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At least 1,369 records · Page 76

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DOE OSTI2022

Extraction of low-energy constants of single- and double-β decays from lattice QCD: A sensitivity analysis

Lattice quantum chromodynamics (LQCD) has the promise of constraining low-energy constants (LECs) of nuclear effective field theories (EFTs) from first-principles calculations that incorporate the dynamics of quarks and gluons. Given the Euclidean and finite-volume nature of LQCD outputs, complex mappings are developed in recent years to obtain the Minkowski and infinite-volume counterparts of LQCD observables. In particular, as LQCD is moving toward computing a set of important few-nucleon matrix elements at the physical values of the quark masses, it is important to investigate whether the anticipated precision of LQCD spectra and matrix elements will be sufficient to guarantee tighter constraints on the relevant LECs than those already obtained from phenomenology, considering the nontrivial mappings involved. With a focus on the leading-order LECs of the pionless EFT, L 1,A and $g$$^{NN}_{v}$, which parametrize, respectively, the strength of the isovector axial two-body current in a single-β decay (and other related processes such pp fusion), and of the isotensor contact two-body operator in the neutrinoless double-β decay within the light neutrino exchange scenario, the expected uncertainty on future extractions of L 1,A and $g$$^{NN}_{v}$ are examined using synthetic data at the physical values of the quark masses. It is observed that achieving small uncertainties in L 1,A will be challenging, and (sub)percent-level precision in the two-nucleon spectra and matrix elements is essential in reducing the uncertainty on this LEC compared to the existing constraints. On the other hand, the short-distance coupling of the neutrinoless double-β decay, $g$$^{NN}_{v}$, is shown to be less sensitive to uncertainties on both LQCD energies and the matrix element, and can likely be constrained with percent-level precision in the upcoming LQCD calculations.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2024

A tetrahedral neptunium(V) complex

Neptunium is an actinide element sourced from anthropogenic production, and, unlike naturally abundant uranium, its coordination chemistry is not well developed in all accessible oxidation states. High-valent neptunium generally requires stabilization from at least one metal–ligand multiple bond, and departing from this structural motif poses a considerable challenge. Here we report a tetrahedral molecular neptunium(V) complex ([Np 5+ (NPC) 4 ][B(ArF 5 ) 4 ], 1-Np) (NPC = [NP t Bu(pyrr) 2 ] − ; t Bu = C(CH 3 ) 3 ; pyrr = pyrrolidinyl (N(C 2 H 4 ) 2 ); B(ArF 5 ) 4 = tetrakis(2,3,4,5,6-pentafluourophenyl)borate). Single-crystal X-ray diffraction, solution-state spectroscopy and density functional theory studies of 1-Np and the product of its proton-coupled electron transfer (PCET) reaction, 2-Np, demonstrate the unique bonding that stabilizes this reactive ion and establishes the thermochemical and kinetic parameters of PCET in a condensed-phase transuranic complex. The isolation of this four-coordinate, neptunium(V) complex reveals a fundamental reaction pathway in transuranic chemistry.

coordination chemistry↗

DOE OSTI2024

Mapping the Binary Covalent Alloy Space to Pursue Superior Nitrogen Reduction Reaction Catalysts

The electrochemical nitrogen reduction reaction (NRR) has the potential to decarbonize industrial ammonia production. However, NRR has poor activity and selectivity versus the competing hydrogen evolution reaction for catalysts that adhere to scaling relations. Overcoming the limitations imposed by scaling relations requires more complex catalyst materials, however, evaluating materials beyond simple metal systems is a large combinatorial problem that requires an improved understanding of the electrocatalyst surface to rationally guide the discovery of superior catalysts. The study uses grand canonical density functional theory to uncover NRR trends on a large and disparate set of binary covalent alloys (BCA) with variable compositions and active-site geometries. The studied BCAs generally follow scaling relations, albeit with larger variance and several systems that significantly break scaling. BCAs with early- to mid-transition metals tend to lie near the volcano peak and activate the N 2 triple bond via a side-on binding configuration. Trends in the BCA space cannot be readily predicted using simple electronic descriptors, which is ascribed to the large geometric variability of the BCA surfaces. Furthermore, it is anticipated that these findings will provide a foundation for the rational design of superior NRR electrocatalysts with increasing material complexity.

36 MATERIALS SCIENCE↗

NASA NTRS1988

Human operator response to error-likely situations in complex engineering systems

The causes of human error in complex systems are examined. First, a conceptual framework is provided in which two broad categories of error are discussed: errors of action, or slips, and errors of intention, or mistakes. Conditions in which slips and mistakes might be expected to occur are identified, based on existing theories of human error. Regarding the role of workload, it is hypothesized that workload may act as a catalyst for error. Two experiments are presented in which humans' response to error-likely situations were examined. Subjects controlled PLANT under a variety of conditions and periodically provided subjective ratings of mental effort. A complex pattern of results was obtained, which was not consistent with predictions. Generally, the results of this research indicate that: (1) humans respond to conditions in which errors might be expected by attempting to reduce the possibility of error, and (2) adaptation to conditions is a potent influence on human behavior in discretionary situations. Subjects' explanations for changes in effort ratings are also explored.

Morris, Nancy M.↗

NASA NTRS1997

Constrained Multipoint Aerodynamic Shape Optimization Using an Adjoint Formulation and Parallel Computers

An aerodynamic shape optimization method that treats the design of complex aircraft configurations subject to high fidelity computational fluid dynamics (CFD), geometric constraints and multiple design points is described. The design process will be greatly accelerated through the use of both control theory and distributed memory computer architectures. Control theory is employed to derive the adjoint differential equations whose solution allows for the evaluation of design gradient information at a fraction of the computational cost required by previous design methods. The resulting problem is implemented on parallel distributed memory architectures using a domain decomposition approach, an optimized communication schedule, and the MPI (Message Passing Interface) standard for portability and efficiency. The final result achieves very rapid aerodynamic design based on a higher order CFD method. In order to facilitate the integration of these high fidelity CFD approaches into future multi-disciplinary optimization (NW) applications, new methods must be developed which are capable of simultaneously addressing complex geometries, multiple objective functions, and geometric design constraints. In our earlier studies, we coupled the adjoint based design formulations with unconstrained optimization algorithms and showed that the approach was effective for the aerodynamic design of airfoils, wings, wing-bodies, and complex aircraft configurations. In many of the results presented in these earlier works, geometric constraints were satisfied either by a projection into feasible space or by posing the design space parameterization such that it automatically satisfied constraints. Furthermore, with the exception of reference 9 where the second author initially explored the use of multipoint design in conjunction with adjoint formulations, our earlier works have focused on single point design efforts. Here we demonstrate that the same methodology may be extended to treat complete configuration designs subject to multiple design points and geometric constraints. Examples are presented for both transonic and supersonic configurations ranging from wing alone designs to complex configuration designs involving wing, fuselage, nacelles and pylons.

Reuther, James↗

DOE OSTI2021

Theory of Trotter Error with Commutator Scaling

The Lie-Trotter formula, together with its higher-order generalizations, provides a simple approach to decomposing the exponential of a sum of operators. Despite significant effort, the error scaling of such product formulas remains poorly understood. We develop a theory of Trotter error that overcomes the limitations of truncating the Baker-Campbell-Hausdorff expansion. Our analysis directly exploits the commutativity of operator summands, producing tighter error bounds for both real- and imaginary-time evolutions. Whereas previous work achieves similar goals for systems with geometric locality or Lie-algebraic structure, our approach holds in general. We give a host of improved algorithms for digital quantum simulation and quantum Monte Carlo methods, nearly matching or even outperforming the best previous results. Our applications include: (i) a simulation of second-quantized plane-wave electronic structure, nearly matching the interaction-picture algorithm of Low and Wiebe; (ii) a simulation of $k$-local Hamiltonians almost with induced one-norm scaling, faster than the qubitization algorithm of Low and Chuang; (iii) a simulation of rapidly decaying power-law interactions, outperforming the Lieb-Robinson-based approach of Tran et al.; (iv) a hybrid simulation of clustered Hamiltonians, dramatically improving the result of Peng, Harrow, Ozols, and Wu; and (v) quantum Monte Carlo simulations of the transverse field Ising model and quantum ferromagnets, tightening previous analyses of Bravyi and Gosset. We obtain further speedups using the fact that product formulas can preserve the locality of the simulated system. Specifically, we show that local observables can be simulated with complexity independent of the system size for power-law interacting systems, which implies a Lieb-Robinson bound nearly matching a recent result of Tran et al. Our analysis reproduces known tight bounds for first- and second-order formulas. We further investigate the tightness of our bounds for higher-order formulas. For quantum simulation of a one-dimensional Heisenberg model with an even-odd ordering of terms, our result overestimates the complexity by only a factor of $5$. Our bound is also close to tight for power-law interactions and other orderings of terms. This suggests that our theory can accurately characterize Trotter error in terms of both the asymptotic scaling and the constant prefactor.

quantum computing, numerical analysis↗

DOE OSTI2022

Estimating mass-absorption cross-section of ambient black carbon aerosols: theoretical, empirical, and machine learning models

The mass-absorption cross-section of black carbon (MAC BC ) is an essential parameter to link the atmospheric concentration of black carbon (BC) with its radiative forcing. When a direct calculation of MAC BC based on observations of aerosol light absorption and BC mass concentration is impossible, we rely on modeling and simulations to estimate MAC BC , but currently, there is no consensus model that can be relied on for accurate predictions across all atmospheric environments when BC particles have different coating thicknesses. Here, we applied five MAC BC prediction models (including three light scattering theories, an empirical model based on observations of particle mass concentrations, and a machine learning model developed in our previous work) to aerosols from three Department of Energy (DOE) Atmospheric Radiation Measurement (ARM) field campaigns. While many studies have found that increasing the complexity of the models helps to constrain biases of the estimated MAC BC , our effort is to evaluate the models based on the criteria of simplicity and accuracy. We find that our machine learning model (support vector machine for regression, SVM) generally performs well across all DOE ARM field campaign data, while the accuracy of core-shell Mie theory depends on the bias correction algorithm applied to filter-based light absorption data. Generally, the empirical model for internally-mixed particles that we considered tends to over-predict MAC BC , while Mie theory for externally-mixed particles tends to under-predict MAC BC . An examination of the influence of coating material on BC cores suggests that the performance of our current SVM model is degraded when the BC is thickly-coated (e.g., it has undergone aging and mixing with other materials in the atmosphere).

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2024

S -matrix positivity without Lorentz invariance: a case study

We investigate the analytic structure of scattering amplitudes in theories in which Lorentz invariance is spontaneously broken. We do so by computing and studying the S-matrix for a simple example: a superfluid described by a complex scalar with quartic interactions. The computation is confined to tree-level, for there are no absolutely stable single-particle states, though the lifetime can be made long by lowering the chemical potential. For the 2 → 2 amplitude in center-of-mass configurations, not only is crossing symmetry violated, there appears a tree level branch cut for unphysical kinematics. Its appearance is a consequence of non-analyticity in the dispersion relation. The branch point defines a new scale in the problem, which scales inversely with the chemical potential. In this example, even derivatives of the forward amplitude are positive while odd derivatives are negative. This pattern can be understood in a general way in the limit of a small chemical potential, or weak Lorentz breaking.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2021

Dioxygen Activation Kinetics over Distinct Cu Site Types in Cu-Chabazite Zeolites

Cu-exchanged zeolites activate dioxygen to form active sites for partial methane oxidation (PMO), nitrogen oxide decomposition, and carbon monoxide oxidation. Apparent rates of O 2 activation depend both on the intrinsic kinetics of distinct Cu site types and the distributions of such sites within a given zeolite, which depend on the density and arrangement of the framework Al atoms. In this work, we use hydrothermal synthesis methods to control the arrangement of framework Al sites in chabazite (CHA) zeolites and, in turn, the distinct Cu site types formed. Time-resolved in situ Raman spectroscopy reveals the kinetics of O 2 adsorption and activation within these well-defined Cu-CHA materials and the concomitant structural evolution of copper–oxygen (Cu x O y ) complexes, which are interpreted alongside Cu(I) oxidation kinetics extracted from in situ X-ray absorption spectroscopy (XAS). Raman spectra of several plausible Cu x O y species simulated using density functional theory suggest that experimental spectra (λ ex = 532 nm) capture the formation of mono(μ-oxo)dicopper species (ZCuOCuZ). Transient experiments show that the timescales required to form Cu x O y structures that no longer change in Raman spectra correspond to the durations of oxidative treatments that maximize CH 3 OH yields in stoichiometric PMO cycles (approximately 2 h). Yet, these periods extend well beyond the timescales for the complete conversion of the initial Cu(I) intermediates to their Cu(II) states (<0.3 h, reflected in X-ray absorption near edge spectroscopy spectra), which demonstrates that Cu x O y complexes continue to evolve structurally following rapid Cu(I) oxidation. The dependence of ZCuOCuZ formation rates on O 2 pressure, H 2 O pressure, and temperature is consistent with a mechanism in which ZCuOH reduces to form ZCu + sites that bind molecular oxygen and form ZCu-O 2 intermediates. Subsequent reaction with proximate ZCu + forms bridging peroxo dicopper complexes that cleave O–O bonds to form ZCuOCuZ in steps facilitated by water. These data and interpretations provide evidence for the chemical processes that link rapid and kinetically irrelevant Cu oxidation steps (frequently probed by XAS and UV–vis spectroscopy) to the relatively slow genesis of reactive Cu complexes that form CH 3 OH during PMO. In doing so, we reveal previously unrecognized complexities in the processes by which Cu ions in zeolites activate O 2 to form active Cu x O y complexes, which underscore the insight afforded by judicious combinations of experimental and theoretical techniques.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2023

Divergent Stabilities of Tetravalent Cerium, Uranium, and Neptunium Imidophosphorane Complexes**

Abstract The study of the redox chemistry of mid‐actinides (U−Pu) has historically relied on cerium as a model, due to the accessibility of trivalent and tetravalent oxidation states for these ions. Recently, dramatic shifts of lanthanide 4+/3+ non‐aqueous redox couples have been established within a homoleptic imidophosphorane ligand framework. Herein we extend the chemistry of the imidophosphorane ligand (NPC=[N=P t Bu(pyrr) 2 ] − ; pyrr=pyrrolidinyl) to tetrahomoleptic NPC complexes of neptunium and cerium ( 1‐M , 2‐M , M=Np, Ce) and present comparative structural, electrochemical, and theoretical studies of these complexes. Large cathodic shifts in the M 4+/3+ (M=Ce, U, Np) couples underpin the stabilization of higher metal oxidation states owing to the strongly donating nature of the NPC ligands, providing access to the U 5+/4+ , U 6+/5+ , and to an unprecedented, well‐behaved Np 5+/4+ redox couple. The differences in the chemical redox properties of the U vs . Ce and Np complexes are rationalized based on their redox potentials, degree of structural rearrangement upon reduction/oxidation, relative molecular orbital energies, and orbital composition analyses employing density functional theory.

Otte, Kaitlyn S.↗

DOE OSTI2023

Divergent Stabilities of Tetravalent Cerium, Uranium, and Neptunium Imidophosphorane Complexes

The study of the redox chemistry of mid-actinides (U–Pu) has historically relied on cerium as a model, due to the accessibility of trivalent and tetravalent oxidation states for these ions. Recently, dramatic shifts of lanthanide 4+/3+ non-aqueous redox couples have been established within a homoleptic imidophosphorane ligand framework. Herein we extend the chemistry of the imidophosphorane ligand (NPC=[N=P t Bu(pyrr) 2 ] – ; pyrr=pyrrolidinyl) to tetrahomoleptic NPC complexes of neptunium and cerium (1-M, 2-M, M=Np, Ce) and present comparative structural, electrochemical, and theoretical studies of these complexes. Large cathodic shifts in the M 4+/3+ (M=Ce, U, Np) couples underpin the stabilization of higher metal oxidation states owing to the strongly donating nature of the NPC ligands, providing access to the U 5+/4+ , U 6+/5+ , and to an unprecedented, well-behaved Np 5+/4+ redox couple. Furthermore, the differences in the chemical redox properties of the U vs. Ce and Np complexes are rationalized based on their redox potentials, degree of structural rearrangement upon reduction/oxidation, relative molecular orbital energies, and orbital composition analyses employing density functional theory.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2020

Local SU(2) × U(1) quark flavor symmetry in the RS bulk

We propose a model of quark flavor based on an additional SU(2) × U(1) local symmetry in a warped extra dimensional bulk. In contrast to other works, we break the additional gauge symmetry in the bulk via two complex scalars which acquire bulk vevs, rather than relying on brane-localized symmetry breaking. A gauge-covariant Kaluza-Klein decomposition of a theory with a bulk spontaneously broken gauge symmetry is performed, and exact expressions for the bulk profiles of all physical particles in such systems are given. The SM quark masses and mixings are then recreated using gauge-covariant bulk quark mass terms and Yukawa-like couplings to the new bulk scalars. A numerical sampling of points in the model parameter space that recreate the quark masses and mixings is performed at a KK scale of M KK = 5 TeV. We then compute the ΔF = 2 4-quark operators arising from our new flavor gauge bosons and scalars, and those arising from Kaluza-Klein modes of SM gauge bosons. By decoupling one of our bulk scalar fields to all quark fields except the right-handed up-like sector, we find that it is possible to greatly suppress tree-level contributions to the highly constrained Kaon mixing parameters. Instead, the dominant constraints on the model emerge from neutral Bd and D meson mixing. These constraints are explored with our numerical sampling of the model parameter space, and the specific contribution of the new flavor gauge bosons and scalars is discussed. We find that for a significant range of realistic flavor gauge couplings, the new gauge bosons compete with the normally dominant gluon flavor-changing currents, but flavor-changing operators emerging from the bulk scalar fields are highly suppressed. Finally, we briefly comment on flavor constraints that are independent of the flavor gauge sector arising from the Z$\overline{b}$ L b L coupling and rare top decays.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2022

Metal-organic framework supported single-site nickel catalysts for butene dimerization

Homotopic sites in a well-controlled environment are not only ideal systems for mechanistic studies, but also allow optimal control of catalytic transformations. Sites having only a single metal cation and sites consisting of metal oxo complexes with few nickel (Ni) cations supported on the nodes of UiO-66 metal-organic framework (Ni-UiO-66) are studied for 1-butene dimerization. Monomeric Ni sites, which bind to the Zr 6 node via two Zr-OH(µ3) linkages, are active and selective for the dimerization of 1-butene to linear and mono-branched C 8 isomers. Ni oxo complexes with few Ni cations show lower activity and promote the oligomerization of transiently formed C 8 isomers. In conclusion, Kohn-Sham density function theory calculations combined with spectroscopic measurements and kinetic analyses indicate that dimerization follows a Cossee-Arlman reaction mechanism.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2019

Unraveling the Mechanism of Water-Mediated Sulfur Tolerance via Operando Surface-Enhanced Raman Spectroscopy

While several proton-conducting anode materials have shown excellent tolerance to sulfur poisoning, the mechanism is still unclear due largely to the inability to probe miniscule amounts of sulfur-containing species using conventional surface characterization techniques. Here we present our findings in unraveling the mechanism of water-mediated sulfur tolerance of a proton conductor under operating conditions empowered by surface-sensitive, operando surface-enhanced Raman spectroscopy (SERS) coupled with impedance spectroscopy. Contrary to the conventional view that surface-adsorbed sulfur is removed mainly by oxygen anions, it is found that –SO 4 groups on the surface of the proton conductor are converted to SO 2 by a water-mediated process, as confirmed by operando SERS analysis and density functional theory (DFT)-based calculations. Furthermore, the combination of operando SERS performed on a model electrode and theoretical computation offers an effective approach to investigate into complex mechanisms of electrode processes in various electrochemical systems, providing information vital to achieve the rational design of better electrode materials.

25 ENERGY STORAGE↗

DOE OSTI2022

Photoelectron spectroscopy in molecular physical chemistry

Photoelectron spectroscopy has long been a powerful method in the toolbox of experimental physical chemistry and molecular physics. Recent improvements in coincidence methods, charged-particle imaging, and electron energy resolution have greatly expanded the variety of environments in which photoelectron spectroscopy can be applied, as well as the range of questions that can now be addressed. In this Perspectives Article, we focus on selected recent studies that highlight these advances and research areas. Furthermore, the topics include reactive intermediates and new thermochemical data, high-resolution comparisons of experiment and theory using methods based on pulsed-field ionisation (PFI), and the application of photoelectron spectroscopy as an analytical tool to monitor chemical reactions in complex environments, like model flames, catalytic or high-temperature reactors.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2020

Octopus, a computational framework for exploring light-driven phenomena and quantum dynamics in extended and finite systems

Over the last few years, extraordinary advances in experimental and theoretical tools have allowed us to monitor and control matter at short time and atomic scales with a high degree of precision. An appealing and challenging route toward engineering materials with tailored properties is to find ways to design or selectively manipulate materials, especially at the quantum level. To this end, having a state-of-the-art ab initio computer simulation tool that enables a reliable and accurate simulation of light-induced changes in the physical and chemical properties of complex systems is of utmost importance. The first principles real-space-based Octopus project was born with that idea in mind, i.e., to provide a unique framework that allows us to describe non-equilibrium phenomena in molecular complexes, low dimensional materials, and extended systems by accounting for electronic, ionic, and photon quantum mechanical effects within a generalized time-dependent density functional theory. This article aims to present the new features that have been implemented over the last few years, including technical developments related to performance and massive parallelism. We also describe the major theoretical developments to address ultrafast light-driven processes, such as the new theoretical framework of quantum electrodynamics density-functional formalism for the description of novel light–matter hybrid states. Those advances, and others being released soon as part of the Octopus package, will allow the scientific community to simulate and characterize spatial and time-resolved spectroscopies, ultrafast phenomena in molecules and materials, and new emergent states of matter (quantum electrodynamical-materials).

74 ATOMIC AND MOLECULAR PHYSICS↗

DOE OSTI2024

Noise reduction of stochastic density functional theory for metals

Density Functional Theory (DFT) has become a cornerstone in the modeling of metals. However, accurately simulating metals, particularly under extreme conditions, presents two significant challenges. First, simulating complex metallic systems at low electron temperatures is difficult due to their highly delocalized density matrix. Second, modeling metallic warm-dense materials at very high electron temperatures is challenging because it requires the computation of a large number of partially occupied orbitals. This study demonstrates that both challenges can be effectively addressed using the latest advances in linear-scaling stochastic DFT methodologies. Despite the inherent introduction of noise into all computed properties by stochastic DFT, this research evaluates the efficacy of various noise reduction techniques under different thermal conditions. Our observations indicate that the effectiveness of noise reduction strategies varies significantly with the electron temperature. Furthermore, we provide evidence that the computational cost of stochastic DFT methods scales linearly with system size for metal systems, regardless of the electron temperature regime.

Chemistry↗

DOE OSTI2025

Exploring the energy landscape of RBMs: reciprocal space insights into bosons, hierarchical learning and symmetry breaking

Deep generative models have become ubiquitous due to their ability to learn and sample from complex distributions. Despite the proliferation of various frameworks, the relationships among these models remain largely unexplored, a gap that hinders the development of a unified theory of AI learning. In this work, we address two central challenges: clarifying the connections between different deep generative models and deepening our understanding of their learning mechanisms. We focus on Restricted Boltzmann Machines (RBMs), a class of generative models known for their universal approximation capabilities for discrete distributions. By introducing a reciprocal space formulation for RBMs, we reveal a connection between these models, diffusion processes, and systems of coupled bosons. Our analysis shows that at initialization, the RBM operates at a saddle point, where the local curvature is determined by the singular values of the weight matrix, whose distribution follows the Marc̆enko-Pastur law and exhibits rotational symmetry. During training, this rotational symmetry is broken due to hierarchical learning, where different degrees of freedom progressively capture features at multiple levels of abstraction. This leads to a symmetry breaking in the energy landscape, reminiscent of Landau’s theory. This symmetry breaking in the energy landscape is characterized by the singular values and the weight matrix eigenvector matrix. We derive the corresponding free energy in a mean-field approximation. We show that in the limit of infinite size RBM, the reciprocal variables are Gaussian distributed. Our findings indicate that in this regime, there will be some modes for which the diffusion process will not converge to the Boltzmann distribution. To illustrate our results, we trained replicas of RBMs with different hidden layer sizes using the MNIST dataset. Our findings not only bridge the gap between disparate generative frameworks but also shed light on the fundamental processes underpinning learning in deep generative models.

97 MATHEMATICS AND COMPUTING↗