Incommensurate two-dimensional checkerboard charge density wave in the low-dimensional superconductor Ta 4 Pd 3 Te 16
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75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
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Results for “Dimensionality”
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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Mixed three-dimensional and two-dimensional perovskites and methods of making the same
An aspect of the present disclosure is a perovskite that includes A(n−1−nw+w)A′(wn−w)A″2BnX(3n−3zn+3z−4e+1)X′(3zn−3z)X″4e, where each of A, A′, A″ are monovalent cations, B is a divalent cation, each of X, X′, and X″ are monovalent anions, 0<w≤1, 0<z≤1, 0<e≤1, and 1≤n≤100000.
Zhu, Kai↗
Grassmannian Diffusion Maps--Based Dimension Reduction and Classification for High-Dimensional Data
This work introduces the Grassmannian diffusion maps (GDMaps), a novel nonlinear dimensionality reduction technique that defines the affinity between points through their representation as low-dimensional subspaces corresponding to points on the Grassmann manifold. Here, the method is designed for applications, such as image recognition and data-based classification of constrained high-dimensional data where each data point itself is a high-dimensional object (i.e., a large matrix) that can be compactly represented in a lower-dimensional subspace. The GDMaps is composed of two stages. The first is a pointwise linear dimensionality reduction wherein each high-dimensional object is mapped onto the Grassmann manifold representing the low-dimensional subspace on which it resides. The second stage is a multipoint nonlinear kernel-based dimension reduction using diffusion maps to identify the subspace structure of the points on the Grassmann manifold. To this end, an appropriate Grassmannian kernel is used to construct the transition matrix of a random walk on a graph connecting points on the Grassmann manifold. Spectral analysis of the transition matrix yields low-dimensional Grassmannian diffusion coordinates embedding the data into a low-dimensional reproducing kernel Hilbert space. Further, a novel data classification/recognition technique is developed based on the construction of an overcomplete dictionary of reduced dimension whose atoms are given by the Grassmannian diffusion coordinates. Three examples are considered. First, a "toy" example shows that the GDMaps can identify an appropriate parametrization of structured points on the unit sphere. The second example demonstrates the ability of the GDMaps to revealing the intrinsic subspace structure of high-dimensional random field data. In the last ex- ample, a face recognition problem is solved considering face images subject to varying illumination conditions, changes in face expressions, and occurrence of occlusions. The technique presented high recognition rates (i.e., 95% in the best case) using a fraction of the data required by conventional methods.
42 ENGINEERING↗
Data Report on Post-Irradiation Dimensional Change of AGC-1 Samples
This report documents the measured post irradiation dimensional change in the AGC-1 samples. The AGC-1 capsule is the first in six planned irradiation capsules comprising the Advanced Graphite Creep (AGC) test series. AGC-1 irradiation began September 5, 2009 in the Advanced Test Reactor (ATR) and was completed on January 8, 2011. The capsule was cooled for 3 months in the ATR Canal, and then shipped to MFC in April 2011 for disassembly and sample extraction. After extraction the samples were shipped to the INL Research Center (IRC) for initial post-irradiation examination (PIE) and storage in the irradiated graphite vault. The AGC-1 capsule design contained “matched pair” samples to ascertain the irradiation-induced dimensional changes and levels of creep experienced in different graphite types. The irradiation-induced dimensional changes and creep levels are determined by comparing the total dimensional change for stressed and unstressed samples of the same type of graphite exposed to the same dose levels and at similar temperatures. Under irradiation creep (i.e. permanent strain due to irradiation, stress, and temperature) the stressed samples should demonstrate more dimensional change than the unstressed samples. This additional dimensional change in the stressed samples is designated as “irradiation-induced creep” in graphite. The data are further presented using the parameters influencing dimensional change in graphite; levels of induced stress, temperature, graphite type, and dose. However, the AGC-1 post-irradiation examination is a significant endeavor and this data report serves to provide irradiation-induced dimensional change data for AGC capsule design refinement as well as a status on the progress of the PIE activities. The dimensional changes of both the samples and graphite body are very important to the design of the future AGC capsules (AGC-3 through AGC-6) and are provided as soon as the data are available in order to determine whether design changes to the next capsule are required. A complete evaluation of the irradiation-induced dimensional change data will be performed for a final AGC-1 PIE report that will include full analysis of pre- and post-irradiation data, with verified AGC-1 irradiation conditions of temperature and dose.
11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
Dynamic realization of emergent high-dimensional optical vortices
The dimensionality of vortical structures has recently been extended beyond two dimensions, providing additional topological complexity and robustness for high-capacity information processing and turbulence control. The generation of high-dimensional vortical structures has mostly been demonstrated in classical systems through the complex interference of fluidic, acoustic, or electromagnetic waves. However, natural materials rarely support three- or higher-dimensional vortical structures and their physical interactions. Here, we experimentally demonstrate a high-dimensional gradient thickness optical cavity (GTOC) in which the optical coupling of planar metal-dielectric multilayers implements topological interactions across multiple dimensions. At non-trivial topological phases, high-dimensional GTOC induces high-dimensional vortical structures in generalized parameter space in three, four dimensions, and beyond. These emergent high-dimensional vortical structures are observed under electro-optic tomography as optical vortex dynamics in two-dimensional real-space, employing the optical thicknesses of dielectric layers as synthetic dimensions. Our findings hold significant promise for emulating high-dimensional physics and developing active topological photonic devices.
36 MATERIALS SCIENCE↗
A co-kurtosis PCA based dimensionality reduction with nonlinear reconstruction using neural networks
For turbulent reacting flow systems, identification of low-dimensional representations of the thermo-chemical state space is vitally important, primarily to significantly reduce the computational cost of device-scale simulations. Principal component analysis (PCA), and its variants, are a widely employed class of methods. Recently, an alternative technique that focuses on higher-order statistical interactions, co-kurtosis PCA (CoK-PCA), has been shown to effectively provide a low-dimensional representation by capturing the stiff chemical dynamics associated with spatiotemporally localized reaction zones. While its effectiveness has only been demonstrated based on a priori analyses with linear reconstruction, in this work, we employ nonlinear techniques to reconstruct the full thermo-chemical state and evaluate the efficacy of CoK-PCA compared to PCA. Specifically, we combine a CoK-PCA-/PCA-based dimensionality reduction (encoding) with an artificial neural network (ANN) based reconstruction (decoding) and examine, a priori, the reconstruction errors of the thermo-chemical state. In addition, we evaluate the errors in species production rates and heat release rates, which are nonlinear functions of the reconstructed state, as a measure of the overall accuracy of the dimensionality reduction technique. We employ four datasets to assess CoK-PCA/PCA coupled with ANN-based reconstruction: zero-dimensional (homogeneous) reactor for autoignition of an ethylene/air mixture that has conventional single-stage ignition kinetics, a dimethyl ether (DME)/air mixture which has two-stage (low and high temperature) ignition kinetics, a one-dimensional freely propagating premixed ethylene/air laminar flame, and a two-dimensional dataset representing turbulent autoignition of ethanol in a homogeneous charge compression ignition (HCCI) engine. Finally, results from the analyses demonstrate the robustness of the CoK-PCA based low-dimensional manifold with ANN reconstruction in accurately capturing the data, specifically from the reaction zones.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Entanglement Cost for Infinite-Dimensional Physical Systems
We prove that the entanglement cost equals the regularized entanglement of formation for any infinite-dimensional quantum state ρ ΑΒ with finite quantum entropy on at least one of the subsystems A or B. This generalizes a foundational result in quantum information theory that was previously formulated only for operations and states on finite-dimensional systems. The extension to infinite-dimensional systems is nontrivial because the conventional tools for establishing both the direct and converse bounds, i.e., strong typicality, monotonicity, and asymptotic continuity, are no longer directly applicable. To address this problem, we construct a new entanglement dilution protocol for infinite-dimensional states implementable by local operations and a finite amount of one-way classical communication (one-way LOCC), using weak and strong typicality multiple times. We also prove the optimality of this protocol among all protocols, even under infinite-dimensional separable operations, by developing an argument based on alternative forms of monotonicity and asymptotic continuity of the entanglement of formation for infinite-dimensional states. Along the way, we derive a new integral representation for the quantum entropy of infinite-dimensional states, which we believe to be of independent interest. Our results allow us to fully characterize an important operational entanglement measure—the entanglement cost—for all infinite-dimensional physical systems.
Complexity↗
Separable physics-informed DeepONet: Breaking the curse of dimensionality in physics-informed machine learning
The deep operator network (DeepONet) has shown remarkable potential in solving partial differential equations (PDEs) by mapping between infinite-dimensional function spaces using labeled datasets. However, in scenarios lacking labeled data, the physics-informed DeepONet (PI-DeepONet) approach, which utilizes the residual loss of the governing PDE to optimize the network parameters, faces significant computational challenges, particularly due to the curse of dimensionality. This limitation has hindered its application to high-dimensional problems, making even standard 3D spatial with 1D temporal problems computationally prohibitive. Additionally, the computational requirement increases exponentially with the discretization density of the domain. Here, to address these challenges and enhance scalability for high-dimensional PDEs, we introduce the Separable physics-informed DeepONet (Sep-PI-DeepONet). This framework employs a factorization technique, utilizing sub-networks for individual one-dimensional coordinates, thereby reducing the number of forward passes and the size of the Jacobian matrix required for gradient computations. By incorporating forward-mode automatic differentiation (AD), we further optimize computational efficiency, achieving linear scaling of computational cost with discretization density and dimensionality, making our approach highly suitable for high-dimensional PDEs. We demonstrate the effectiveness of Sep-PI-DeepONet through three benchmark PDE models: the viscous Burgers’ equation, Biot’s consolidation theory, and a parameterized heat equation. Our framework maintains accuracy comparable to the conventional PI-DeepONet while reducing training time by two orders of magnitude. Notably, for the heat equation solved as a 4D problem, the conventional PI-DeepONet was computationally infeasible (estimated 289.35 h), while the Sep-PI-DeepONet completed training in just 2.5 h. These results underscore the potential of Sep-PI-DeepONet in efficiently solving complex, high-dimensional PDEs, marking a significant advancement in physics-informed machine learning.
Neural operator↗
Cost function for low-dimensional manifold topology assessment
Abstract In reduced-order modeling, complex systems that exhibit high state-space dimensionality are described and evolved using a small number of parameters. These parameters can be obtained in a data-driven way, where a high-dimensional dataset is projected onto a lower-dimensional basis. A complex system is then restricted to states on a low-dimensional manifold where it can be efficiently modeled. While this approach brings computational benefits, obtaining a good quality of the manifold topology becomes a crucial aspect when models, such as nonlinear regression, are built on top of the manifold. Here, we present a quantitative metric for characterizing manifold topologies. Our metric pays attention to non-uniqueness and spatial gradients in physical quantities of interest, and can be applied to manifolds of arbitrary dimensionality. Using the metric as a cost function in optimization algorithms, we show that optimized low-dimensional projections can be found. We delineate a few applications of the cost function to datasets representing argon plasma, reacting flows and atmospheric pollutant dispersion. We demonstrate how the cost function can assess various dimensionality reduction and manifold learning techniques as well as data preprocessing strategies in their capacity to yield quality low-dimensional projections. We show that improved manifold topologies can facilitate building nonlinear regression models.
42 ENGINEERING↗
Dimensional Control over Metal Halide Perovskite Crystallization Guided by Active Learning
Metal halide perovskite (MHP) derivatives, a promising class of optoelectronic materials, have been synthesized with a range of dimensionalities that govern their optoelectronic properties and determine their applications. We demonstrate a data-driven approach combining active learning and high-throughput experimentation to discover, control, and understand the formation of phases with different dimensionalities in the morpholinium (morph) lead iodide system. Using a robot-assisted workflow, we synthesized and characterized two novel MHP derivatives that have distinct optical properties: a one-dimensional (1D) morphPbI 3 phase ([C 4 H 10 NO][PbI 3 ]) and a two-dimensional (2D) (morph) 2 PbI 4 phase ([C 4 H 10 NO] 2 [PbI 4 ]). To efficiently acquire the data needed to construct a machine learning (ML) model of the reaction conditions where the 1D and 2D phases are formed, data acquisition was guided by a diverse-mini-batch-sampling active learning algorithm, using prediction confidence as a stopping criterion. Querying the ML model uncovered the reaction parameters that have the most significant effects on dimensionality control. Based on these insights, we discuss possible reaction schemes that may selectively promote the formation of morph-Pb-I phases with different dimensionalities. The data-driven approach presented here, including the use of additives to manipulate dimensionality, will be valuable for controlling the crystallization of a range of materials over large reaction-composition spaces.
36 MATERIALS SCIENCE↗
Deep learning for characterizing the self-assembly of three-dimensional colloidal systems
Creating a systematic framework to characterize the structural states of colloidal self-assembly systems is crucial for unraveling the fundamental understanding of these systems’ stochastic and non-linear behavior. The most accurate characterization methods create high-dimensional neighborhood graphs that may not provide useful information about structures unless these are well-defined reference crystalline structures. Dimensionality reduction methods are thus required to translate the neighborhood graphs into a low-dimensional space that can be easily interpreted and used to characterize non-reference structures. We investigate a framework for colloidal system state characterization that employs deep learning methods to reduce the dimensionality of neighborhood graphs. The framework next uses agglomerative hierarchical clustering techniques to partition the low-dimensional space and assign physically meaningful classifications to the resulting partitions. In this work, we first demonstrate the proposed colloidal self-assembly state characterization framework on a three-dimensional in silico system of 500 multi-flavored colloids that self-assemble under isothermal conditions. We next investigate the generalizability of the characterization framework by applying the framework to several independent self-assembly trajectories, including a three-dimensional in silico system of 2052 colloidal particles that undergo evaporation-induced self-assembly.
36 MATERIALS SCIENCE↗
Sampling Low-Dimensional Markovian Dynamics for Preasymptotically Recovering Reduced Models from Data with Operator Inference
This work introduces a method for learning low-dimensional models from data of high-dimensional black-box dynamical systems. The novelty is that the learned models are exactly the reduced models that are traditionally constructed with classical projection-based model reduction techniques. Thus, the proposed approach learns models that are guaranteed to have the well-studied properties of reduced models known from model reduction, without requiring full knowledge of the governing equations and without requiring the operators of the high-dimensional systems. The key ingredient is a new data sampling scheme to obtain re-projected trajectories of high-dimensional systems that correspond to Markovian dynamics in low-dimensional subspaces. The exact recovery of reduced models from these re-projected trajectories is guaranteed pre-asymptotically under certain conditions for finite amounts of data and for a large class of systems with polynomial nonlinear terms. Numerical results demonstrate that the low-dimensional models learned with the proposed approach match reduced models from traditional model reduction up to numerical errors in practice. In conclusion, the numerical results further indicate that low-dimensional models fitted to re-projected trajectories are predictive even in situations where models fitted to trajectories without re-projection are inaccurate and unstable.
97 MATHEMATICS AND COMPUTING↗
Hutchinson Trace Estimation for high-dimensional and high-order Physics-Informed Neural Networks
Physics-Informed Neural Networks (PINNs) have proven effective in solving partial differential equations (PDEs), especially when some data are available by seamlessly blending data and physics. However, extending PINNs to high-dimensional and even high-order PDEs encounters significant challenges due to the computational cost associated with automatic differentiation in the residual loss function calculation. Herein, we address the limitations of PINNs in handling high-dimensional and high-order PDEs by introducing the Hutchinson Trace Estimation (HTE) method. Starting with the second-order high-dimensional PDEs, which are ubiquitous in scientific computing, HTE is applied to transform the calculation of the entire Hessian matrix into a Hessian vector product (HVP). This approach not only alleviates the computational bottleneck via Taylor-mode automatic differentiation but also significantly reduces memory consumption from the Hessian matrix to an HVP’s scalar output. We further showcase HTE’s convergence to the original PINN loss and its unbiased behavior under specific conditions. Comparisons with the Stochastic Dimension Gradient Descent (SDGD) highlight the distinct advantages of HTE, particularly in scenarios with significant variability and variance among dimensions. We further extend the application of HTE to higher-order and higher-dimensional PDEs, specifically addressing the biharmonic equation. By employing tensor-vector products (TVP), HTE efficiently computes the colossal tensor associated with the fourth-order high-dimensional biharmonic equation, saving memory and enabling rapid computation. The effectiveness of HTE is illustrated through experimental setups, demonstrating comparable convergence rates with SDGD under memory and speed constraints. Additionally, HTE proves valuable in accelerating the Gradient-Enhanced PINN (gPINN) version as well as the Biharmonic equation. Overall, HTE opens up a new capability in scientific machine learning for tackling high-order and high-dimensional PDEs.
Curse of dimensionality↗
A survey of unsupervised learning methods for high-dimensional uncertainty quantification in black-box-type problems
Constructing surrogate models for uncertainty quantification (UQ) on complex partial differential equations (PDEs) having inherently high-dimensional O(10 n ), n ≥ 2, stochastic inputs (e.g., forcing terms, boundary conditions, initial conditions) poses tremendous challenges. The “curse of dimensionality” can be addressed with suitable unsupervised learning techniques used as a pre-processing tool to encode inputs onto lower-dimensional subspaces while retaining its structural information and meaningful properties. In this work, we review and investigate thirteen dimension reduction methods including linear and nonlinear, spectral, blind source separation, convex and non-convex methods and utilize the resulting embeddings to construct a mapping to quantities of interest via polynomial chaos expansions (PCE). Here, we refer to the general proposed approach as manifold PCE (m-PCE), where manifold corresponds to the latent space resulting from any of the studied dimension reduction methods. To investigate the capabilities and limitations of these methods we conduct numerical tests for three physics-based systems (treated as black-boxes) having high-dimensional stochastic inputs of varying complexity modeled as both Gaussian and non-Gaussian random fields to investigate the effect of the intrinsic dimensionality of input data. We demonstrate both the advantages and limitations of the unsupervised learning methods and we conclude that a suitable m-PCE model provides a cost-effective approach compared to alternative algorithms proposed in the literature, including recently proposed expensive deep neural network-based surrogates and can be readily applied for high-dimensional UQ in stochastic PDEs.
42 ENGINEERING↗
Bias-Variance Trade-Off in Physics-Informed Neural Networks with Randomized Smoothing for High-Dimensional PDEs
Physics-Informed Neural Networks (PINNs) have triggered a paradigm shift in scientific computing, leveraging mesh-free properties and robust approximation capabilities. While proving effective for low-dimensional partial differential equations (PDEs), the computational cost of PINNs remains a hurdle in high-dimensional scenarios. This is particularly pronounced when computing high-order and high-dimensional derivatives in the physics-informed loss. Randomized Smoothing PINN (RS-PINN) introduces Gaussian noise for stochastic smoothing of the original neural net model, enabling the use of Monte Carlo methods for derivative approximation, which eliminates the need for costly automatic differentiation. Despite its computational efficiency, especially in the approximation of high-dimensional derivatives, RS-PINN introduces biases in both loss and gradients, negatively impacting convergence, especially when coupled with stochastic gradient descent (SGD) algorithms. We present a comprehensive analysis of biases in RS-PINN, attributing them to the nonlinearity of the Mean Squared Error (MSE) loss as well as the intrinsic nonlinearity of the PDE itself. We propose tailored bias correction techniques, delineating their application based on the order of PDE nonlinearity. The derivation of an unbiased RS-PINN allows for a detailed examination of its advantages and disadvantages compared to the biased version. Specifically, the biased version has a lower variance and runs faster than the unbiased version, but it is less accurate due to the bias. To optimize the bias-variance trade-off, we combine the two approaches in a hybrid method that balances the rapid convergence of the biased version with the high accuracy of the unbiased version. In addition to methodological contributions, we present an enhanced implementation of RS-PINN. Extensive experiments on diverse high-dimensional PDEs, including Fokker-Planck, Hamilton-Jacobi-Bellman (HJB), viscous Burgers’, Allen-Cahn, and Sine-Gordon equations, illustrate the bias-variance trade-off and highlight the effectiveness of the hybrid RS-PINN. Empirical guidelines are provided for selecting biased, unbiased, or hybrid versions, depending on the dimensionality and nonlinearity of the specific PDE problem.
97 MATHEMATICS AND COMPUTING↗
High-dimensional data analytics in civil engineering: A review on matrix and tensor decomposition
Recent developments in sensing and monitoring techniques have led to the generation of high-dimensional data in the field of civil engineering. High-dimensional data analytics methods have thus been developed to interpret such complex data. Among the different high-dimensional data analytics techniques, matrix and tensor decomposition methods have acquired a notable interest in the civil engineering community over the past decade. Due to their unique ability to deal with highly redundant and correlated data, these methods are establishing themselves as promising and efficient tools to analyze high-dimensional data in the civil engineering arena. In this paper, high-dimensional data is referred to as a data set in which the number of features is comparable or larger than the number of observations. This review paper aims to summarize the applications of matrix and tensor decomposition methods in civil engineering over the last decade. The survey begins with a general overview of matrix and tensor decomposition followed by highlighting their significance in the field. Afterward, various applications of these high-dimensional data analytics methods in civil engineering are presented, while the advantages offered by these methods are discussed. Lastly, challenges and potential research avenues for employing matrix and tensor decomposition and future emerging trends for their novel use are highlighted.
42 ENGINEERING↗
Uncovering interpretable relationships in high-dimensional scientific data through function preserving projections
Abstract In many fields of science and engineering, we frequently encounter experiments or simulations datasets that describe the behavior of complex systems and uncovering human interpretable patterns between their inputs and outputs via exploratory data analysis is essential for building intuition and facilitating discovery. Often, we resort to 2D embeddings for examining these high-dimensional relationships (e.g. dimensionality reduction). However, most existing embedding methods treat the dimensions as coordinates for samples in a high-dimensional space, which fail to capture the potential functional relationships, and the few methods that do take function into consideration either only focus on linear patterns or produce non-linear embeddings that are hard to interpret. To address these challenges, we proposed function preserving projections (FPP), which construct 2D linear embeddings optimized to reveal interpretable yet potentially non-linear patterns between the domain and the range of a high-dimensional function. The intuition here is that humans are good at understanding potentially non-linear patterns in 2D but unable to interpret non-linear mapping from high-dimensional space to 2D. Therefore, we should restrict the projection to linear but not the pattern we are seeking. Using FPP on real-world datasets, one can obtain fundamentally new insights about high-dimensional relationships in extremely large datasets that could not be processed with existing dimension reduction methods.
97 MATHEMATICS AND COMPUTING↗
Plasmons in two-dimensional topological insulators
Here, we analyze collective excitations in models of two-dimensional topological insulators using the random phase approximation. In a two-dimensional extension of the Su-Schrieffer-Heeger model, edge plasmonic excitations with induced charge-density distributions localized at the boundaries of the system are found in the topologically nontrivial phase, dispersing similarly as one-dimensional bulk plasmons in the conventional Su-Schrieffer-Heeger chain. For two-dimensional bulk collective modes, we reveal regimes of enhanced interband wave function correlations, leading to characteristic hardening and softening of inter- and intraband bulk plasmonic branches, respectively. In the two-dimensional Haldane Chern insulator model, chiral, unidirectional edge plasmons in nano-ribbon architectures are observed, which can be characterized by an effective Coulomb interaction cross section. Bulk collective excitations in the two-dimensional Haldane model are shown to be originated by single-particle band structure details in different topological phases.
2-dimensional systems↗