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DOE OSTI2024

Computationally efficient and error aware surrogate construction for numerical solutions of subsurface flow through porous media

Limiting the injection rate to restrict the pressure below a threshold at a critical location can be an important goal of simulations that model the subsurface pressure between injection and extraction wells. The pressure is approximated by the solution of Darcy’s partial differential equation for a given permeability field. The subsurface permeability is modeled as a random field since it is known only up to statistical properties. This induces uncertainty in the computed pressure. Solving the partial differential equation for an ensemble of random permeability simulations enables estimating a probability distribution for the pressure at the critical location. These simulations are computationally expensive, and practitioners often need rapid online guidance for real-time pressure management. An ensemble of numerical partial differential equation solutions is used to construct a Gaussian process regression model that can quickly predict the pressure at the critical location as a function of the extraction rate and permeability realization. The Gaussian process surrogate analyzes the ensemble of numerical pressure solutions at the critical location as noisy observations of the true pressure solution, enabling robust inference using the conditional Gaussian process distribution. Our first novel contribution is to identify a sampling methodology for the random environment and matching kernel technology for which fitting the Gaussian process regression model scales as O ( n log n ) instead of the typical O ( n 3 ) rate in the number of samples n used to fit the surrogate. The surrogate model allows almost instantaneous predictions for the pressure at the critical location as a function of the extraction rate and permeability realization. Our second contribution is a novel algorithm to calibrate the uncertainty in the surrogate model to the discrepancy between the true pressure solution of Darcy’s equation and the numerical solution. Finally, although our method is derived for building a surrogate for the solution of Darcy’s equation with a random permeability field, the framework broadly applies to solutions of other partial differential equations with random coefficients.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2022

Phylogenetic structure of specialization: A new approach that integrates partner availability and phylogenetic diversity to quantify biotic specialization in ecological networks

Abstract Biotic specialization holds information about the assembly, evolution, and stability of biological communities. Partner availabilities can play an important role in enabling species interactions, where uneven partner availabilities can bias estimates of biotic specialization when using phylogenetic diversity indices. It is therefore important to account for partner availability when characterizing biotic specialization using phylogenies. We developed an index, phylogenetic structure of specialization (PSS), that avoids bias from uneven partner availabilities by uncoupling the null models for interaction frequency and phylogenetic distance. We incorporate the deviation between observed and random interaction frequencies as weights into the calculation of partner phylogenetic α‐diversity. To calculate the PSS index, we then compare observed partner phylogenetic α‐diversity to a null distribution generated by randomizing phylogenetic distances among the same number of partners. PSS quantifies the phylogenetic structure (i.e., clustered, overdispersed, or random) of the partners of a focal species. We show with simulations that the PSS index is not correlated with network properties, which allows comparisons across multiple systems. We also implemented PSS on empirical networks of host–parasite, avian seed‐dispersal, lichenized fungi–cyanobacteria, and hummingbird pollination interactions. Across these systems, a large proportion of taxa interact with phylogenetically random partners according to PSS, sometimes to a larger extent than detected with an existing method that does not account for partner availability. We also found that many taxa interact with phylogenetically clustered partners, while taxa with overdispersed partners were rare. We argue that species with phylogenetically overdispersed partners have often been misinterpreted as generalists when they should be considered specialists. Our results highlight the important role of randomness in shaping interaction networks, even in highly intimate symbioses, and provide a much‐needed quantitative framework to assess the role that evolutionary history and symbiotic specialization play in shaping patterns of biodiversity. PSS is available as an R package at https://github.com/cjpardodelahoz/pss .

59 BASIC BIOLOGICAL SCIENCES↗

DOE OSTI2021

Southwest Pacific tropical cyclone development classification utilizing machine learning and synoptic composites

This study evaluates the ability of machine learning algorithms to classify tropical depressions (TDs) and tropical storms (TSs) in the western region of the southwest Pacific Ocean (SWPO). Decision rules are generated to predict the environment required for a depression to fully develop into a mature storm, and the most influential predictors in the classification decision are ranked. TD and TS are discriminated based on a maximum sustained wind speed threshold (≥17 ms -1 ). Various aerosol, thermodynamic, and dynamic parameters are extracted closest to the initiation point of each non-developing and developing sample. The covariates associated with each labelled sample are used to train a decision tree and random forest model. Results using a testing dataset suggest the random forest approach more accurately distinguishes between non-developing and developing samples. The classification accuracy of the decision tree and random forest are 72% and 91%, respectively. Random forest outperformed the decision tree by providing higher accuracy in test data. The most important variables for binary classification are sea salt aerosol optical depth (AOD), 1,000 mb relative humidity, and sea surface temperature. AOD is a quantitative estimate of the aerosols presents in the air through the extinction of a ray of light as it passes through the atmosphere. Mean composite maps constructed in an unsupervised manner have been created for the most important variables identified by the random forest classifier during TD and TS events to highlight the difference in geophysical and aerosol variables' climatology during the two different classifications. This work will advance the risk management strategies for northeastern Australia and other SWPO basin islands to control their tropical cyclone related losses through prioritizing forecasting variables that are the strongest predictors of the strengthening of tropical depressions into tropical cyclones.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2021

Probabilistic learning on manifolds constrained by nonlinear partial differential equations for small datasets

A novel extension of the Probabilistic Learning on Manifolds (PLoM) is presented. It makes it possible to synthesize solutions to a wide range of nonlinear stochastic boundary value problems described by partial differential equations (PDEs) for which a stochastic computational model (SCM) is available and which depend on a vector-valued random control parameter. The cost of a single numerical evaluation of this SCM is assumed to be such that only a limited number of points can be computed for constructing the training dataset (small data). Each point of the training dataset is made up of realizations from a vector-valued stochastic process (the stochastic solution) and the associated random control parameter on which it depends. The presented PLoM constrained by PDE allows for generating a large number of learned realizations of the stochastic process and its corresponding random control parameter. These learned realizations are generated so as to minimize the vector-valued random residual of the PDE in the mean-square sense. Appropriate novel methods are developed to solve this challenging problem. Three applications are presented. The first one is a simple uncertain nonlinear dynamical system with a nonstationary stochastic excitation. The second one concerns the 2D nonlinear unsteady Navier–Stokes equations for incompressible flows in which the Reynolds number is the random control parameter. Here, the last one deals with the nonlinear dynamics of a 3D elastic structure with uncertainties. The results obtained make it possible to validate the PLoM constrained by stochastic PDE but also provide further validation of the PLoM without constraint.

Machine learning↗

DOE OSTI2021

A modular methodology for time-domain stochastic seismic wave propagation

Presented here is a modular methodology for time-domain stochastic seismic wave propagation analysis. Presented methodology is designed to analyse uncertain seismic motions as an input, propagating through uncertain material. Traditional approach for uncertain wave propagation relies on models that include deep bedrock, local soil site, and their random process and random field information. Such models can become quite large and computationally intractable. The modular approach proposed herein features two step approach that allows separate consideration of the deep bedrock and local site along with corresponding random field information. In this work, the first step considers an auxiliary stochastic motions problem in the bedrock. Stochastic local site response can then be simulated in a reduced domain within certain depth from the surface. Application of uncertain seismic motions at depth, for local uncertain site response is done using stochastic effective forces developed through the Domain Reduction Method. By using Hermite polynomial chaos expansion to represent the non-Gaussian random field of material parameters and non-stationary random process of seismic motion, the proposed modular methodology is formulated using intrusive stochastic Galerkin approach, as seen in the Stochastic Elastic–Plastic Finite Element Method (SEPFEM). Developed modular methodology is illustrated using a 1-D stochastic seismic wave propagation analysis with three cases, and simulation results are also verified with results from conventional approach.

58 GEOSCIENCES↗

DOE OSTI2021

Flow-driven spectral chaos (FSC) method for simulating long-time dynamics of arbitrary-order non-linear stochastic dynamical systems

Uncertainty quantification techniques such as the time-dependent generalized polynomial chaos (TD-gPC) use an adaptive orthogonal basis to better represent the stochastic part of the solution space (aka random function space) in time. However, because the random function space is constructed using tensor products, TD-gPC-based methods are known to suffer from the curse of dimensionality. Here, we introduce a new numerical method called the flow-driven spectral chaos (FSC) which overcomes this curse of dimensionality at the random-function-space level. The proposed method is not only computationally more efficient than existing TD-gPC-based methods but is also far more accurate. The FSC method uses the concept of enriched stochastic flow maps to track the evolution of a finite-dimensional random function space efficiently in time. To transfer the probability information from one random function space to another, two approaches are developed and studied herein. In the first approach, the probability information is transferred in the mean-square sense, whereas in the second approach the transfer is done exactly using a new theorem that was developed for this purpose. The FSC method can quantify uncertainties with high fidelity, especially for the long-time response of stochastic dynamical systems governed by ODEs of arbitrary order. Six representative numerical examples, including a nonlinear problem (the Van-der-Pol oscillator), are presented to demonstrate the performance of the FSC method and corroborate the claims of its superior numerical properties. Finally, a parametric, high-dimensional stochastic problem is used to demonstrate that when the FSC method is used in conjunction with Monte Carlo integration, the curse of dimensionality can be overcome altogether.

(nonlinear) stochastic dynamical systems↗

DOE OSTI2025

Neural chaos: A spectral stochastic neural operator

Building surrogate models for operators with uncertainty quantification capabilities is essential for many engineering applications where randomness–such as variability in material properties, boundary conditions, and initial conditions–is unavoidable. Polynomial Chaos Expansion (PCE) is widely recognized as a go-to method for constructing stochastic surrogates in both intrusive and non-intrusive ways, and it has recently been used in the context of operator learning. However, its application becomes challenging for complex or high-dimensional processes, as achieving accuracy requires higher-order polynomials, which can increase computational demand and/or the risk of overfitting. Furthermore, PCE requires specialized treatments to manage random variables that are not independent, and these treatments may be problem-dependent or may fail with increasing complexity. Here, in this work, we adopt the same formalism as the spectral expansion used in PCE; however, we replace the classical polynomial basis functions with neural network (NN) basis functions to leverage their expressivity. To achieve this, we propose an algorithm that identifies NN-parameterized basis functions in a purely data-driven manner, without any prior assumptions about the joint distribution of the random variables involved, whether independent or dependent, or about their marginal distributions. The proposed algorithm identifies each NN-parameterized basis function sequentially, ensuring they are orthogonal with respect to the data distribution. The basis functions are constructed directly on the joint stochastic variables without requiring a tensor product structure or assuming independence of the random variables. This approach may offer greater flexibility for complex stochastic models, while simplifying implementation compared to the tensor product structures typically used in PCE to handle random vectors. This is particularly advantageous given the current state of open-source packages, where building and training neural networks can be done with just a few lines of code and extensive community support. We demonstrate the effectiveness of the proposed scheme through several numerical examples of varying complexity and provide comparisons with classical PCE.

Polynomial chaos expansion↗

DOE OSTI2020

Building Durable Multimetallic Electrocatalysts from Intermetallic Seeds

When combined with earth-abundant metals, Pt-based alloy nanoparticles (NPs) can be cost-effective electrocatalysts. However, these NPs can experience leaching of non-noble-metal components under harsh electrocatalytic conditions. In this work, the Skrabalak group has demonstrated a novel NP construct in which Pt-based random alloy surfaces are stabilized against non-noble-metal leaching by their deposition onto intermetallic seeds. These core@shell NPs are highly durable electrocatalysts, with the ability to tune catalytic performance by the core@shell architecture, surface alloy composition, and NP shape. This versatility was demonstrated in a model system in which random alloy (ra-) PtM surfaces were deposited onto ordered intermetallic (i-) PdCu seeds using seed-mediated co-reduction (SMCR). In the initial demonstration, ra-PtCu shells were deposited on i-PdCu seeds, with these core@shell NPs exhibiting higher specific and mass activities for the oxygen reduction reaction (ORR) when compared to similarly sized ra-PtCu NPs. These NPs also showed outstanding durability, maintaining ~85% in specific activity after 5000 cycles. Characterization of the NPs after use revealed minimal loss of Cu. The activity enhancement was attributed to the strained surface that arises from the lattice mismatch between the intermetallic core and random alloy surface. The outstanding durability was attributed to the ordered structure of the intermetallic core. The origin of this durability enhancement was investigated by classical molecular dynamics simulations, where Pt atoms were found to have a lower potential energy when deposited on an intermetallic core than when deposited on a random alloy core. Also, ordering of Cu atoms at the core@shell interface appears to enhance the overall binding between the core and the shell materials. Inspired by this initial demonstration, SMCR has been used to achieve shells of different random alloy compositions, PtM (M = Ni, Co, Cu, or Fe). This advance is significant because ligand effects vary as a function of PtM identity and Pt/M ratio. These features also influence the degree of surface strain imparted from the lattice mismatch between the core and shell materials. Like the initial demonstration, standout features of these core@shell NPs were high durability and resistance to non-noble metal leaching. Moving forward, efforts have been directed toward integrating shape-control to this core@shell NP construct. This integration is motivated by the shape-dependent catalytic performance of NPs derived from the selective expression of specific facets. Considering the initial i-PdCu@ra-PtCu system, NPs with a cubic shape have been achieved by judicious selection of capping ligands during SMCR. Evaluation of these NPs as catalysts for the electrooxidation of formic acid found that the nanocubic shape enhances catalytic performance compared to similar core@shell NPs with a spherical morphology. We envision that SMCR can be applied to other NP systems to achieve highly durable catalysts as the syntheses of monodisperse and shape-controlled intermetallic seeds are advanced. This Account highlights the role of intermetallic cores in providing more durable electrocatalysts. More broadly, the versatility of SMCR is highlighted as a route to integrate architecture, alloy surfaces, and shape within one NP system, and how this achievement is inspiring new high-performance and robust catalysts is discussed.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2023

Benchmarking quantum logic operations relative to thresholds for fault tolerance

Contemporary methods for benchmarking noisy quantum processors typically measure average error rates or process infidelities. However, thresholds for fault-tolerant quantum error correction are given in terms of worst-case error rates—defined via the diamond norm—which can differ from average error rates by orders of magnitude. One method for resolving this discrepancy is to randomize the physical implementation of quantum gates, using techniques like randomized compiling (RC). In this work, we use gate set tomography to perform precision characterization of a set of two-qubit logic gates to study RC on a superconducting quantum processor. We find that, under RC, gate errors are accurately described by a stochastic Pauli noise model without coherent errors, and that spatially correlated coherent errors and non-Markovian errors are strongly suppressed. We further show that the average and worst-case error rates are equal for randomly compiled gates, and measure a maximum worst-case error of 0.0197(3) for our gate set. Our results show that randomized benchmarks are a viable route to both verifying that a quantum processor’s error rates are below a fault-tolerance threshold, and to bounding the failure rates of near-term algorithms, if—and only if—gates are implemented via randomization methods which tailor noise.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

Milky Way mass with K giants and BHB stars using LAMOST, SDSS/SEGUE, and Gaia : 3D spherical Jeans equation and tracer mass estimator

ABSTRACT We measure the enclosed Milky Way mass profile to Galactocentric distances of ∼70 and ∼50 kpc using the smooth, diffuse stellar halo samples of Bird et al. The samples are Large Sky Area Multi-Object Fiber Spectroscopic Telescope (LAMOST) and Sloan Digital Sky Survey/Sloan Extension for Galactic Understanding and Exploration (SDSS/SEGUE) K giants (KG) and SDSS/SEGUE blue horizontal branch (BHB) stars with accurate metallicities. The 3D kinematics are available through LAMOST and SDSS/SEGUE distances and radial velocities and Gaia DR2 proper motions. Two methods are used to estimate the enclosed mass: 3D spherical Jeans equation and Evans et al. tracer mass estimator (TME). We remove substructure via the Xue et al. method based on integrals of motion. We evaluate the uncertainties on our estimates due to random sampling noise, systematic distance errors, the adopted density profile, and non-virialization and non-spherical effects of the halo. The tracer density profile remains a limiting systematic in our mass estimates, although within these limits we find reasonable agreement across the different samples and the methods applied. Out to ∼70 and ∼50 kpc, the Jeans method yields total enclosed masses of 4.3 ± 0.95 (random) ±0.6 (systematic) × 1011 M⊙ and 4.1 ± 1.2 (random) ±0.6 (systematic) × 1011 M⊙ for the KG and BHB stars, respectively. For the KG and BHB samples, we find a dark matter virial mass of $M_{200}=0.55^{+0.15}_{-0.11}$ (random) ±0.083 (systematic) × 1012 M⊙ and $M_{200}=1.00^{+0.67}_{-0.33}$ (random) ±0.15 (systematic) × 1012 M⊙, respectively.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2025

Operator dynamics in Floquet many-body systems

We study operator dynamics in many-body quantum systems, focusing on generic features of systems that are ergodic, spatially extended, and lack conserved densities. Quantum circuits of various types provide simple models for such systems. We focus on Floquet quantum circuits, comparing their behavior with what has been found previously for circuits that are random in time. Floquet circuits, which have discrete time-translation symmetry, represent an intermediate case between circuits that are random in time and lack any symmetry, and systems with a time-independent Hamiltonian and continuous time-translation invariance. By making this comparison, one of our aims is to identify signatures of time-translation symmetry in Floquet operator dynamics. To characterize behavior we examine a variety of quantities in solvable models and numerically: operator autocorrelation functions; the partial spectral form factor; the out-of-time-order correlator (OTOC); and the paths in operator space that make the dominant contributions to the ensemble-averaged autocorrelation functions. Our most striking result is that ensemble-averaged autocorrelation functions show behavior that is distinctively different in Floquet systems compared to systems in which successive time-steps are independent. Specifically, while average autocorrelation functions decay on a microscopic timescale for circuits that are random in time, in Floquet systems they have a late-time tail with a duration that grows parametrically with the size of the operator support. In the simplest models this tail is separated from the initial decay by a minimum, so that the average autocorrelation function has an intermediate-time peak. The existence of these tails provides a way to understand deviations of the spectral form factor from random matrix behavior at times shorter than the Thouless time. In contrast to this feature in autocorrelation functions, we find no new aspects to the behavior of OTOCs for Floquet models compared to random-in-time circuits. We show that this difference between averaged autocorrelation functions and OTOCs can be understood in terms of the paths in operator space that contribute to the two quantities: paths for the former retain a limited support at late times, while paths for the latter are dominated by operator spreading. Published by the American Physical Society 2025

Yoshimura, Takato (ORCID:0000000309159846)↗

DOE OSTI2024

Quantifying Quantum Chaos through Microcanonical Distributions of Entanglement

A characteristic feature of “quantum chaotic” systems is that their eigenspectra and eigenstates display universal statistical properties described by random matrix theory (RMT). However, eigenstates of local systems also encode structure beyond RMT. To capture this feature, we introduce a framework that allows us to compare the properties of eigenstates in local systems with those of pure random states. In particular, our framework defines a notion of distance between quantum state ensembles that utilizes the Kullback-Leibler divergence to compare the microcanonical distribution of entanglement entropy (EE) of eigenstates with a reference RMT distribution generated by pure random states (with appropriate constraints). This notion gives rise to a quantitative metric for quantum chaos that not only accounts for averages of the distributions but also higher moments. The differences in moments are compared on a highly resolved scale set by the standard deviation of the RMT distribution, which is exponentially small in system size. As a result, the metric can distinguish between chaotic and integrable behaviors and, in addition, quantify and compare the of chaos (in terms of proximity to RMT behavior) between two systems that are assumed to be chaotic. We implement our framework in local, minimally structured, Floquet random circuits, as well as a canonical family of many-body Hamiltonians, the mixed-field Ising model (MFIM). Importantly, for Hamiltonian systems, we find that the reference random distribution must be appropriately constrained to incorporate the effect of energy conservation in order to describe the ensemble properties of midspectrum eigenstates. The metric captures deviations from RMT across all models and parameters, including those that have been previously identified as strongly chaotic, and for which other diagnostics of chaos such as level spacing statistics look strongly thermal. In Floquet circuits, the dominant source of deviations is the second moment of the distribution, and this persists for all system sizes. For the MFIM, we find significant variation of the KL divergence in parameter space. Notably, we find a small region where deviations from RMT are minimized, suggesting that “maximally chaotic” Hamiltonians may exist in fine-tuned pockets of parameter space. Published by the American Physical Society 2024

Physics↗

DOE OSTI2020

UQ Toolkit v 2.0

The Uncertainty Quantification (UQ) Toolkit is a software library for the characterizaton and propagation of uncertainties in computational models. For the characterization of uncertainties, Bayesian inference tools are provided to infer uncertain model parameters, as well as Bayesian compressive sensing methods for discovering sparse representations of high-dimensional input-output response surfaces, and also Karhunen-Loève expansions for representing stochastic processes. Uncertain parameters are treated as random variables and represented with Polynomial Chaos expansions (PCEs). The library implements several spectral basis function types (e.g. Hermite basis functions in terms of Gaussian random variables or Legendre basis functions in terms of uniform random variables) that can be used to represent random variables with PCEs. For propagation of uncertainty, tools are provided to propagate PCEs that describe the input uncertainty through the computational model using either intrusive methods (Galerkin projection of equations onto basis functions) or non-intrusive methods (perform deterministic operation at sampled values of the random values and project the obtained results onto basis functions).

Safta, Cosmin↗

DOE OSTI2023

Self-assembly of co-continuous nanostructured copolymer templates with compositional and architectural dispersity

In our previous DOE-supported project (DE-SC0016208), we demonstrated that a particular architecture dubbed ‘randomly end-linked copolymer networks’ (RECNs) led to especially robust self-assembly of disordered cocontinuous nanostructures. Such structures, wherein both microphases are fully percolating in three dimensions, are attractive for a wide-variety of energy-relevant applications including purification membranes, separators in batteries and fuel cells, catalyst supports, and high surface area electrodes. In the current project we sought to substantially extend our fundamental understanding of how different types of randomness in blocky copolymer architecture contribute to the robust formation of disordered cocontinuous nanostructures. To do so, we filled in the sizable gap between the previously studied cases of randomly linked linear multiblock copolymers and RECNs by studying two intermediate cases: (i) random miktoarm star polymers, which contain dispersity in the number of the two chemically-incompatible strands located at each junction and therefore the preferred interfacial curvature of each polymer, and (ii) randomly branched copolymers, which also possess bridging strands that stretch across domains and apply local ‘elastic’ forces to the interfaces. In addition, we studied a different network architecture, statistically crosslinked copolymer networks, that is more straightforward to prepare using chain growth polymerization methods, enabling more facile formation of cocontinuous nanostructures. Finally, we extended our approaches from model systems based on polystyrene and poly(lactic acid) to copolymer architectures containing either (i) polysulfone, an engineering polymer that provides excellent mechanical properties for membrane applications, or (ii) polyacrylonitrile, providing a convenient route to nanoporous carbon materials.

36 MATERIALS SCIENCE↗

DOE OSTI2025

Aided Active Learning (AAL) for Enhanced Critical Heat Flux Prediction

Accurate prediction of critical heat flux (CHF) is crucial for the safe and efficient operation of nuclear reactors. Traditional CHF modeling methods often require extensive experimental data, which are hard to obtain. This study introduces the Aided Active Learning (AAL) framework, which strategically minimizes data requirements without sacrificing model accuracy. Unlike conventional Active Learning (AL), AAL introduces an additional step of randomly selecting a subset from the sample pool before applying the query strategy. To evaluate the performance of AAL, two query strategies—uncertainty-based sampling and error-reduction sampling—were evaluated across the following models: random forest (RF), feedforward neural network (FNN), and variational feedforward neural network (vFNN). The proposed framework demonstrated that AAL effectively reduces the number of training samples needed to achieve comparable predictive accuracy. For the RF model, AL required only 710 samples to achieve an R2 score of 0.98, as compared to the 4,785 samples needed by random sampling. Similarly, the FNN model achieved the same R2 score with just 355 samples when using AL, a significant improvement over the 825 samples required by random sampling. In case of uncertainty-based sampling strategy, vFNN attained an R2 of 0.98 with 3,420 samples, reducing the sample requirement by 47% relative to the 6,440 samples needed for random sampling. Its performance suggests that larger training data are required to fully leverage its uncertainty quantification capabilities.

22 - GENERAL STUDIES OF NUCLEAR REACTORS↗

DOE OSTI2021

Quantum magnetism on small-world networks

While classical spin systems in random networks have been intensively studied, much less is known about quantum magnets in random graphs. In this work, we investigate interacting quantum spins on small-world networks, building on mean-field theory and extensive quantum Monte Carlo simulations. Starting from one-dimensional (1D) rings, we consider two situations: All-to-all interacting and long-range interactions randomly added. The effective infinite dimension of the lattice leads to a magnetic ordering at finite temperature Tc with mean-field criticality. Nevertheless, in contrast to the classical case, we find two distinct power-law behaviors for Tc versus the average strength of the extra couplings. This is controlled by a competition between a characteristic length scale of the random graph and the thermal correlation length of the underlying 1D system, thus challenging mean-field theories. We also investigate the fate of a gapped 1D spin chain against the small-world effect.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2024

Gradient Coding With Iterative Block Leverage Score Sampling

Gradient coding is a method for mitigating straggling servers in a centralized computing network that uses erasure-coding techniques to distributively carry out first-order optimization methods. Randomized numerical linear algebra uses randomization to develop improved algorithms for large-scale linear algebra computations. In this study, we propose a method for distributed optimization that combines gradient coding and randomized numerical linear algebra. The proposed method uses a randomized ℓ 2 -subspace embedding and a gradient coding technique to distribute blocks of data to the computational nodes of a centralized network, and at each iteration the central server only requires a small number of computations to obtain the steepest descent update. The novelty of our approach is that the data is replicated according to importance scores, called block leverage scores, in contrast to most gradient coding approaches that uniformly replicate the data blocks. Furthermore, we do not require a decoding step at each iteration, avoiding a bottleneck in previous gradient coding schemes. We show that our approach results in a valid ℓ 2 -subspace embedding, and that our resulting approximation converges to the optimal solution.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Void formation in operator growth, entanglement, and unitarity

The structure of the Heisenberg evolution of operators plays a key role in explaining diverse processes in quantum many-body systems. In this paper, we discuss a new universal feature of operator evolution: an operator can develop a void during its evolution, where its nontrivial parts become separated by a region of identity operators. Such processes are present in both integrable and chaotic systems, and are required by unitarity. We show that void formation has important implications for unitarity of entanglement growth and generation of mutual information and multipartite entanglement. We study explicitly the probability distributions of void formation in a number of unitary circuit models, and conjecture that in a quantum chaotic system the distribution is given by the one we find in random unitary circuits, which we refer to as the random void distribution. We also show that random unitary circuits lead to the same pattern of entanglement growth for multiple intervals as in (1 + 1)-dimensional holographic CFTs after a global quench, which can be used to argue that the random void distribution leads to maximal entanglement growth.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗