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DOE OSTI2022

Grassmannian diffusion maps based surrogate modeling via geometric harmonics

Abstract A novel surrogate model based on the Grassmannian diffusion maps (GDMaps) and utilizing geometric harmonics (GH) is developed for predicting the response of complex physical phenomena. The method utilizes GDMaps to obtain a low‐dimensional representation of the underlying behavior of physical/mathematical systems with respect to uncertain input parameters. Using this representation, GH, an out‐of‐sample extension technique, is employed to create a global map from the input parameter space to a Grassmannian diffusion manifold. GH is further employed to locally map points on the diffusion manifold onto the tangent space of a Grassmann manifold. The exponential map is then used to project the points in the tangent space onto the Grassmann manifold, where reconstruction of the full solution is performed. The performance of the proposed surrogate model is verified with three examples. The first problem is a toy example used to illustrate the technique. In the second example, errors associated with the various mappings are assessed by studying response predictions of the electric potential of a dielectric cylinder in a homogeneous electric field. The last example applies the method for uncertainty prediction in the strain field evolution in a model amorphous material using the shear transformation zone theory of plasticity.

42 ENGINEERING↗

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 OSTI2026

Enabling probabilistic learning on manifolds through double diffusion maps

Here, we present a generative learning framework for probabilistic sampling that extends Probabilistic Learning on Manifolds (PLoM), which is designed to generate statistically consistent realizations of a random vector in a finite-dimensional Euclidean space, informed by a (representative) set of observations. In its original form, PLoM constructs a reduced-order probabilistic model by combining three main components: (a) kernel density estimation to approximate the underlying probability measure, (b) Diffusion Maps to characterize the manifold of the data, and (c) a reduced-order Itô Stochastic Differential Equation (ISDE) to sample from the learned distribution. However, its sampling dynamics are posed in the ambient space and the retained number of reduced coordinates is chosen by projection-reconstruction error. In practice, this often (i) requires more coordinates than the data’s intrinsic dimension to achieve stable sampling and (ii) lacks a smooth, basis-independent lifting back to the data domain; moreover, standard Diffusion Maps emphasize harmonic eigenfunctions and can miss non-harmonic latent structure. We address these limitations by decoupling geometry learning from sampling: a first Diffusion Maps pass identifies non-harmonic coordinates on which we formulate a full-order ISDE directly in the latent space, while Double Diffusion Maps captures multiscale geometric features and Geometric Harmonics (GH) learns a smooth lifting map to the ambient variables that is independent of the particular diffusion basis. This hybrid design preserves the system’s dynamical richness with a compact geometric representation and enables principled out-of-sample inference. The effectiveness and robustness of the proposed method are illustrated through two numerical studies: one based on data generated from two-dimensional Hermite polynomial functions and another based on high-fidelity simulations of a detonation wave in a reactive flow.

Double diffusion maps↗

DOE OSTI2023

Machine learning-assisted crystal engineering of a zeolite

Abstract It is shown that Machine Learning (ML) algorithms can usefully capture the effect of crystallization composition and conditions (inputs) on key microstructural characteristics (outputs) of faujasite type zeolites (structure types FAU, EMT, and their intergrowths), which are widely used zeolite catalysts and adsorbents. The utility of ML (in particular, Geometric Harmonics) toward learning input-output relationships of interest is demonstrated, and a comparison with Neural Networks and Gaussian Process Regression, as alternative approaches, is provided. Through ML, synthesis conditions were identified to enhance the Si/Al ratio of high purity FAU zeolite to the hitherto highest level (i.e., Si/Al = 3.5) achieved via direct (not seeded), and organic structure-directing-agent-free synthesis from sodium aluminosilicate sols. The analysis of the ML algorithms’ results offers the insight that reduced Na 2 O content is key to formulating FAU materials with high Si/Al ratio. An acid catalyst prepared by partial ion exchange of the high-Si/Al-ratio FAU (Si/Al = 3.5) exhibits improved proton reactivity (as well as specific activity, per unit mass of catalyst) in propane cracking and dehydrogenation compared to the catalyst prepared from the previously reported highest Si/Al ratio (Si/Al = 2.8).

36 MATERIALS SCIENCE↗

DOE OSTI2024

On learning what to learn: Heterogeneous observations of dynamics and establishing possibly causal relations among them

Abstract Before we attempt to (approximately) learn a function between two sets of observables of a physical process, we must first decide what the inputs and outputs of the desired function are going to be. Here we demonstrate two distinct, data-driven ways of first deciding “the right quantities” to relate through such a function, and then proceeding to learn it. This is accomplished by first processing simultaneous heterogeneous data streams (ensembles of time series) from observations of a physical system: records of multiple observation processes of the system. We determine (i) what subsets of observables are common between the observation processes (and therefore observable from each other, relatable through a function); and (ii) what information is unrelated to these common observables, therefore particular to each observation process, and not contributing to the desired function. Any data-driven technique can subsequently be used to learn the input–output relation—from k-nearest neighbors and Geometric Harmonics to Gaussian Processes and Neural Networks. Two particular “twists” of the approach are discussed. The first has to do with the identifiability of particular quantities of interest from the measurements. We now construct mappings from a single set of observations from one process to entire level sets of measurements of the second process, consistent with this single set. The second attempts to relate our framework to a form of causality: if one of the observation processes measures “now,” while the second observation process measures “in the future,” the function to be learned among what is common across observation processes constitutes a dynamical model for the system evolution.

Sroczynski, David W.↗

DOE OSTI2021

Field Measurement Results of the 15 T Nb 3 Sn Dipole Demonstrator MDPCT1b

A 15 T Nb 3 Sn dipole with cos-theta-type coils was developed to demonstrate a possible magnet design for a post-LHC proton-proton Collider. The magnet has 4-layer coils with 60-mm aperture, and has graded current density between the inner and outer layers to maximize performance. The coils are constrained by vertically-split, thick iron laminations, connected by aluminum Iclamps, and a thick stainless-steel skin. The magnet was previously tested at the Vertical Magnet Test Facility (VMTF) at Fermilab with reduced pre-load, and achieved record 14.1 T fields at maximum current. Now the magnet has been reassembled with full pre-load and has been cryogenically retested at VMTF. We report here on the magnetic measurements results of these tests, including field strength, geometrical harmonics, coil magnetization, and iron saturation characteristics of the demonstrator.

43 PARTICLE ACCELERATORS↗

DOE OSTI2022

Manifold Learning-Based Polynomial Chaos Expansions for High-Dimensional Surrogate Models

In this work we introduce a manifold learning-based method for uncertainty quantification (UQ) in systems describing complex spatiotemporal processes. Our first objective is to identify the embedding of a set of high-dimensional data representing quantities of interest of the computational or analytical model. For this purpose, we employ Grassmannian diffusion maps, a two-step nonlinear dimension reduction technique which allows us to reduce the dimensionality of the data and identify meaningful geometric descriptions in a parsimonious and inexpensive manner. Polynomial chaos expansion is then used to construct a mapping between the stochastic input parameters and the diffusion coordinates of the reduced space. An adaptive clustering technique is proposed to identify an optimal number of clusters of points in the latent space. The similarity of points allows us to construct a number of geometric harmonic emulators which are finally utilized as a set of inexpensive pretrained models to perform an inverse map of realizations of latent features to the ambient space and thus perform accurate out-of-sample predictions. Thus, the proposed method acts as an encoder-decoder system which is able to automatically handle very high-dimensional data while simultaneously operating successfully in the small-data regime. The method is demonstrated on two benchmark problems and on a system of advection-diffusion-reaction equations which model a first-order chemical reaction between two species. In all test cases, the proposed method is able to achieve highly accurate approximations which ultimately lead to the significant acceleration of UQ tasks.

42 ENGINEERING↗

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 OSTI2025

Generative learning for slow manifolds and bifurcation diagrams

In dynamical systems characterized by separation of time scales, the approximation of so called “slow manifolds”, on which the long term dynamics lie, is a useful step for model reduction. Initializing on such slow manifolds is a useful step in modeling, since it circumvents fast transients, and is crucial in multiscale algorithms (like the equation-free approach) alternating between fine scale (fast) and coarser scale (slow) simulations. In a similar spirit, when one studies the infinite time dynamics of systems depending on parameters, the system attractors (e.g., its steady states) lie on bifurcation diagrams (curves for one-parameter continuation, and more generally, on manifolds in state parameter space. Sampling these manifolds gives us representative attractors (here, steady states of ODEs or PDEs) at different parameter values. Algorithms for the systematic construction of these manifolds (slow manifolds, bifurcation diagrams) are required parts of the “traditional” numerical nonlinear dynamics toolkit. In more recent years, as the field of Machine Learning develops, conditional score-based generative models (cSGMs) have been demonstrated to exhibit remarkable capabilities in generating plausible data from target distributions that are conditioned on some given label. It is tempting to exploit such generative models to produce samples of data distributions (points on a slow manifold, steady states on a bifurcation surface) conditioned on (consistent with) some quantity of interest (QoI, observable). In this work, we present a framework for using cSGMs to quickly (a) initialize on a low-dimensional (reduced-order) slow manifold of a multi-time-scale system consistent with desired value(s) of a QoI (a “label”) on the manifold, and (b) approximate steady states in a bifurcation diagram consistent with a (new, out-of-sample) parameter value. This conditional sampling can help uncover the geometry of the reduced slow-manifold and/or approximately “fill in” missing segments of steady states in a bifurcation diagram. Finally, the quantity of interest, which determines how the sampling is conditioned, is either known a priori or identified using manifold learning-based dimensionality reduction techniques applied to the training data.

Dynamical systems↗

DOE OSTI2026

Nonlocal Metasurface with Resonance-Selective Nonlinear Wavefront Shaping

We demonstrate a nonlinear nonlocal metasurface supporting quasi-trapped modes that enable helicity-dependent wavefront shaping at third-harmonic wavelengths. The geometric phase is selectively modified near resonance, revealing a mechanism for polarization-dependent nonlinear phase control.

Sedeh, Hooman [Duke University]↗

DOE OSTI2026

Interpretation of high-harmonic fast-wave propagation in the scrape-off layer of NSTX-U as a geometrically bounded-waveguide mode

This paper revisits the onset of high-harmonic fast-wave (HHFW) propagation in the scrape-off layer (SOL) plasma of the NSTX/NSTX-U spherical tokamak, motivated by past HHFW heating and current drive experiments and modelling. Previously, the fast-wave propagation in the SOL was correlated with the opening (suppression) of the fast-wave right-hand cutoff layer in front of the antenna. In this work, the SOL propagation is interpreted as a geometric waveguide mode guided by the SOL geometry, whose radial width can be comparable to the wave’s perpendicular wavelength in the HHFW regime. A two-dimensional circular model is first employed using full-wave solvers to characterise the poloidal eigenmode structures and to clarify their relationship to annulus resonance. By progressively adding a tokamak-like magnetic-field configuration, starting from a uniform axial field, the effects of magnetic-field gradients and pitch on the SOL waveguide mode are characterised. The poloidal mode numbers supported by the SOL plasma agree well with analytic estimates, indicating that an anisotropic plasma in a bounded geometry selectively supports and amplifies the resonant poloidal mode number. Additionally, two-dimensional axisymmetric NSTX-U simulations demonstrate that the SOL eigenmode features identified in the circular model persist in the experimentally relevant configurations. A control approach based on lengthening the wave perpendicular wavelength relative to the SOL width, including a higher magnetic (B)-field operation, is discussed. The analysis here shows the key role of a bounded geometry in interpreting HHFW eigenmode coupling and propagation in the NSTX-U SOL plasma.

plasma heating↗

DOE OSTI2026

A tutorial on high-order harmonic generation in atoms, molecules, and condensed matter

This tutorial introduces strong-field-driven high-order harmonics, their experimental generation and characterization techniques, and their main applications including attosecond pulse generation and ultrafast spectroscopy of the target material. We begin from the use of atomic targets, where the first high-order harmonic generation (HHG) experiments were realized in the late 1980s. Then, we briefly discuss the basics of the microscopic generation mechanism and how various steps of the mechanism were exploited in applications such as generating isolated attosecond pulses and probing molecular orbitals. We introduce and describe the standard experimental approaches for condensed phase HHG, where we discuss unique technical challenges of the use of solid-state materials, such as the mitigation of plasma formation and laser damage. We cover the fundamentals of high-harmonic spectroscopy in condensed matter systems, such as wide bandgap dielectrics, semiconductors, liquid media, and 2D-crystals. We provide some examples of rapidly emerging spectroscopic capabilities, such as for probing crystal symmetries, Berry phases, and associated non-trivial topological properties of the source material. Finally, we provide an overview of the research field, including some of the challenges, opportunities, and open questions.

Attosecond pulses↗

DOE OSTI2022

Causality and dimensionality in geometric scattering

We report the scattering matrix which describes low-energy, non-relativistic scattering of spin-1/2 fermions interacting via finite-range potentials can be obtained from a geometric action principle in which space and time do not appear explicitly. In the case of zero-range forces, causality leads to constraints on scattering trajectories in the geometric picture. The effect of spatial dimensionality is also investigated by considering scattering in two and three dimensions. In the geometric formulation it is found that dimensionality is encoded in the phase of the harmonic potential that appears in the geometric action.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2021

Equivalence of a harmonic oscillator to a free particle and Eisenhart lift

Highlights: • Equivalence of a harmonic oscillator to a free particle is considered. • The Eisenhart lift is applied to most general harmonic and linear potentials. • In geometric language, equivalence is the result of the conformal flatness of the Eisenhart metric. It is widely known in quantum mechanics that solutions of the Schrödinger equation (SE) for a linear potential are in one-to-one correspondence with the solutions of the free SE. The physical reason for this correspondence is Einstein’s principle of equivalence. What is usually not so widely known is that solutions of the Schrödinger equation with harmonic potential can also be mapped to the solutions of the free Schrödinger equation. The physical understanding of this equivalence is not known as precisely as in the case of the equivalence principle. We present a geometric picture that will link both of the above equivalences with one constraint on the Eisenhart metric.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2022

Binding kinetics of harmonically confined random walkers

Diffusion-mediated binding of molecules under the influence of discrete spatially confining potentials is a commonly encountered scenario in systems subjected to explicit fields or implicit fields arising from tethering restraints. Here, in this work, we derive analytical expressions for the mean binding time of two random walkers geometrically confined by means of two harmonic potentials in one- and two-dimensional systems, which show excellent agreement with Brownian dynamics simulations. As a demonstration of its utility, we use this theory to maximize the communication speed in existing DNA walkers, obtaining quantitative agreement with previously reported experimental findings. The analytical expressions derived in this paper are broadly applicable to diverse systems, providing ways to characterize communication processes and optimize the rate of signal propagation for sensing and computing applications at the nanoscale.

36 MATERIALS SCIENCE↗

DOE OSTI2026

Polarization Control via Artificial Optical Nonlinearity in Dielectric Metasurfaces

Nonlinear optical phenomena are generally governed by geometry in matter systems, as they depend on the spatial arrangement of atoms within materials or molecules. Metasurfaces, through precisely designed geometries on a subwavelength scale, allow the optical response of a material to be tailored far beyond its natural properties. Therefore, metasurfaces are highly appealing for enabling the engineering of nonlinear optical interactions. Current studies of nonlinear metasurfaces predominantly focus on the phase control of the generated light. Nonetheless, investigating the tensorial nature of the nonlinearity of metasurfaces and its effect on the polarization of the generated light is critical to fully unlocking a range of applications, such as nonlinear vector beam generation and nonlinear polarization imaging. Here, we study the artificial optical nonlinearity of a dielectric metasurface originating from its meta-atom symmetry and describe the third-order nonlinear behavior by considering the polarization degree of freedom. We establish an effective nonlinear medium model that serves as a design toolbox for developing amorphous silicon-based geometric metasurfaces with customizable features for third-harmonic generation. We further extract quantitative values of the artificial nonlinear susceptibility tensor elements related to the investigated nonlinear process and geometry. The implemented functional devices demonstrate the versatility of dielectric metasurfaces in shaping the emitted light in terms of amplitude, phase, and polarization for the precise engineering of advanced nonlinear architectures targeting applications in nonlinear imaging and complex light generation.

Diffraction↗

DOE OSTI2024

Ab initio exploration of low–lying electronic states of linear and bent MNX + (M = Ca, Sr, Ba, Ra; X = O, S, Se, Te, Po) and their origins

High-level multireference and coupled cluster quantum calculations were employed to analyze low-lying electronic states of linear-MNX + and side-bonded-M[NX] + (M = Ca, Sr, Ba, Ra; X = O, S, Se, Te, Po) species. Their full potential energy curves (PECs), dissociation energies (D e s), geometric parameters, excitation energies (T e s), and harmonic vibrational frequencies (ω e s) are reported. The first three chemically bound electronic states of MNX + and M[NX] + are 3 Σ – , 1 Δ, 1 Σ + and 3 A", 1 A', 1 A", respectively. The 3 Σ – , 1 Δ, 1 Σ + of MNX + originate from the M + ( 2 D) + NX( 2 Π) fragments, whereas the 3 A", 1 A', 1A" states of M[NX] + dissociate to M + ( 2 S) + NX( 2 Π) as a result of avoided crossings. The MNX + and M[NX] + are real minima on the potential energy surface and their interconversions are possible. The M 2+ NX – /M 2+ [NX] – ionic structure is an accurate representation for their low-lying electronic states. The D e s of MNX + species were found to depend on the dipole moment (μ) of the corresponding NX ligands and a linear relationship between these two parameters was observed.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2025

Nonlinear van der Waals Metasurfaces with Resonantly Enhanced Light Generation

Efficient nonlinear wave mixing is of paramount importance for a wide range of applications. However, weak optical nonlinearities pose significant challenges for accessing nonlinear light–matter interaction in compact systems. Here, we experimentally study second harmonic generation in deeply subwavelength 3R-MoS 2 metasurfaces (<λ/13 thick). Our measurements, supported by theoretical analysis, reveal a complex interplay and coupling between geometric resonances, optical extinction, and exciton-driven strong nonlinear susceptibility dispersion. We further demonstrate >150-fold enhancement in second harmonic signal at 740 nm mediated by the A exciton resonance. Additionally, our theoretical studies predict an enhancement of more than 10 6 in second harmonic generation in <100 nm thick structures exhibiting bound states in the continuum resonance. These findings provide insight into accessing and harnessing the unprecedented 3R-MoS 2 nonlinearities at a subwavelength scale, paving the way to ultracompact nonlinear photonic devices.

77 NANOSCIENCE AND NANOTECHNOLOGY↗