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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 1,171 records · Page 65
All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

Green synthesis of cellulose formate and its efficient conversion into 5-hydroxymethylfurfural

5-hydroxymethylfurfural (HMF) production from cellulose is a hot topic, while the rigidity of cellulose limits its efficient conversion into HMF. Cellulose formate has exhibited the great potential for HMF production but it’s synthesis via direct formylation of cellulose in formic acid is difficult. We reported a green synthetic route of cellulose formate and its efficient conversion into HMF. Ball milled cellulose formate (BCF) was synthesized via ball milling followed by formylation in formic acid at 80 °C for 0.5 h. The structure of BCF was thoroughly characterized and BCF was subsequently converted to HMF catalyzed by AlCl3 and HCl in dimethyl sulfoxide (DMSO)-H 2 O media. Results showed that ball milling of cellulose for 1–1.5 h efficiently deconstructed its crystallinity and degree of polymerization, boosting its further formylation and catalytic conversion. An overall 66.9% HMF yield from cellulose was achieved at 150 °C for 20 min using BCF derived from cellulose ball milled for 1.5 h as the substrate. BCF exhibits high reactivity toward HMF due to its excellent solubility and accessibility in DMSO-H 2 O media, and the release of formic acid in reaction further facilitates HMF production. BCF synthesized with different ball milling times shows different degrees of substitution of formyl groups, which could serve as an indicator and regulate the HMF production. The green synthesis of cellulose formate creates an efficient pathway for HMF production.

5-hydroxymethylfurfural↗
DOE OSTI2026

Systematic analysis of melt pool dynamics in laser processing of mixed powder feedstocks

Functionally graded materials (FGMs) fabricated via additive manufacturing of blended powders offer the potential to spatially tailor properties for new technologies, such as fusion first-wall systems, turbine blades, and spacecraft. However, processing these materials is difficult due to the multiplicity of processing parameters to optimize, all of which must be changed as substrate material, powder feedstock compositions, and melt pool dynamics evolve. Here, this work systematically evaluates the qualitative and quantitative effects of these variables on the melt pool size, shape, composition, and particle distribution in an exemplar Ti-Ta system, and connects the experimental results to Marangoni flow behavior and phenomena observed in other systems. Increasing laser power linearly increases melt pool size and layer thickness, driving engineering considerations such as part/geometrical tolerances. Decreasing laser velocity changes the melt pool shape from lenticular to convex and reduces chemical homogeneity due to extreme thermal and compositional gradients between the melt pool center and boundaries. Thermophysical property differences between the powder feedstock and substrate material, as well as the directionality of the gradient, affect dilution and melt pool dynamics, which in turn affect the melt pool boundary characteristics, shape, and uniformity. Mixed powder feedstocks of intermediate compositions do not behave according to linear interpolations between single-material endpoints, instead building taller and wider melt pools. As such, it is recommended to quantify process maps for at least one intermediate composition in the FGM or multi-material system of interest to ensure optimized processing parameters, predictable melt pool sizes and shapes, and compositional and spatial precision.

Dissimilar↗
DOE OSTI2020

Elastic moduli of high-density, sintered monoliths of yttrium dihydride

Yttrium dihydride has been considered as potential nuclear reactor moderator for micro-reactors operating in the thermal energy spectrum due to their ability to retain hydrogen to high temperatures while having a relatively low impact on neutron economy. Despite these advantages, it is difficult to prepare yttrium dihydride in high-density, near net-shape monoliths for moderator applications due to volume expansion upon hydriding and the pyrophoric nature of the material. In this work, high-density monoliths of yttrium dihydride were prepared by direct hydriding and using powder metallurgical methods. Resonant ultrasound spectroscopy was used to measure the elastic moduli of the sintered monoliths, while nanoindentation was used to determine the Young’s modulus and hardness of the directly-hydrided yttrium and the sintered monoliths. Single crystal elastic constants and related properties of yttrium dihydride were also determined using density functional theory (DFT) for comparison. We report mechanical properties of materials produced by both methods were observed to be consistent with the values from literature for yttrium dihydride. Due to the novelty of producing yttrium dihydride by powder metallurgy, this result indicated that the monoliths produced by powder metallurgy were of high quality. To that end, powder metallurgy is believed to be a viable method for large-scale production of yttrium dihydride monoliths for nuclear reactor moderator applications.

36 MATERIALS SCIENCE↗
DOE OSTI2020

Thermophysical properties of high-density, sintered monoliths of yttrium dihydride in the range 373–773 K

Yttrium dihydride is a promising nuclear reactor moderator for high-temperature, thermal micro-reactors due to its ability to retain hydrogen to high temperatures while having a relatively low impact on neutron economy. However, it is difficult to fabricate yttrium dihydride in high-density, near net-shape monoliths for moderator applications due to challenges associated with the hydrogen absorption process. Here, high-density monoliths of yttrium dihydride were prepared by powder metallurgical methods. The lattice strain and thermal expansion coefficient were measured using neutron diffraction of powders of directly-hydrided yttrium, while the molar heat capacity and thermal diffusivity of sintered yttrium dihydride were measured using differential scanning calorimetry (DSC) and laser flash analysis (LFA), respectively. The molar heat capacity of yttrium dihydride was also calculated using density functional theory (DFT) for comparison. These parameters were used to then calculate the thermal conductivity and resistivity of yttrium dihydride as a function of temperature. The thermophysical properties of materials produced by both methods were observed to be consistent with the values from literature for yttrium dihydride. Due to the novelty of producing yttrium dihydride by powder metallurgical methods, this result indicated that the sintered monoliths were of high quality and that powder metallurgy is a viable method for large-scale production of yttrium dihydride monoliths for nuclear reactor moderator applications.

36 MATERIALS SCIENCE↗
DOE OSTI2020

Dislocation cross-slip in precipitation hardened Mg–Nd alloys

An integrated experimental characterization and molecular dynamics (MD) simulation approach was used to explore the dislocation-precipitate interactions in a dilute Magnesium–Neodymium (Mg–Nd) precipitation hardened alloy. In situ indentation in a transmission electron microscope (TEM) and postmortem TEM characterization of deformed samples and MD simulations revealed that basal < a > type dislocations interacted differently with β 1 (Mg 3 Nd) and β"(Mg 3-7 Nd) precipitates. For β" precipitates, the basal dislocations directly shear the precipitates. For β 1 precipitates, such shearing becomes much more difficult because it requires the creation of antiphase boundaries in the ordered lattice of β 1 precipitates. Screw dislocations were observed to cross-slip from basal to the prismatic plane, which could be parallel to the broad facet of β 1 precipitates. It is postulated that double cross-slip (basal to prismatic to basal) via the Hirsch mechanism may enable screw dislocations to overcome the β 1 precipitates. MD simulations also revealed that an edge dislocation is unable to bypass the precipitate via the generation of screw dislocation segments plus the double cross-slip mechanism. The edge dislocation can bow around the precipitate, and the segment that adopts a screw character can cross-slip to the prismatic plane but with increasing applied stress, cross-slips back to the original basal plane to continue glide via precipitate shearing. Finally, the implication of glide dislocation - β 1 -precipitate interaction mechanisms on the strength and ductility of β 1 precipitate dominant microstructures is discussed.

36 MATERIALS SCIENCE↗
DOE OSTI2021

Deep learning inversion of gravity data for detection of CO 2 plumes in overlying aquifers

In this work, we developed an effective U-Net based deep learning (DL) model for inversion of surface gravity data on a rectangular grid to predict 2-D high-resolution subsurface CO 2 distribution along a vertical cross-section due to CO 2 leakage through a wellbore within a deep CO 2 storage reservoir. We used synthetic data to model two types of CO 2 leakage scenarios: one CO 2 plume in a shallow aquifer (single plume case), and two plumes present at different depths (double plume case). The 3-D synthetic plume samples were created by sampling among predetermined CO 2 plume depths, saturations, and volumes. The corresponding surface gravity data on a rectangular grid were generated by a 3-D forward model. The U-Net model detected 72% of single-plume samples, and one or both plumes in 75% of double-plume samples. Most of the undetected single plumes have small gravity field strengths below the typical noise level of 5 μGal. This model generated reproducible, reliable predictions with acceptable errors and demonstrated improved spatial resolution over the conventional least-squares inversion. In contrast to the conventional least-squares inversion, which often overestimates the size of its target and underestimates its density, this U-Net model accurately delineated the boundary of a target. Furthermore, this DL inversion detected deep, small, or low saturation CO 2 plumes that are often more difficult to resolve with conventional gravity inversion methods. We note the limitations of this feasibility study, including the use of synthetic data with regular CO 2 plume shapes, and the prediction of a 2-D plume cross-section rather than the full 3-D plume, as well, we recognize the lower detection fraction for double-plume scenarios. Nevertheless, this study demonstrates that DL gravity inversion is a promising and potentially superior method to conventional least-squares inversion. Our U-Net based deep learning inversion approach may be adapted for inversion of other types of geophysical data. DL inversion can facilitate near real-time monitoring of geologic carbon sequestration to provide site operators with prompt information about subsurface CO 2 distribution for risk management and mitigation.

58 GEOSCIENCES↗
DOE OSTI2020

Consequences of product inhibition in the quantification of kinetic parameters

While the potential for product inhibition in catalytic reactions is well known, the impact of neglected inhibition on measured kinetic parameters is often overlooked. The presence of product inhibition, most often caused by the competitive adsorption of products with reactants on catalytic active sites, is difficult to determine a priori for an arbitrary catalytic system. The significance of product inhibition relies on the concentration of the products, their adsorption thermodynamics on catalytically relevant sites, and process parameters such as temperature and pressure. When inhibition is significant, however, apparent activation energies and reaction orders vary from the differential-reactor apparent activation energy by a factor of 1/(1 - δ), where δ is the total inhibition order (e.g., the factor (1 - δ) = 1.6 for a system with product inhibition of -0.6 order). This is illustrated here with the kinetics of NO oxidation over Cu ion clusters (Cu x O y ) in Cu-SSZ-13, for which the product NO 2 inhibits the forward reaction. Furthermore, in the presence of inhibition, when only reactants are fed to a flow reactor or placed in a batch reactor, there is often no practical conversion that is low enough to guarantee differential behavior. Inclusion of products in the feed solves this problem, allowing accurate determination of kinetic parameters such as apparent reaction orders and activation energies. We also demonstrate that evaluation of the necessity of co-feeding products to assure measurement of differential-reactor data in a given catalytic system is straightforward from a plot of the log of the rate (or conversion) versus the log of the space time in a flow reactor or elapsed time in a batch reactor. We encourage inclusion of this test in all kinetic analyses that are reasonably approximated by power law rate expressions.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2022

Molybdenum oxide and sulfide active sites for isobutane dehydrogenation with methanol as a probe molecule

Molybdenum oxides and sulfides are alternative catalysts for several reactions, including alkane dehydrogenation. Correlations have been developed between reactivity and structure, but it is difficult to compare different catalysts without determining numbers of active sites. Here, methanol temperature programmed surface reaction (TPSR) is applied to bulk MoS 2 and a series of supported molybdenum oxides and sulfides used for isobutane dehydrogenation. Rates and deactivation behavior are similar between supported molybdenum oxides and sulfides. Coking increases with loading and correlates well with the fraction of Mo-based CO 2 -forming sites from methanol TPSR. Normalizing isobutane dehydrogenation rates by formaldehyde-forming sites unifies rates over oxide and sulfide materials. Further, these measurements confirm earlier hypotheses that MoS 2 and higher loadings of oxides and sulfides are intrinsically fast catalysts, but they have few active sites and are prone to coking. In general, methanol TPSR is proposed as an accessible route to count sites on both supported oxides and sulfides.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2025

A Markov chain Monte Carlo (MCMC) Bayesian inference approach to analyze apparent activation barriers and reaction orders from microreactor data

Statistical analysis of steady-state catalytic kinetic data is often limited by data sparsity due to the slow pace at which the data is collected. Data sparsity and limitations in statistical analysis make it difficult to differentiate between mechanistic models and catalytic sites. A Bayesian inference tool is reported for catalysis researchers to estimate error in the determination of reaction orders from steady state microreactor data. The benefits of a Bayesian inference approach are discussed, as an alternative to the more common frequentist approach. The approach incorporates prior knowledge of the system and the data collected to form an error estimate on reaction orders. We investigated the effects of three distinct data treatments—individual fitting of trials, pooled analysis, and constrained regression methods—on the precision and uncertainty of reaction order determinations. To assess the robustness of our findings, we conducted sensitivity analyses to evaluate the influence of Bayesian parameters on uncertainty estimation. Additionally, we utilized synthetic data to illustrate how data quality impacts the precision of uncertainty assessments. We show Bayesian analysis can obtain a more precise estimation of error with a sparse data set than a frequentist analysis. Finally, this work provides strong evidence that the adoption of Bayesian analysis of kinetic data may help researchers make more precise arguments as to the strength of their evidence for a particular mechanistic hypothesis, or in comparing across different catalysts.

42 ENGINEERING↗
DOE OSTI2020

Correlating inter-particle forces and particle shape to shear-induced aggregation/fragmentation and rheology for dilute anisotropic particle suspensions: A complementary study via capillary rheometry and in-situ small and ultra-small angle X-ray scattering

Predicting the stability of suspensions composed of anisotropic particles is difficult due to the complexity and interplay of hydrodynamic and colloidal forces. For example, it is not well understood how non-spherical particle shapes affect suspension-rheological responses. This study presents evidence showing how the stacking of boehmite platelets dispersed in aqueous solution leads to moderate shear thinning, even at low boehmite mass fractions. Wide angle X-ray, small angle X-ray, and ultra-small angle X-ray scattering were used to simultaneously monitor changes to the size and fractal dimensions of boehmite aggregates from 6 – 10,000 Å as the sample was recirculated through an in situ capillary rheometer. The latter also provided simultaneous measurements of the viscosity of the suspensions. Computational fluid dynamics modeling of the apparatus was used to provide a more rigorous analysis of the fluid flow. In addition to shear-induced aggregation/fragmentation correlated to a complicated balance between hydrodynamic and colloidal forces, the results indicate that orientationally-dependent interactions, arising from non-spherical boehmite aggregate sub-units, may explain the relatively large experimental viscosity when the hydrodynamic force is small compared to colloidal forces. Ultimately, the results from this study can provide a foundation for future work investigating particle shape effects on colloidal and hydrodynamics forces.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2020

Exact and locally implicit source term solvers for multifluid-Maxwell systems

Recently, a family of models that couple multifluid systems to the full Maxwell equations have been used in laboratory, space, and astrophysical plasma modeling. These models are more complete descriptions of the plasma than reduced models like magnetohydrodynamic (MHD) since they are derived more closely from the full kinetic Vlasov-Maxwell system, without assumptions like quasi-neutrality, negligible electron mass, etc. Thus these models naturally retain non-ideal MHD effects like electron inertia, Hall term, pressure anisotropy/nongyrotropy, displacement current, among others. One obstacle to broader application of these model is that an explicit treatment of their source terms leads to the need to resolve rapid processes like plasma oscillation and electron cyclotron motion, even when these are not important. In this paper, we suggest two ways to address this issue. First, we derive the analytic solutions to the source update equations, which can be implemented as a practical, but less generic solver. We then develop a time-centered, locally implicit algorithm to update the source terms, allowing stepping over the fast kinetic time-scales. For a plasma with S species, the locally implicit algorithm involves inverting a local (3 S + 3) × (3 S + 3) matrix only, thus is very efficient. The performance can be further increased by using the direct update formulas to skip null calculations. In this paper, we present benchmarks illustrating the exact energy-conservation of the locally implicit solver, as well as its efficiency and robustness for both small-scale, idealized problems and largescale, complex systems. The locally implicit algorithm can be also easily extended to include other local sources, like collisions and ionization, which are difficult to solve analytically.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2021

Adjoint DSMC for nonlinear Boltzmann equation constrained optimization

Applications for kinetic equations such as optimal design and inverse problems often involve finding unknown parameters through gradient-based optimization algorithms. Based on the adjoint-state method, we derive two different frameworks for approximating the gradient of an objective functional constrained by the nonlinear Boltzmann equation. While the forward problem can be solved by the DSMC method, it is difficult to efficiently solve the high-dimensional continuous adjoint equation obtained by the “optimize-then-discretize” approach. This challenge motivates us to propose an adjoint DSMC method following the “discretize-then-optimize” approach for Boltzmann-constrained optimization. We also analyze the properties of the two frameworks and their connections. Here, several numerical examples are presented to demonstrate their accuracy and efficiency.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Physics-informed neural networks for solving forward and inverse flow problems via the Boltzmann-BGK formulation

We report the Boltzmann equation with the Bhatnagar-Gross-Krook collision model (Boltzmann-BGK equation) has been employed to describe multiscale flows, i.e., from the hydrodynamic limit to free molecular flow. In this study we employ physics-informed neural networks (PINNs) to solve forward and inverse problems via the Boltzmann-BGK formulation (PINN-BGK), enabling PINNs to model flows in both the continuum and rarefied regimes. In particular, the PINN-BGK is composed of three sub-networks, i.e., the first for approximating the equilibrium distribution function, the second for approximating the non-equilibrium distribution function, and the third one for encoding the Boltzmann-BGK equation as well as the corresponding boundary/initial conditions. By minimizing the residuals of the governing equations and the mismatch between the predicted and provided boundary/initial conditions, we can approximate the Boltzmann-BGK equation for both continuous and rarefied flows. For forward problems, the PINN-BGK is utilized to solve various benchmark flows given boundary/initial conditions, e.g. Kovasznay flow, Taylor-Green flow, cavity flow, and micro Couette flow for Knudsen number up to 5. For inverse problems, we focus on rarefied flows in which accurate boundary conditions are difficult to obtain. We employ the PINN-BGK to infer the flow field in the entire computational domain given a limited number of interior scattered measurements on the velocity without using the (unknown) boundary conditions. Results for the two-dimensional micro Couette and micro cavity flows with Knudsen numbers ranging from 0.1 to 10 indicate that the PINN-BGK can infer the velocity field in the entire domain with good accuracy. Finally, we also present some results on using transfer learning to accelerate the training process. Specifically, we can obtain a three-fold speedup comparted to the standard training process (e.g., Adam plus L-BFG-B) for the two-dimensional flow problems considered in our work.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

An explicitly solvable energy-conserving algorithm for pitch-angle scattering in magnetized plasmas

In this work, we develop an Explicitly Solvable Energy-Conserving (ESEC) algorithm for the Stochastic Differential Equation (SDE) describing the pitch-angle scattering process in magnetized plasmas. The Cayley transform is used to calculate both the deterministic gyromotion and stochastic scattering, affording the algorithm to be explicitly solvable and exactly energy conserving. An unusual property of the SDE for pitch-angle scattering is that its coefficients diverge at the zero velocity and do not satisfy the global Lipschitz condition. Consequently, when standard numerical methods, such as the Euler-Maruyama (EM), are applied, numerical convergence is difficult to establish. For the proposed ESEC algorithm, its energy-preserving property enables us to overcome this obstacle. We rigorously prove that the ESEC algorithm is order 1/2 strongly convergent. This result is confirmed by detailed numerical studies. For the case of pitch-angle scattering in a magnetized plasma with a constant magnetic field, the numerical solution is benchmarked against the analytical solution, and excellent agreements are found.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2022

Efficient hybrid explicit-implicit learning for multiscale problems

Splitting method is a powerful method to handle application problems by splitting physics, scales, domain, and so on. Many splitting algorithms have been designed for efficient temporal discretization. Here, in this paper, our goal is to use temporal splitting concepts in designing machine learning algorithms and, at the same time, help splitting algorithms by incorporating data and speeding them up. We propose a machine learning assisted splitting scheme which improves the efficiency of the scheme meanwhile preserves the accuracy. We consider a recently introduced multiscale splitting algorithms, where the multiscale problem is solved on a coarse grid. To approximate the dynamics, only a few degrees of freedom are solved implicitly, while others explicitly. This splitting concept allows identifying degrees of freedom that need implicit treatment. In this paper, we use this splitting concept in machine learning and propose several strategies. First, the implicit part of the solution can be learned as it is more difficult to solve, while the explicit part can be computed. This provides a speed-up and data incorporation for splitting approaches. Secondly, one can design a hybrid neural network architecture because handling explicit parts requires much fewer communications among neurons and can be done efficiently. Thirdly, one can solve the coarse grid component via PDEs or other approximation methods and construct simpler neural networks for the explicit part of the solutions. We discuss these options and implement one of them by interpreting it as a machine translation task. This interpretation of the splitting scheme successfully enables us using the Transformer since it can perform model reduction for multiple time series and learn the connection between them. We also find that the splitting scheme is a great platform to predict the coarse solution with insufficient information of the target model: the target problem is partially given and we need to solve it through a known problem which approximates the target. Our machine learning model can incorporate and encode the given information from two different problems and then solve the target problems. We conduct four numerical examples and the results show that our method is stable and accurate.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

High-order dimensionally-split Cartesian embedded boundary method for non-dissipative schemes

Centered finite-difference schemes are commonly used for high-fidelity turbulent flow simulations in canonical configurations because of their non-dissipative property and computational efficiency. However, their use in flow simulations over complex geometries is limited by the requirements of a structured grid and a stable boundary treatment in the absence of artificial (numerical) dissipation. Cartesian embedded boundary (EB) approaches provide an efficient structured-grid framework to apply difference schemes over complex domains. However, they are often restricted to low orders of accuracy because of numerical instabilities at the embedded boundaries and the issues of small-cell problem that are difficult to address with high-order accuracy. The present work discusses a systematic approach to obtain high-order EB methods with non-dissipative centered schemes in the interior. This approach, based on satisfying the primary and secondary conservation conditions, is employed to derive EB schemes that are up to sixth-order accurate in the interior and fourth-order accurate globally for hyperbolic, parabolic as well as incompletely parabolic problems. The proposed finite-difference discretization is, by construction, dimensionally split and addresses the small-cell problem without any cell/geometry transformations, thus, highly simplifying implementation in a flow solver. Various linear and non-linear numerical tests are performed to evaluate the stability and the accuracy of the proposed EB schemes.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Explicitly antisymmetrized neural network layers for variational Monte Carlo simulation

The combination of neural networks and quantum Monte Carlo methods has arisen as a promising path forward for highly accurate electronic structure calculations. Previous proposals have combined equivariant neural network layers with a final antisymmetric layer in order to satisfy the antisymmetry requirements of the electronic wavefunction. However, to date it is unclear if one can represent antisymmetric functions of physical interest, and it is difficult to precisely measure the expressiveness of the antisymmetric layer. Here, this work attempts to address this problem by introducing explicitly antisymmetrized universal neural network layers. This approach has a computational cost which increases factorially with respect to the system size, but we are nonetheless able to apply it to small systems to better understand how the structure of the antisymmetric layer affects its performance. We first introduce a generic antisymmetric (GA) neural network layer, which we use to replace the entire antisymmetric layer of the highly accurate ansatz known as the FermiNet. We demonstrate that the resulting FermiNet-GA architecture can yield effectively the exact ground state energy for small atoms and molecules. We then consider a factorized antisymmetric (FA) layer which more directly generalizes the FermiNet by replacing the products of determinants with products of antisymmetrized neural networks. We find, interestingly, that the resulting FermiNet-FA architecture does not significantly outperform the FermiNet. This strongly suggests that the sum of products of antisymmetries is a key limiting aspect of the FermiNet architecture. To explore this further, we investigate a slight modification of the FermiNet, called the full determinant mode, which replaces each product of determinants with a single combined determinant. We find that the full single-determinant FermiNet closes a large part of the gap between the standard single-determinant FermiNet and FermiNet-GA on small atomic and molecular problems. Surprisingly, on the nitrogen molecule at a dissociating bond length of 4.0 Bohr, the full single-determinant FermiNet can outperform the largest standard FermiNet calculation with 64 determinants.

97 MATHEMATICS AND COMPUTING↗