DOE OSTI2024

A finite difference informed random walker (FDiRW) solver for strongly inhomogeneous diffusion problems

In nature, many complex multi-physics coupling problems exhibit strong diffusivity inhomogeneity. For instance, in the context of radionuclide absorption by porous wasteform materials within a flowing waste stream, the difference of species’ diffusivity in solid and liquid phases spans by 3~8 orders of magnitude. To solve the diffusion equations with strongly inhomogeneous diffusivity, traditional discretization-based methods, such as the Finite Difference Method (FDM), require infinitesimally small time steps (<10 -10 ) as high spatial resolutions are employed in most microstructure evolution processes, leading to prohibitively high computational costs. Here, this work developed an integrated numerical approach (FDiRW: Finite Difference informed Random Walk) to tackle this challenge. The idea is that utilizing the Random Walk concept, the fast diffusion is modeled as a superposition of point source’s solution for a concentration distribution while FDM is used to obtain the point source’s solution at each node. A mesh-coarsening algorithm is developed to generate an exclusive coarse mesh for FDiRW approach to maximize its efficiency. The effectiveness of the coarse mesh-based FDiRW approach is validated by benchmarking Finite Difference solutions. Numerical results demonstrated that FDiRW achieves a remarkable 1000x computational efficiency improvement over FDM while preserving desired accuracy for a medium-sized model of 192 × 192 × 192 grids. Finally, as models scale up, a floating-point operations (PLOPs) analysis of the FDiRW algorithm reveals that its computational complexity grows quadratically in terms of the number of nodes employed in computation.

36 MATERIALS SCIENCE↗

DOE OSTI2025

A provably stable numerical method for the anisotropic diffusion equation in confined magnetic fields

We present a novel numerical method for solving the anisotropic diffusion equation in magnetic fields confined to a periodic box which is accurate and provably stable. We derive energy estimates of the solution of the continuous initial boundary value problem. A discrete formulation is presented using operator splitting in time with the summation by parts finite difference approximation of spatial derivatives for the perpendicular diffusion operator. Weak penalty procedures are derived for implementing both boundary conditions and parallel diffusion operator obtained by field line tracing. We prove that the fully-discrete approximation is unconditionally stable. Discrete energy estimates are shown to match the continuous energy estimate given the correct choice of penalty parameters. A nonlinear penalty parameter is shown to provide an effective method for tuning the parallel diffusion penalty and significantly minimises rounding errors. Several numerical experiments, using manufactured solutions, the “NIMROD benchmark” problem and a single island problem, are presented to verify numerical accuracy, convergence, and asymptotic preserving properties of the method. Finally, we present a magnetic field with chaotic regions and islands and show the contours of the anisotropic diffusion equation reproduce key features in the field.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

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↗

DOE OSTI2026

Enabling probabilistic learning on manifolds through double diffusion maps

Here, we present a generative learning framework for probabilistic sampling that extends Probabilistic Learning on Manifolds (PLoM), which is designed to generate statistically consistent realizations of a random vector in a finite-dimensional Euclidean space, informed by a (representative) set of observations. In its original form, PLoM constructs a reduced-order probabilistic model by combining three main components: (a) kernel density estimation to approximate the underlying probability measure, (b) Diffusion Maps to characterize the manifold of the data, and (c) a reduced-order Itô Stochastic Differential Equation (ISDE) to sample from the learned distribution. However, its sampling dynamics are posed in the ambient space and the retained number of reduced coordinates is chosen by projection-reconstruction error. In practice, this often (i) requires more coordinates than the data’s intrinsic dimension to achieve stable sampling and (ii) lacks a smooth, basis-independent lifting back to the data domain; moreover, standard Diffusion Maps emphasize harmonic eigenfunctions and can miss non-harmonic latent structure. We address these limitations by decoupling geometry learning from sampling: a first Diffusion Maps pass identifies non-harmonic coordinates on which we formulate a full-order ISDE directly in the latent space, while Double Diffusion Maps captures multiscale geometric features and Geometric Harmonics (GH) learns a smooth lifting map to the ambient variables that is independent of the particular diffusion basis. This hybrid design preserves the system’s dynamical richness with a compact geometric representation and enables principled out-of-sample inference. The effectiveness and robustness of the proposed method are illustrated through two numerical studies: one based on data generated from two-dimensional Hermite polynomial functions and another based on high-fidelity simulations of a detonation wave in a reactive flow.

Double diffusion maps↗