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At least 73 records · Page 4
All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

Heterogeneous Mixtures of Dictionary Functions to Approximate Subspace Invariance in Koopman Operators: Why Deep Koopman Operators Work

Abstract Koopman operators model nonlinear dynamics as a linear dynamic system acting on a nonlinear function as the state. This nonstandard state is often called a Koopman observable and is usually approximated numerically by a superposition of functions drawn from a dictionary . In a widely used algorithm, extended dynamic mode decomposition (EDMD), the dictionary functions are drawn from a fixed class of functions. Deep learning combined with EDMD has been used to learn novel dictionary functions in an algorithm called deep dynamic mode decomposition (deepDMD). The learned representation both (1) accurately models and (2) scales well with the dimension of the original nonlinear system. In this paper, we analyze the learned dictionaries from deepDMD and explore the theoretical basis for their strong performance. We explore State-Inclusive Logistic Lifting (SILL) dictionary functions to approximate Koopman observables. Error analysis of these dictionary functions show they satisfy a property of subspace approximation, which we define as uniform finite approximate closure. Typically, a Koopman dictionary’s nonlinear functions are homogeneous. In this paper, we discover that structured mixing of heterogeneous dictionary functions drawn from different classes of nonlinear functions achieve the same accuracy and dimensional scaling as the deep-learning-based deepDMD algorithm Yeung et al. ( In: 2019 American Control Conference (ACC), 2019). We specifically show this by building a heterogeneous dictionary comprised of SILL functions and conjunctive radial basis functions (RBFs). This mixed dictionary achieves similar accuracy and dimensional scaling to deepDMD with an order of magnitude reduction in parameters, while maintaining geometric interpretability. These results strengthen the viability of dictionary-based Koopman models to solving high-dimensional nonlinear learning problems.

Johnson, Charles A.↗
DOE OSTI2020

Hole–hole Tamm–Dancoff-approximated density functional theory: A highly efficient electronic structure method incorporating dynamic and static correlation

The study of photochemical reaction dynamics requires accurate as well as computationally efficient electronic structure methods for the ground and excited states. While time-dependent density functional theory (TDDFT) is not able to capture static correlation, complete active space self-consistent field methods neglect much of the dynamic correlation. Hence, inexpensive methods that encompass both static and dynamic electron correlation effects are of high interest. Here, we revisit hole–hole Tamm–Dancoff approximated (hh-TDA) density functional theory for this purpose. The hh-TDA method is the hole–hole counterpart to the more established particle–particle TDA (pp-TDA) method, both of which are derived from the particle–particle random phase approximation (pp-RPA). In hh-TDA, the N-electron electronic states are obtained through double annihilations starting from a doubly anionic (N+2 electron) reference state. In this way, hh-TDA treats ground and excited states on equal footing, thus allowing for conical intersections to be correctly described. Furthermore, the treatment of dynamic correlation is introduced through the use of commonly employed density functional approximations to the exchange-correlation potential. Additionally, we show that hh-TDA is a promising candidate to efficiently treat the photochemistry of organic and biochemical systems that involve several low-lying excited states—particularly those with both low-lying ππ* and nπ* states where inclusion of dynamic correlation is essential to describe the relative energetics. In contrast to the existing literature on pp-TDA and pp-RPA, we employ a functional-dependent choice for the response kernel in pp- and hh-TDA, which closely resembles the response kernels occurring in linear response and collinear spin-flip TDDFT.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2020

DFTB+, a software package for efficient approximate density functional theory based atomistic simulations

DFTB+ is a versatile community developed open source software package offering fast and efficient methods for carrying out atomistic quantum mechanical simulations. By implementing various methods approximating density functional theory (DFT), such as the density functional based tight binding (DFTB) and the extended tight binding method, it enables simulations of large systems and long timescales with reasonable accuracy while being considerably faster for typical simulations than the respective ab initio methods. Based on the DFTB framework, it additionally offers approximated versions of various DFT extensions including hybrid functionals, time dependent formalism for treating excited systems, electron transport using non-equilibrium Green’s functions, and many more. DFTB+ can be used as a user-friendly standalone application in addition to being embedded into other software packages as a library or acting as a calculation-server accessed by socket communication. We give an overview of the recently developed capabilities of the DFTB+ code, demonstrating with a few use case examples, discuss the strengths and weaknesses of the various features, and also discuss on-going developments and possible future perspectives.

74 ATOMIC AND MOLECULAR PHYSICS↗
DOE OSTI2016

Push it to the limit: Characterizing the convergence of common sequences of basis sets for intermolecular interactions as described by density functional theory

With the aim of systematically characterizing the convergence of common families of basis sets such that general recommendations for basis sets can be made, we have tested a wide variety of basis sets against complete-basis binding energies across the S22 set of intermolecular interactions—noncovalent interactions of small and medium-sized molecules consisting of first- and second-row atoms—with three distinct density functional approximations: SPW92, a form of local-density approximation; B3LYP, a global hybrid generalized gradient approximation; and B97M-V, a meta-generalized gradient approximation with nonlocal correlation. We have found that it is remarkably difficult to reach the basis set limit; for the methods and systems examined, the most complete basis is Jensen’s pc-4. The Dunning correlation-consistent sequence of basis sets converges slowly relative to the Jensen sequence. The Karlsruhe basis sets are quite cost effective, particularly when a correction for basis set superposition error is applied: counterpoise-corrected def2-SVPD binding energies are better than corresponding energies computed in comparably sized Dunning and Jensen bases, and on par with uncorrected results in basis sets 3-4 times larger. These trends are exhibited regardless of the level of density functional approximation employed. A sense of the magnitude of the intrinsic incompleteness error of each basis set not only provides a foundation for guiding basis set choice in future studies but also facilitates quantitative comparison of existing studies on similar types of systems.

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

Multi-fidelity Bayesian neural networks: Algorithms and applications

Here we propose a new class of Bayesian neural networks (BNNs) that can be trained using noisy data of variable fidelity, and we apply them to learn function approximations as well as to solve inverse problems based on partial differential equations (PDEs). These multi-fidelity BNNs consist of three neural networks: The first is a fully connected neural network, which is trained following the maximum a posteriori probability (MAP) method to fit the low-fidelity data; the second is a Bayesian neural network employed to capture the cross-correlation with uncertainty quantification between the low- and high-fidelity data; and the last one is the physics-informed neural network, which encodes the physical laws described by PDEs. For the training of the last two neural networks, we first employ the mean-field variational inference (VI) to maximize the evidence lower bound (ELBO) to obtain informative prior distributions for the hyperparameters in the BNNs, and subsequently we use the Hamiltonian Monte Carlo (HMC) method to estimate accurately the posterior distributions for the corresponding hyperparameters. We demonstrate the accuracy of the present method using synthetic data as well as real measurements. Specifically, we first approximate a one- and four-dimensional function, and then infer the reaction rates in one- and two-dimensional diffusion-reaction systems. Moreover, we infer the sea surface temperature (SST) in the Massachusetts and Cape Cod Bays using satellite images and in-situ measurements. Taken together, our results demonstrate that the present method can capture both linear and nonlinear correlation between the low- and high-fidelity data adaptively, identify unknown parameters in PDEs, and quantify uncertainties in predictions, given a few scattered noisy high-fidelity data. Finally, we demonstrate that we can effectively and efficiently reduce the uncertainties and hence enhance the prediction accuracy with an active learning approach, using as examples a specific one-dimensional function approximation and an inverse PDE problem.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Exact Gaussian processes for massive datasets via non-stationary sparsity-discovering kernels

Abstract A Gaussian Process (GP) is a prominent mathematical framework for stochastic function approximation in science and engineering applications. Its success is largely attributed to the GP’s analytical tractability, robustness, and natural inclusion of uncertainty quantification. Unfortunately, the use of exact GPs is prohibitively expensive for large datasets due to their unfavorable numerical complexity of $$O(N^3)$$ O ( N 3 ) in computation and $$O(N^2)$$ O ( N 2 ) in storage. All existing methods addressing this issue utilize some form of approximation—usually considering subsets of the full dataset or finding representative pseudo-points that render the covariance matrix well-structured and sparse. These approximate methods can lead to inaccuracies in function approximations and often limit the user’s flexibility in designing expressive kernels. Instead of inducing sparsity via data-point geometry and structure, we propose to take advantage of naturally-occurring sparsity by allowing the kernel to discover—instead of induce—sparse structure. The premise of this paper is that the data sets and physical processes modeled by GPs often exhibit natural or implicit sparsities, but commonly-used kernels do not allow us to exploit such sparsity. The core concept of exact, and at the same time sparse GPs relies on kernel definitions that provide enough flexibility to learn and encode not only non-zero but also zero covariances. This principle of ultra-flexible, compactly-supported, and non-stationary kernels, combined with HPC and constrained optimization, lets us scale exact GPs well beyond 5 million data points.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Mechanical properties of zeolite-templated carbons from approximate density functional theory calculations

Zeolite-templated carbon (ZTC) is a unique porous carbonaceous material whose structure is ordered at the nanometre scale, enabling a representative periodic description at the atomistic level. Utilizing an existing, well-defined reference model for ZTCs, a structural library of varying compositions was developed by refinement using density-functional tight-binding (DFTB) potentials parameterized for materials science applications. We first determined the quantum chemical-refined structures of models with CH, CHO, CHON, CHOB, and CHOBN compositions with various degrees of heteroatom substitution. These structural models comprise the characteristic morphological features of highly porous carbon materials, such as open-blade surfaces, edges, saddles, and closed-strut formations, spanning a range of curvatures and characteristic sizes. Second, we carried out alternating compression and expansion of the CHO model unit cell to determine the lowest energy structure as well as to obtain its bulk modulus in order to demonstrate a close connection between macroscopic observations and atomic-scale structures. Further, the agreement between experimental measurements and the computational model is remarkable and demonstrates the power of approximate density functional theory as a cost-effective computational tool with chemical accuracy for the investigation of structure/property relationships in real-world carbon-based solids.

03 NATURAL GAS↗
DOE OSTI2016

mBEEF-vdW: Robust fitting of error estimation density functionals

Here, we propose a general-purpose semilocal/nonlocal exchange-correlation functional approximation, named mBEEF-vdW. The exchange is a meta generalized gradient approximation, and the correlation is a semilocal and nonlocal mixture, with the Rutgers-Chalmers approximation for van der Waals (vdW) forces. The functional is fitted within the Bayesian error estimation functional (BEEF) framework. We improve the previously used fitting procedures by introducing a robust MM-estimator based loss function, reducing the sensitivity to outliers in the datasets. To more reliably determine the optimal model complexity, we furthermore introduce a generalization of the bootstrap 0.632 estimator with hierarchical bootstrap sampling and geometric mean estimator over the training datasets. Using this estimator, we show that the robust loss function leads to a 10% improvement in the estimated prediction error over the previously used least-squares loss function. The mBEEF-vdW functional is benchmarked against popular density functional approximations over a wide range of datasets relevant for heterogeneous catalysis, including datasets that were not used for its training. Overall, we find that mBEEF-vdW has a higher general accuracy than competing popular functionals, and it is one of the best performing functionals on chemisorption systems, surface energies, lattice constants, and dispersion. We also show the potential-energy curve of graphene on the nickel(111) surface, where mBEEF-vdW matches the experimental binding length. mBEEF-vdW is currently available in gpaw and other density functional theory codes through Libxc, version 3.0.0.

36 MATERIALS SCIENCE↗
DOE OSTI2025

Massively parallel and universal approximation of nonlinear functions using diffractive processors

Nonlinear computation is essential for a wide range of information processing tasks, yet implementing nonlinear functions using optical systems remains a challenge due to the weak and power-intensive nature of optical nonlinearities. Overcoming this limitation without relying on nonlinear optical materials could unlock unprecedented opportunities for ultrafast and parallel optical computing systems. Here, we demonstrate that large-scale nonlinear computation can be performed using linear optics through optimized diffractive processors composed of passive phase-only surfaces. In this framework, the input variables of nonlinear functions are encoded into the phase of an optical wavefront—e.g., via a spatial light modulator (SLM)—and transformed by an optimized diffractive structure with spatially varying point-spread functions to yield output intensities that approximate a large set of unique nonlinear functions–all in parallel. We provide proof establishing that this architecture serves as a universal function approximator for an arbitrary set of bandlimited nonlinear functions, also covering wavelength-multiplexed nonlinear functions as well as multi-variate and complex-valued functions that are all-optically cascadable. Our analysis also indicates the successful approximation of typical nonlinear activation functions commonly used in neural networks, including the sigmoid, tanh, ReLU (rectified linear unit), and softplus. We numerically demonstrate the parallel computation of one million distinct nonlinear functions, accurately executed at wavelength-scale spatial density at the output of a diffractive optical processor. Furthermore, we experimentally validated this framework using in situ optical learning and approximated 35 unique nonlinear functions in a single shot using a compact setup consisting of an SLM and an image sensor. These results establish diffractive optical processors as a scalable platform for massively parallel universal nonlinear function approximation, paving the way for new capabilities in analog optical computing based on linear materials.

Rahman, Md Sadman Sakib [University of California,↗
DOE OSTI2014

Kinetic and electron-electron energies for convex sums of ground state densities with degeneracies and fractional electron number

Properties of exact density functionals provide useful constraints for the development of new approximate functionals. This paper focuses on convex sums of ground-level densities. It is observed that the electronic kinetic energy of a convex sum of degenerate ground-level densities is equal to the convex sum of the kinetic energies of the individual degenerate densities. (The same type of relationship holds also for the electron-electron repulsion energy.) This extends a known property of the Levy-Valone Ensemble Constrained-Search and the Lieb Legendre-Transform refomulations of the Hohenberg-Kohn functional to the individual components of the functional. Moreover, we observe that the kinetic and electron-repulsion results also apply to densities with fractional electron number (even if there are no degeneracies), and we close with an analogous point-wise property involving the external potential. Examples where different degenerate states have different kinetic energy and electron-nuclear attraction energy are given; consequently, individual components of the ground state electronic energy can change abruptly when the molecular geometry changes. These discontinuities are predicted to be ubiquitous at conical intersections, complicating the development of universally applicable density-functional approximations.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2019

Approximate Green’s Function Coupled Cluster A Method Employing Effective Dimension Reduction

The Green’s function coupled cluster (GFCC) method, originally proposed in the early 1990s, is a powerful many-body tool for computing and analyzing the electronic structures of molecular structures and periodic systems, especially when electrons are strongly correlated. However, in order for the GFCC method to become an approach that can be routinely used in the electronic structure calculations, more robust linear algebraic techniques and approximations need to be employed to reduce its extremely high computational overhead. In our recent studies, we showed that the GFCC equation can be numerically solved by seeking solutions of multiple linear systems in the frequency regime of interest, which can be easily distributed in a massively parallel environment, and broaden the avenue for more state-of-the-art numerical linear algebraic tools associated with multiple linear systems to be kicked in to this field. In the present pursuit of this practice, we show a successful application of model order reduction (MOR) techniques in the GFCC framework. Briefly speaking, for a frequency regime of interest that requires a high resolution of spectral function, instead of solving GFCC linear equation in the original dimension for every single frequency point, an easily-solvable linear system model with reduced dimension can be built upon projecting the original GFCC linear system into a subspace, from which the interpolation and extrapolation can give a reasonable approximation to the results of the original GFCC linear equations. Here, we show that the subspace can be properly constructed in an iterative manner from the auxiliary vectors of the GFCC linear equations at some select frequencies inside the frequency regime. During the iterations, the quality of the subspace, as well as the linear system model, can be systematically improved. The approach is tested in this work in terms of the efficiency and accuracy of computing spectral functions for some typical molecular systems including carbon monoxide, 1,3-butadiene, benzene, and adenine molecules. As a byproduct, the high quality linear system model is also found to be able to provide high quality initial guess for the existing linear solver to improve the convergence performance.

Peng, Bo↗
DOE OSTI2024

A numerical Poisson solver with improved radial solutions for a self-consistent locally scaled self-interaction correction method

Abstract The universal applicability of density functional approximations is limited by self-interaction error made by these functionals. Recently, a novel one-electron self-interaction-correction (SIC) method that uses an iso-orbital indicator to apply the SIC at each point in space by scaling the exchange-correlation and Coulomb energy densities was proposed. The locally scaled SIC (LSIC) method is exact for the one-electron densities, and unlike the well-known Perdew–Zunger SIC (PZSIC) method recovers the uniform electron gas limit of the uncorrected density functional approximation, and reduces to PZSIC method as a special case when isoorbital indicator is set to the unity. Here, we present a numerical scheme that we have adopted to evaluate the Coulomb potential of the electron density scaled by the iso-orbital indicator required for the self-consistent LSIC calculations. After analyzing the behavior of the finite difference method (FDM) and the green function solution to the radial part of the Poisson equation, we adopt a hybrid approach that uses the FDM for the Coulomb potential due to the monopole and the GF for all higher-order terms. The performance of the resultant hybrid method is assessed using a variety of systems. The results show improved accuracy than earlier numerical schemes. We also find that, even with a generic set of radial grid parameters, accurate energy differences can be obtained using a numerical Coulomb solver in standard density functional studies.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2025

Excited-State Densities from Time-Dependent Density Functional Response Theory

While the variational principle for excited-state energies leads to a route to obtaining excited-state densities from time-dependent density functional theory, relatively little attention has been paid to the quality of the resulting densities in real space obtained with different exchange-correlation functional approximations or how nonadiabatic approximations developed for energies of states of double-excitation character perform for their densities. Here we derive an expression directly in real space for the excited-state density, which includes the case of nonadiabatic kernels and consequently is able, for the first time, to yield densities of states of double-excitation character. Under some well-defined simplifications, we compare the performance of the local-density approximation and exact-exchange approximation, which are in a sense at the opposite extremes of the fundamental functional approximations, on local and charge-transfer excitations in one-dimensional model systems and show that the dressed Time-Dependent Density Functional Theory (TDDFT) approach gives good densities of double excitations.

approximation↗
DOE OSTI2024

Frontiers in Magnetic Materials

Magnetism is crucial to many modern technologies, a driver for condensed matter physics research and one of the most remarkable and diverse properties of matter. We propose to develop understanding of novel magnetism and magnetic related behavior in materials and use this to accelerate the discovery of forefront magnetic materials. The approach is via the connection of magnetic properties to specific structures and materials. Topics that will be addressed are (1) Metallic magnetic materials with unusually low carrier concentrations and/or moments (2) Magnetism arising from unusual chemistry including 4d and 5d magnetism and (3) Materials with strong spin-fluctuations, which can lead to quantum criticality, spin-fluctuation induced superconductivity and other novel quantum behavior. These topics overlap, for example, the 4d ruthenates include ferromagnets (perovskite SrRuO 3 ), extremely high ordering temperature antiferromagnets (honeycomb lattice SrRu 2 O 6 ) and well as quantum materials with strong spin fluctuations (layered perovskite Sr 2 RuO 4 and Sr 3 Ru 2 O 7 ). We will use of density calculations to connect magnetic properties with chemistry and structure and employ phenomenological theories to extend these results to properties that are not directly given by direct first principles methods and we will conduct tests to explore the limitations of density functional approximations and new functionals.

36 MATERIALS SCIENCE↗
DOE OSTI2016

Cosmic reionization on computers: the faint end of the galaxy luminosity function

Using numerical cosmological simulations completed under the “Cosmic Reionization On Computers” project, I explore theoretical predictions for the faint end of the galaxy UV luminosity functions at z≳6. A commonly used Schechter function approximation with the magnitude cut at M{sub cut}∼−13 provides a reasonable fit to the actual luminosity function of simulated galaxies. When the Schechter functional form is forced on the luminosity functions from the simulations, the magnitude cut M{sub cut} is found to vary between −12 and −14 with a mild redshift dependence. An analytical model of reionization from Madau et al., as used by Robertson et al., provides a good description of the simulated results, which can be improved even further by adding two physically motivated modifications to the original Madau et al. equation.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2018

Sierra/RKPM: Current State and Applications

The Reproducing Kernel Particle Method (RKPM), a meshfree method, has been implemented in Sandia's Sierra/Solid Mechanics in a collaboration between Sandia and the University of California San Diego's Center for Extreme Events Research (UCSD/CEER). Meshfree methods, like RKPM, have an advantage over mesh-based methods, like the Finite Element Method (FEM), in applications where achieving or maintaining a quality mesh becomes burdensome or impractical. For example, using FEM for problems with very large deformations will result in poor element Jacobians which causes problems with the parametric mapping. RKPM constructs the approximation functions in the physical domain, circumventing the parametric mapping issue. Also, reconstructing the approximation functions at very large deformations is straight-forward. RKPM has an advantage over traditional meshfree methods such as Smoothed-Particle Hydrodynamics (SPH) due to its ability to reproduce linear or higher-order functions exactly. This removes the tensile instabilities that are present in SPH and allows preservation of angular momentum. The point of this memo is to explore the capabilities and limitations of the current implementation by testing it on three different applications: 1) a quasi-static ductile shearing problem 2) a dynamic concrete panel perforation problem and 3) a set of dynamic metal panel perforation problems. In summary, areas where RKPM appears to be a promising alternative to current methods have been identified. Also, outstanding inefficiencies and issues (bugs) with code are noted, ways to improve the capabilities using material from literature are mentioned and areas deserving of new research are highlighted.

42 ENGINEERING↗
DOE OSTI2015

Generalized gradient approximation exchange energy functional with correct asymptotic behavior of the corresponding potential

A new non-empirical exchange energy functional of the generalized gradient approximation (GGA) type, which gives an exchange potential with the correct asymptotic behavior, is developed and explored. In combination with the Perdew-Burke-Ernzerhof (PBE) correlation energy functional, the new CAP-PBE (CAP stands for correct asymptotic potential) exchange-correlation functional gives heats of formation, ionization potentials, electron affinities, proton affinities, binding energies of weakly interacting systems, barrier heights for hydrogen and non-hydrogen transfer reactions, bond distances, and harmonic frequencies on standard test sets that are fully competitive with those obtained from other GGA-type functionals that do not have the correct asymptotic exchange potential behavior. Distinct from them, the new functional provides important improvements in quantities dependent upon response functions, e.g., static and dynamic polarizabilities and hyperpolarizabilities. CAP combined with the Lee-Yang-Parr correlation functional gives roughly equivalent results. Consideration of the computed dynamical polarizabilities in the context of the broad spectrum of other properties considered tips the balance to the non-empirical CAP-PBE combination. Intriguingly, these improvements arise primarily from improvements in the highest occupied and lowest unoccupied molecular orbitals, and not from shifts in the associated eigenvalues. Those eigenvalues do not change dramatically with respect to eigenvalues from other GGA-type functionals that do not provide the correct asymptotic behavior of the potential. Unexpected behavior of the potential at intermediate distances from the nucleus explains this unexpected result and indicates a clear route for improvement.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗