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Results for “Nonlinear dimensionality reduction”

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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All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

Online learning of quadratic manifolds from streaming data for nonlinear dimensionality reduction and nonlinear model reduction

Here, this work introduces an online greedy method for constructing quadratic manifolds from streaming data, designed to enable in situ analysis of numerical simulation data on the Petabyte scale. Unlike traditional batch methods, which require all data to be available upfront and take multiple passes over the data, the proposed online greedy method incrementally updates quadratic manifolds in one pass as data points are received, eliminating the need for expensive disk input/output operations as well as storing and loading data points once they have been processed. A range of numerical examples demonstrate that the online greedy method learns accurate quadratic manifold embeddings while being capable of processing data that far exceed common disk input/output capabilities and volumes as well as main-memory sizes.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Neural Active Manifolds: Nonlinear Dimensionality Reduction for Uncertainty Quantification

We present a new approach for nonlinear dimensionality reduction, specifically designed for computationally expensive mathematical models. We leverage autoencoders to discover a one-dimensional neural active manifold (NeurAM) capturing the model output variability, through the aid of a simultaneously learnt surrogate model with inputs on this manifold. Our method only relies on model evaluations and does not require the knowledge of gradients. The proposed dimensionality reduction framework can then be applied to assist outer loop many-query tasks in scientific computing, like sensitivity analysis and multifidelity uncertainty propagation. In particular, we prove, both theoretically under idealized conditions, and numerically in challenging test cases, how NeurAM can be used to obtain multifidelity sampling estimators with reduced variance by sampling the models on the discovered low-dimensional and shared manifold among models. Several numerical examples illustrate the main features of the proposed dimensionality reduction strategy and highlight its advantages with respect to existing approaches in the literature.

Autoencoders↗
DOE OSTI2023

Energy conserving quadrature based dimensionality reduction for nonlinear hydrodynamics problems

Projection based dimensionality reduction is used to approximate the full discretiza tion of a PDE problem in order to reduce the cost of its numerical simulation while staying faithful to the full dynamics. For nonlinear problems, hyperreduction methods are additionally needed to remove the dependence of the nonlinear terms on the full problem’s size. In this project, we develop an energy conserving hyperreduction method and apply it to Eulerian-Lagrangian hydrodynamics problems, demonstrating that the reduced model successfully accelerates the numerical simulations while conserving the problem’s total energy with high accuracy.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Kernel Manifolds: Nonlinear‐Augmentation Dimensionality Reduction Using Reproducing Kernel Hilbert Spaces

This paper generalizes recent advances on quadratic manifold (QM) dimensionality reduction by developing kernel methods-based nonlinear-augmentation dimensionality reduction. QMs, and more generally feature map-based nonlinear corrections, augment linear dimensionality reduction with a nonlinear correction term in the reconstruction map to overcome approximation accuracy limitations of purely linear approaches. While feature map-based approaches typically learn a least squares optimal polynomial correction term, we generalize this approach by learning an optimal nonlinear correction from a user-defined reproducing kernel Hilbert space. Our approach allows one to impose arbitrary nonlinear structure on the correction term, including polynomial structure, and includes feature map and radial basis function-based corrections as special cases. Furthermore, our method has relatively low training cost and has monotonically decreasing error as the latent space dimension increases. In conclusion, we compare our approach to proper orthogonal decomposition and several recent QM approaches on data from several example problems.

kernel methods↗
DOE OSTI2023

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↗
DOE OSTI2024

Interpretation of autoencoder-learned collective variables using Morse–Smale complex and sublevelset persistent homology: An application on molecular trajectories

Dimensionality reduction often serves as the first step toward a minimalist understanding of physical systems as well as the accelerated simulations of them. In particular, neural network-based nonlinear dimensionality reduction methods, such as autoencoders, have shown promising outcomes in uncovering collective variables (CVs). However, the physical meaning of these CVs remains largely elusive. In this work, we constructed a framework that (1) determines the optimal number of CVs needed to capture the essential molecular motions using an ensemble of hierarchical autoencoders and (2) provides topology-based interpretations to the autoencoder-learned CVs with Morse–Smale complex and sublevelset persistent homology. Furthermore, this approach was exemplified using a series of n-alkanes and can be regarded as a general, explainable nonlinear dimensionality reduction method.

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

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↗
DOE OSTI2024

Improved multifidelity Monte Carlo estimators based on normalizing flows and dimensionality reduction techniques

Here, we study the problem of multifidelity uncertainty propagation for computationally expensive models. In particular, we consider the general setting where the high-fidelity and low-fidelity models have a dissimilar parameterization both in terms of number of random inputs and their probability distributions, which can be either known in closed form or provided through samples. We derive novel multifidelity Monte Carlo estimators which rely on a shared subspace between the high-fidelity and low-fidelity models where the parameters follow the same probability distribution, i.e., a standard Gaussian. We build the shared space employing normalizing flows to map different probability distributions into a common one, together with linear and nonlinear dimensionality reduction techniques, active subspaces and autoencoders, respectively, which capture the subspaces where the models vary the most. We then compose the existing low-fidelity model with these transformations and construct modified models with an increased correlation with the high-fidelity model, which therefore yield multifidelity estimators with reduced variance. A series of numerical experiments illustrate the properties and advantages of our approaches.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Disentangling Ferroelectric Wall Dynamics and Identification of Pinning Mechanisms via Deep Learning

Field-induced domain-wall dynamics in ferroelectric materials underpins multiple applications ranging from actuators to information technology devices and necessitates a quantitative description of the associated mechanisms including giant electromechanical couplings, controlled nonlinearities, or low coercive voltages. While the advances in dynamic piezoresponse force microscopy measurements over the last two decades have rendered visualization of polarization dynamics relatively straightforward, the associated insights into the local mechanisms have been elusive. This work explores the domain dynamics in model polycrystalline materials using a workflow combining deep-learning-based segmentation of the domain structures with nonlinear dimensionality reduction using multilayer rotationally invariant autoencoders (rVAE). The former allows unambiguous identification and classification of the ferroelectric and ferroelastic domain walls. Here, the rVAE discovers the latent representations of the domain wall geometries and their dynamics, consequently providing insight into the intrinsic mechanisms of polarization switching, that can further be compared to simple physical models. The rVAE disentangles the factors affecting the pinning efficiency of ferroelectric walls, offering insights into the correlation of ferroelastic wall distribution and ferroelectric wall pinning.

36 MATERIALS SCIENCE↗
DOE OSTI2025

Enhancing Interpretability in Generative Modeling: Statistically Disentangled Latent Spaces Guided by Generative Factors in Scientific Datasets

This study addresses the challenge of statistically extracting generative factors from complex, high-dimensional datasets in unsupervised or semi-supervised settings. We investigate encoder-decoder-based generative models for nonlinear dimensionality reduction, focusing on disentangling low-dimensional latent variables corresponding to independent physical factors. Introducing Aux-VAE, a novel architecture within the classical Variational Autoencoder framework, we achieve disentanglement with minimal modifications to the standard VAE loss function by leveraging prior statistical knowledge through auxiliary variables. These variables guide the shaping of the latent space by aligning latent factors with learned auxiliary variables. We validate the efficacy of Aux-VAE through comparative assessments on multiple datasets, including astronomical simulations.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Geometric learning for computational mechanics Part II: Graph embedding for interpretable multiscale plasticity

The history-dependent behaviors of classical plasticity models are often driven by internal variables evolved according to phenomenological laws. The difficulty to interpret how these internal variables represent a history of deformation, the lack of direct measurement of these internal variables for calibration and validation, and the weak physical underpinning of those phenomenological laws have long been criticized as barriers to creating realistic models. In this work, geometric machine learning on graph data (e.g. finite element solutions) is used as a means to establish a connection between nonlinear dimensional reduction techniques and plasticity models. Geometric learning-based encoding on graphs allows the embedding of rich time-history data onto a low-dimensional Euclidean space such that the evolution of plastic deformation can be predicted in the embedded feature space. Finally, a corresponding decoder can then convert these low-dimensional internal variables back into a weighted graph such that the dominating topological features of plastic deformation can be observed and analyzed.

42 ENGINEERING↗
DOE OSTI2025

Understanding latent timescales in neural ordinary differential equation models of advection-dominated dynamical systems

The neural ordinary differential equation (ODE) framework has shown considerable promise in recent years in developing highly accelerated surrogate models for complex physical systems characterized by partial differential equations (PDEs). For PDE-based systems, state-of-the-art neural ODE strategies leverage a two-step procedure to achieve this acceleration: a nonlinear dimensionality reduction step provided by an autoencoder, and a time integration step provided by a neural-network based model for the resultant latent space dynamics (the neural ODE). This work explores the applicability of such autoencoder-based neural ODE strategies for PDEs in which advection terms play a critical role. More specifically, alongside predictive demonstrations, physical insight into the sources of model acceleration (i.e., how the neural ODE achieves its acceleration) is the scope of the current study. Such investigations are performed by quantifying the effects of both autoencoder and neural ODE components on latent system time-scales using eigenvalue analysis of dynamical system Jacobians. To this end, the sensitivity of various critical training parameters – de-coupled versus end-to-end training, latent space dimensionality, and the role of training trajectory length, for example – to both model accuracy and the discovered latent system timescales is quantified. Furthermore, this work specifically uncovers the key role played by the training trajectory length (the number of rollout steps in the loss function during training) on the latent system timescales: larger trajectory lengths correlate with an increase in limiting neural ODE time-scales, and optimal neural ODEs are found to recover the largest time-scales of the full-order (ground-truth) system. Demonstrations are performed across fundamentally different unsteady fluid dynamics configurations influenced by advection: (1) the Kuramoto–Sivashinsky equations (2) Hydrogen-Air channel detonations (the compressible reacting Navier–Stokes equations with detailed chemistry), and (3) 2D Atmospheric flow.

Advection-dominated dynamical systems↗
DOE OSTI2023

Comparing storm resolving models and climates via unsupervised machine learning

Global storm-resolving models (GSRMs) have gained widespread interest because of the unprecedented detail with which they resolve the global climate. However, it remains difficult to quantify objective differences in how GSRMs resolve complex atmospheric formations. This lack of comprehensive tools for comparing model similarities is a problem in many disparate fields that involve simulation tools for complex data. To address this challenge we develop methods to estimate distributional distances based on both nonlinear dimensionality reduction and vector quantization. Our approach automatically learns physically meaningful notions of similarity from low-dimensional latent data representations that the different models produce. This enables an intercomparison of nine GSRMs based on their high-dimensional simulation data (2D vertical velocity snapshots) and reveals that only six are similar in their representation of atmospheric dynamics. Furthermore, we uncover signatures of the convective response to global warming in a fully unsupervised way. Our study provides a path toward evaluating future high-resolution simulation data more objectively.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2021

Reduced-order modeling of advection-dominated systems with recurrent neural networks and convolutional autoencoders

A common strategy for the dimensionality reduction of nonlinear partial differential equations (PDEs) relies on the use of the proper orthogonal decomposition (POD) to identify a reduced subspace and the Galerkin projection for evolving dynamics in this reduced space. However, advection-dominated PDEs are represented poorly by this methodology since the process of truncation discards important interactions between higher-order modes during time evolution. In this study, we demonstrate that encoding using convolutional autoencoders (CAEs) followed by a reduced-space time evolution by recurrent neural networks overcomes this limitation effectively. We demonstrate that a truncated system of only two latent space dimensions can reproduce a sharp advecting shock profile for the viscous Burgers equation with very low viscosities, and a six-dimensional latent space can recreate the evolution of the inviscid shallow water equations. Additionally, the proposed framework is extended to a parametric reduced-order model by directly embedding parametric information into the latent space to detect trends in system evolution. Furthermore, our results show that these advection-dominated systems are more amenable to low-dimensional encoding and time evolution by a CAE and recurrent neural network combination than the POD-Galerkin technique.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Structure Tuning, Strong Second Harmonic Generation Response, and High Optical Stability of the Polar Semiconductors Na 1- x K x AsQ 2

We report the mixed cation compounds Na 1-x K x AsSe 2 (x = 0.8, 0.65, 0.5) and Na 0.1 K 0.9 AsS 2 crystallize in the polar noncentrosymmetric space group Cc. The AAsQ(2) (A = alkali metals, Q = S, Se) family features one-dimensional (1D) 1 / ∞ [AQ 2 - ] chains comprising corner-sharing pyramidal AQ 3 units in which the packing of these chains is dependent on the alkali metals. The parallel 1 / ∞ [AQ(2) - ] chains interact via short As ∙∙∙Se contacts, which increase in length when the fraction of K atoms is increased. The increase in the As ∙∙∙Se interchain distance increases the band gap from 1.75 eV in γ-NaAsSe 2 to 2.01 eV in Na 0.35 K 0.65 AsSe 2 , 2.07 eV in Na 0.2 K 0.8 AsSe 2 , and 2.18 eV in Na 0.1 K 0.9 AsS 2 . The Na 1-x K x AsSe 2 (x = 0.8, 0.65) compounds melt congruently at approximately 316 °C. Wavelength-dependent second harmonic generation (SHG) measurements on powder samples of Na 1-x K x AsSe 2 (x = 0.8, 0.65, 0.5) and Na 0.1 K 0.9 AsS 2 suggest that Na 0.2 K 0.8 AsSe 2 and Na 0.1 K 0.9 AsS 2 have the highest SHG response and exhibit significantly higher laser-induced damage thresholds (LIDTs). Theoretical SHG calculations on Na 0.5 K 0.5 AsSe 2 confirm its SHG response with the highest value of d 33 = 22.5 pm/V χ 333 (2) = 45.0 pm/V). The effective nonlinearity for a randomly oriented powder is calculated to be d eff = 18.9 pm/V χ eff (2) = 37.8 pm/V), which is consistent with the experimentally obtained value of d eff = 16.5 pm/V χ eff (2) = 33.0 pm/V). Three-photon absorption is the dominant mechanism for the optical breakdown of the compounds under intense excitation at 1580 nm, with Na 0.2 K 0.8 AsSe 2 exhibiting the highest stability.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗