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Results for “differentiable manifold”

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

cymyc: $\underline{C}$alabi-$\underline{Y}$au $\underline{M}$etrics, $\underline{Y}$ukawas, and $\underline{C}$urvature

We introduce cymyc, a high-performance Python library for numerical investigation of the geometry of a large class of string compactification manifolds and their associated moduli spaces. We develop a well-defined geometric ansatz to numerically model tensor fields of arbitrary degree on a large class of Calabi-Yau manifolds. cymyc includes a machine learning component which incorporates this ansatz to model tensor fields of interest on these spaces by finding an approximate solution to the system of partial differential equations they should satisfy.

differential and algebraic geometry↗
DOE OSTI2023

Mediating Photochemical Reaction Rates at Lewis Acidic Rare Earths by Selective Energy Loss to 4f-Electron States

Manifesting chemical differences in individual rare earth (RE) element complexes is challenging due to the similar sizes of the tripositive cations and the corelike 4f shell. In this work, we disclose a new strategy for differentiating between similarly sized Dy 3+ and Y 3+ ions through a tailored photochemical reaction of their isostructural complexes in which the f-electron states of Dy 3+ act as an energy sink. Complexes RE(hfac) 3 (NMMO) 2 (RE = Dy (2-Dy) and Y (2-Y), hfac = hexafluoroacetylacetonate, and NMMO = N-methylmorpholine-N-oxide) showed variable rates of oxygen atom transfer (OAT) to triphenylphosphine under ultraviolet (UV) irradiation, as monitored by 1 H and 19 F NMR spectroscopies. Ultrafast transient absorption spectroscopy (TAS) identified the excited state(s) responsible for the photochemical OAT reaction or lack thereof. Competing sensitization pathways leading to excited-state deactivation in 2-Dy through energy transfer to the 4f electron manifold ultimately slows the OAT reaction at this metal cation. The measured rate differences between the open-shell Dy 3+ and closed-shell Y 3+ complexes demonstrate that using established principles of 4f ion sensitization may deliver new, selective modalities for differentiating the RE elements that do not depend on cation size.

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

Generative learning of densities on manifolds

A generative modeling framework is proposed that combines diffusion models and manifold learning to efficiently sample data densities on manifolds. The approach utilizes Diffusion Maps to uncover possible low-dimensional underlying (latent) spaces in the high-dimensional data (ambient) space. Two approaches for sampling from the latent data density are described. The first is a score-based diffusion model, which is trained to map a standard normal distribution to the latent data distribution using a neural network. The second one involves solving an Itô stochastic differential equation in the latent space. Additional realizations of the data are generated by lifting the samples back to the ambient space using Double Diffusion Maps , a recently introduced technique typically employed in studying dynamical system reduction; here the focus lies in sampling densities rather than system dynamics. The proposed approaches enable sampling high dimensional data densities restricted to low-dimensional, a priori unknown manifolds. The efficacy of the proposed framework is demonstrated through a benchmark problem and a material with multiscale structure.

Double diffusion maps↗
DOE OSTI2021

A Tailored Convolutional Neural Network for Nonlinear Manifold Learning of Computational Physics Data Using Unstructured Spatial Discretizations

In this work, we propose a nonlinear manifold learning technique based on deep convolutional autoencoders that is appropriate for model order reduction of physical systems in complex geometries. Convolutional neural networks have proven to be highly advantageous for compressing data arising from systems demonstrating a slow-decaying Kolmogorov n-width. However, these networks are restricted to data on structured meshes. Unstructured meshes are often required for performing analyses of real systems with complex geometry. Our custom graph convolution operators based on the available differential operators for a given spatial discretization effectively extend the application space of deep convolutional autoencoders to systems with arbitrarily complex geometry that are typically discretized using unstructured meshes. We propose sets of convolution operators based on the spatial derivative operators for the underlying spatial discretization, making the method particularly well suited to data arising from the solution of partial differential equations. We demonstrate the method using examples from heat transfer and fluid mechanics and show better than an order of magnitude improvement in accuracy over linear methods.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

An adaptive and stability-promoting layerwise training approach for sparse deep neural network architecture

This work presents a two-stage adaptive framework for progressively developing deep neural network (DNN) architectures that generalize well for a given training data set. In the first stage, a layerwise training approach is adopted where a new layer is added each time and trained independently by freezing parameters in the previous layers. We impose desirable structures on the DNN by employing manifold regularization, sparsity regularization, and physics-informed terms. We introduce a ε – δ – stability-promoting concept as a desirable property for a learning algorithm and show that employing manifold regularization yields a ε – δ stability-promoting algorithm. Further, we also derive the necessary conditions for the trainability of a newly added layer and investigate the training saturation problem. In the second stage of the algorithm (post-processing), a sequence of shallow networks is employed to extract information from the residual produced in the first stage, thereby improving the prediction accuracy. Numerical investigations on prototype regression and classification problems demonstrate that the proposed approach can outperform fully connected DNNs of the same size. Moreover, by equipping the physics-informed neural network (PINN) with the proposed adaptive architecture strategy to solve partial differential equations, we numerically show that adaptive PINNs not only are superior to standard PINNs but also produce interpretable hidden layers with provable stability. As a result, we also apply our architecture design strategy to solve inverse problems governed by elliptic partial differential equations.

42 ENGINEERING↗
DOE OSTI2021

Generalising G 2 geometry: involutivity, moment maps and moduli

We analyse the geometry of generic Minkowski N = 1, D = 4 flux compactifications in string theory, the default backgrounds for string model building. In M-theory they are the natural string theoretic extensions of G2 holonomy manifolds. In type II theories, they extend the notion of Calabi-Yau geometry and include the class of flux backgrounds based on generalised complex structures first considered by Graña et al. (GMPT). Using E 7(7) × $\mathbb{R}$ + generalised geometry we show that these compactifications are characterised by an SU(7) ⊂ E 7(7) structure defining an involutive subbundle of the generalised tangent space, and with a vanishing moment map, corresponding to the action of the diffeomorphism and gauge symmetries of the theory. The Kähler potential on the space of structures defines a natural extension of Hitchin’s G 2 functional. Using this framework we are able to count, for the first time, the massless scalar moduli of GMPT solutions in terms of generalised geometry cohomology groups. It also provides an intriguing new perspective on the existence of G 2 manifolds, suggesting possible connections to Geometrical Invariant Theory and stability.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

Geometry optimization speedup through a geodesic approach to internal coordinates

We present a new geodesic-based method for geometry optimization in a basis set of redundant internal coordinates. Overall, our method updates the molecular geometry by following the geodesic generated by a displacement vector on the internal coordinate manifold, which dramatically reduces the number of steps required to converge to a minimum. Our method can be implemented in any existing optimization code, requiring only implementation of derivatives of the Wilson B-matrix and the ability to numerically solve an ordinary differential equation.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2021

A general framework for substructuring‐based domain decomposition methods for models having nonlocal interactions

Abstract A mathematical framework is provided for a substructuring‐based domain decomposition (DD) approach for nonlocal problems that features interactions between points separated by a finite distance. Here, by substructuring it is meant that a traditional geometric configuration for local partial differential equation (PDE) problems is used in which a computational domain is subdivided into non‐overlapping subdomains. In the nonlocal setting, this approach is substructuring‐based in the sense that those subdomains interact with neighboring domains over interface regions having finite volume, in contrast to the local PDE setting in which interfaces are lower dimensional manifolds separating abutting subdomains. Key results include the equivalence between the global, single‐domain nonlocal problem and its multi‐domain reformulation, both at the continuous and discrete levels. These results provide the rigorous foundation necessary for the development of efficient solution strategies for nonlocal DD methods.

Capodaglio, Giacomo↗
DOE OSTI2022

Approximating the linear response of physical chaos

Abstract Parametric derivatives of statistics are highly desired quantities in prediction, design optimization and uncertainty quantification. In the presence of chaos, the rigorous computation of these quantities is certainly possible, but mathematically complicated and computationally expensive. Based on Ruelle’s formalism, this paper shows that the sophisticated linear response algorithm can be dramatically simplified in higher-dimensional systems featuring statistical homogeneity in the physical space. We argue that the contribution of the SRB (Sinai–Ruelle–Bowen) measure gradient, which is an integral yet the most cumbersome part of the full algorithm, is negligible if the objective function is appropriately aligned with unstable manifolds. This abstract condition could potentially be satisfied by a vast family of real-world chaotic systems, regardless of the physical meaning and mathematical form of the objective function and perturbed parameter. We demonstrate several numerical examples that support these conclusions and that present the use and performance of a simplified linear response algorithm. In the numerical experiments, we consider physical models described by differential equations, including Lorenz 96 and Kuramoto–Sivashinsky.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2026

ssys

`ssys` is a Python toolkit for exact algebraic transformation of ordinary differential equation (ODE) models into canonical S-system or Generalized Mass Action (GMA) form. Given a model in Antimony or SBML format, `ssys` produces a mathematically equivalent representation. The transformation introduces auxiliary variables as needed to decompose a broad class of nonlinearities into products of power-law terms. The recast is exact: the original and transformed systems have identical dynamics on the invariant constraint manifold defined by auxiliary variable definitions, given consistent initial conditions.

Hlavacek, William [Los Alamos National Laboratory]↗
DOE OSTI2024

Machine learning for the identification of phase transitions in interacting agent-based systems: A Desai-Zwanzig example

Deriving closed-form analytical expressions for reduced-order models, and judiciously choosing the closures leading to them, has long been the strategy of choice for studying phase- and noise-induced transitions for agent-based models (ABMs). In this paper, we propose a data-driven framework that pinpoints phase transitions for an ABM—the Desai-Zwanzig model—in its mean-field limit, using a smaller number of variables than traditional closed-form models. To this end, we use the manifold learning algorithm Diffusion Maps to identify a parsimonious set of data-driven latent variables, and we show that they are in one-to-one correspondence with the expected theoretical order parameter of the ABM. We then utilize a deep learning framework to obtain a conformal reparametrization of the data-driven coordinates that facilitates, in our example, the identification of a single parameter-dependent ordinary differential equation (ODE) in these coordinates. Additionally, we identify this ODE through a residual neural network inspired by a numerical integration scheme (forward Euler). We then use the identified ODE—enabled through an odd symmetry transformation—to construct the bifurcation diagram exhibiting the phase transition.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

LaSDI: Parametric Latent Space Dynamics Identification

Enabling fast and accurate physical simulations with data has become an important area of computational physics to aid in inverse problems, design-optimization, uncertainty quantification, and other various decision-making applications. This paper presents a data-driven framework for parametric latent space dynamics identification procedure that enables fast and accurate simulations. The parametric model is achieved by building a set of local latent space model and designing an interaction among them. An individual local latent space dynamics model achieves accurate solution in a trust region. By letting the set of trust region to cover the whole parameter space, our model shows an increase in accuracy with an increase in training data. Herein we introduce two different types of interaction mechanisms, i.e., point-wise and region-based approach. Both linear and nonlinear data compression techniques are used. We illustrate the framework of Latent Space Dynamics Identification (LaSDI) enable a fast and accurate solution process on various partial differential equations, i.e., Burgers’ equations, radial advection problem, and nonlinear heat conduction problem, achieving 0 (100)x speed-up and 0 (1)% relative error with respect to the corresponding full order models.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Time-series forecasting using manifold learning, radial basis function interpolation, and geometric harmonics

We address a three-tier numerical framework based on nonlinear manifold learning for the forecasting of high-dimensional time series, relaxing the “curse of dimensionality” related to the training phase of surrogate/machine learning models. At the first step, we embed the high-dimensional time series into a reduced low-dimensional space using nonlinear manifold learning (local linear embedding and parsimonious diffusion maps). Then, we construct reduced-order surrogate models on the manifold (here, for our illustrations, we used multivariate autoregressive and Gaussian process regression models) to forecast the embedded dynamics. Finally, we solve the pre-image problem, thus lifting the embedded time series back to the original high-dimensional space using radial basis function interpolation and geometric harmonics. The proposed numerical data-driven scheme can also be applied as a reduced-order model procedure for the numerical solution/propagation of the (transient) dynamics of partial differential equations (PDEs). In conclusion, we assess the performance of the proposed scheme via three different families of problems: (a) the forecasting of synthetic time series generated by three simplistic linear and weakly nonlinear stochastic models resembling electroencephalography signals, (b) the prediction/propagation of the solution profiles of a linear parabolic PDE and the Brusselator model (a set of two nonlinear parabolic PDEs), and (c) the forecasting of a real-world data set containing daily time series of ten key foreign exchange rates spanning the time period 3 September 2001–29 October 2020.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2023

Data-driven Whitney forms for structure-preserving control volume analysis

Control volume analysis models physics via the exchange of generalized fluxes between subdomains. Here, we introduce a scientific machine learning framework adopting a partition of unity architecture to identify physically-relevant control volumes, with generalized fluxes between subdomains encoded via Whitney forms. The approach provides a differentiable parameterization of geometry which may be trained in an end-to-end fashion to extract reduced models from full field data while exactly preserving physics. The architecture admits a data-driven finite element exterior calculus allowing discovery of mixed finite element spaces with closed form quadrature rules. An equivalence between Whitney forms and graph networks reveals that the geometric problem of control volume learning is equivalent to an unsupervised graph discovery problem. The framework is developed for manifolds in arbitrary dimension, with examples provided for H(div) problems in $\mathbb{R}$ establishing convergence and structure preservation properties. Finally, we consider a lithium-ion battery problem where we discover a reduced finite element space encoding transport pathways from high-fidelity microstructure resolved simulations. The approach reduces the 5.89M finite element simulation to 136 elements while reproducing pressure to under 0.1% error and preserving conservation.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Decorated TQFTs and their Hilbert spaces

We discuss topological quantum field theories that compute topological invariants which depend on additional structures (or decorations) on three-manifolds. The q-series invariant $\hat{Z}$(q) proposed by Gukov, Pei, Putrov, and Vafa is an example of such an invariant. We describe how to obtain these decorated invariants by cutting and gluing and make a proposal for Hilbert spaces that are assigned to two-dimensional surfaces in the $\hat{Z}$-TQFT.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2024

Short-Wave Infrared Upconverting Nanoparticles

Optical technologies enable real-time, noninvasive analysis of complex systems but are limited to discrete regions of the optical spectrum. While wavelengths in the short-wave infrared (SWIR) window (typically, 1700-3000 nm) should enable deep subsurface penetration and reduced photodamage, there are few luminescent probes that can be excited in this region. Here, we report the discovery of lanthanide-based upconverting nanoparticles (UCNPs) that efficiently convert 1740 or 1950 nm excitation to wavelengths compatible with conventional silicon detectors. Screening of Ln3+ ion combinations by differential rate equation modeling identifies Ho3+/Tm3+ or Tm3+ dopants with strong visible or NIR-I emission following SWIR excitation. Experimental upconverted photoluminescence excitation (U-PLE) spectra find that 10% Tm3+-doped NaYF4 core/shell UCNPs have the strongest 800 nm emission from SWIR wavelengths, while UCNPs with an added 2% or 10% Ho3+ show the strongest red emission when excited at 1740 or 1950 nm. Mechanistic modeling shows that addition of a low percentage of Ho3+ to Tm3+-doped UCNPs shifts their emission from 800 to 652 nm by acting as a hub of efficient SWIR energy acceptance and redistribution up to visible emission manifolds. Parallel experimental and computational analysis shows rate equation models are able to predict compositions for specific wavelengths of both excitation and emission. These SWIR-responsive probes open a new IR bioimaging window, and are responsive at wavelengths important for vision technologies.

Qi, Xiao↗
DOE OSTI2022

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↗
DOE OSTI2025

Inverse Trans Influence and Uranium-Arene σ-Bonding Drive Molecular Geometry: Ligand Modification from Hard to Soft Flips the Oxide from Axial to Equatorial

A rare example of an equatorially bound terminal uranium(V) oxo complex in a chelating sulfur-based ligand environment, namely [(mes( Me,Ad ArS) 3 )U V (O eq )(THF)] (2), is presented. Octahedrally coordinated 2 is obtained by reaction of the mesitylene-anchored tris-thiophenolate-coordinated uranium(III) complex [U III ((SAr Ad,Me ) 3 mes)] (1) with the oxygen-atom transfer reagent N 2 O. The observed, equatorially bound oxo ligand in 2 is in stark contrast to its known tris-aryloxide analog, [(mes( Me,Ad ArO) 3 )U V (O ax )(THF)] (A), where the oxo ligand occupies the typically observed axial coordination site. Complexes 1 and 2 are characterized by single-crystal X-ray diffraction analyses and spectroscopic and magnetochemical methods, including 1 H NMR, UV/vis/NIR electronic absorption, as well as EPR spectroscopy and SQUID magnetometry, thus confirming the C S symmetry and the pentavalent oxidation state of 2. Encompassing quantum chemical calculations (DFT and CASPT2) on 2 and its tris-phenolate analog A, support and rationalize the structural and electronic differences. The molecular orbital pictures show that a stabilizing σ-bonding interaction arising from the U–O eq inverse trans influence (ITI) is present in 2 but missing in A. In 2, the sulfur 3p orbitals are closer in energy to the uranium 5f manifold than the arene π-system, leading to an ITI, while U–arene σ- or δ-bonding is not observed. Although the arene orbitals remain separated from the uranium 5f orbitals in A, the absence of an ITI allows the arene a 2u orbital to engage in a σ-type interaction with the metal. Thus, incorporating a tris-thiophenolate to an arene anchor introduces a new design concept in molecular f-element chemistry. This approach stabilizes an equatorially bound U(V) oxo center, contrasting with its tris-phenolate counterpart, where oxo coordination is axial. The observed geometric divergence, driven by competing ITI and U–arene interactions, not only tunes electronic structure but also leads to differentiated reactivity: only the phenolate analogs activate H 2 O, while the thiolates do not.

Hydrocarbons↗