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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 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 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 OSTI2026

Toward Chemical Accuracy for Chemi- and Physisorption with an Efficient Density Functional

Understanding molecular adsorption on surfaces underpins many problems in chemistry and materials science. Accurately and efficiently describing the adsorption has been a challenging task for first-principles methods as the process can involve both short-range chemical bond formations and long-range physical interactions, e.g., van der Waals (vdW) interaction. Density functional theory presents an appealing choice for modeling adsorption reactions, although calculations with many exchange-correlation density functional approximations struggle to accurately describe both chemical and physical molecular adsorptions. Here, we propose an efficient density functional approximation that is accurate for both chemical and physical adsorption by concurrently optimizing its semilocal component and the long-range vdW correction against the prototypical adsorption CO/Pt(111) and Ar 2 binding energy curve. The resulting function opens the door to accurate and efficient modeling of general molecular adsorption.

adsorption↗
DOE OSTI2025

Coherency-Constrained Spectral Clustering for Power Network Reduction

This paper presents a methodology for reducing the complexity of large-scale power network models using spectral clustering, aggregation of electrical components, and cost function approximation. Two approaches are explored using unconstrained and constrained spectral clustering to determine areas for effective system reduction. Once the system areas are determined, both loads and generators by type are aggregated, and their new cost function is approximated through polynomial curve-fitting or statistical methods. The performance of reduced networks is evaluated in terms of their ability to follow the true daily cost of the original system over a 24-hour period considering a set of several days. Two test systems are taken as test beds. Application of the methodology to a modified version of the IEEE 39-bus system reduces it from 17 generators to a 4-bus system and 9 generators with about 93% of accuracy. Similarly, the IEEE 118-bus system is reduced from 19 generators to a 3-bus system with three aggregated units achieving over 99% of accuracy. These findings address scalability challenges and enhance accuracy for high and mid-loading level conditions, and by aggregating thermal units with similar cost functions.

42 ENGINEERING↗
DOE OSTI2026

Selecting Critical Scenarios of DER Adoption in Distribution Grids Using Bayesian Optimization

We develop a new methodology to select scenarios of DER adoption most critical for distribution grids. Anticipating risks of future voltage and line flow violations due to additional PV adopters is central for utility investment planning but continues to rely on deterministic or ad hoc scenario selection. We propose a highly efficient search framework based on multi-objective Bayesian Optimization. We treat underlying grid stress metrics as computationally expensive black-box functions, approximated via Gaussian Process surrogates and design an acquisition function based on probability of scenarios being Pareto-critical across a collection of line- and bus-based violation objectives. Our approach provides a statistical guarantee and offers an order of magnitude speed-up relative to a conservative exhaustive search. Case studies on realistic feeders with 200-400 buses demonstrate the effectiveness and accuracy of our approach.

Mulkin, Olivier↗
DOE OSTI2025

Free-energy orbital-free density functional theory: recent developments, perspective, and outlook

By summarizing the constraint-based development of orbital-free free-energy density functional approximations, we provide a perspective on progress over the last 15 years, the limitations of existing functionals, and the challenges awaiting resolution. We outline the chronology of the development of noninteracting and exchange-correlation free-energy orbital-free functionals and summarize the theoretical basis of existing local density approximation, second-order approximation, generalized gradient approximation (GGA), and meta-GGAs. We discuss limitations and challenges such as problems with thermodynamic derivatives, free-energy nonadditivity and the closely related issue of all-electron versus valence-only local pseudo-potential performance.

74 ATOMIC AND MOLECULAR PHYSICS↗
DOE OSTI2025

Untangling Sources of Error in the Density-Functional Many-Body Expansion

The many-body expansion provides a framework for data-driven applications of electronic structure theory, including parametrization of classical force fields and machine learning. In this article, we demonstrate that its use significantly amplifies quadrature grid errors when modern density-functional approximations are employed. Standard grids that work well in conventional density-functional calculations result in runaway error accumulation when used with the many-body expansion. At the same time, delocalization error is also exacerbated, leading to exaggerated estimates of nonadditive n-body interactions. This is illustrated for anion–water clusters using the SCAN, r2SCAN, ωB97X-V and ωB97M-V functionals. By employing dense quadrature grids, the inherent self-interaction error is exposed, which can then be mitigated using a variety of other strategies.

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

Gold-Standard Chemical Database 137 (GSCDB137): A Diverse Set of Accurate Energy Differences for Assessing and Developing Density Functionals

We present GSCDB137, a rigorously curated benchmark library of 137 data sets (8377 entries) covering main-group and transition-metal reaction energies and barrier heights, (intra- and intermolecular) noncovalent interactions, dipole moments, polarizabilities, electric-field response energies, and vibrational frequencies. Legacy data from GMTKN55 and MGCDB84 have been updated to today's best reference values; redundant or low-quality points were removed, and many new, property-focused sets were added. Testing 29 popular density functional approximations (DFAs) confirms the expected Jacob's-ladder hierarchy overall but also reveals notable exceptions: functional performance for frequencies and electric-field properties correlates poorly with that for other ground-state energetics. ωB97M-V and ωB97X-V are the most balanced hybrid meta-GGA and hybrid GGA, respectively; B97M-V and revPBE-D4 lead the meta-GGA and GGA classes. Double hybrids lower mean errors by about 30% versus their hybrid analogues but demand careful frozen-core, basis set, and spin contamination treatment. GSCDB137 offers a comprehensive, openly documented platform for rigorous validation of DFA and universal machine learning potentials, and training of the next generation of exchange-correlation functionals.

Liang, Jiashu [University of California, Berkeley,↗
DOE OSTI2026

Local Spin Density Approximation Strongly Improved by a Better-Informed Local Scaling of Its Self-Interaction Correction

The Perdew−Zunger self-interaction correction (PZSIC) makes density functional approximations (DFAs) exact for all one-electron densities. However, it overcorrects in manyelectron regions, introducing errors for the uniform-density limit, where uncorrected DFAs are exact. The locally scaled PZSIC (LSIC), based on the iso-orbital indicator zσ [which distinguishes single-orbital and slowly varying density regions and is used with the local spin density approximation (LSDA)], restores the uniform-density limit and significantly improves results for many properties, including chemical reaction barrier heights, atomization energies, and ionization potentials. Yet, LSIC performs poorly for weakly bonded systems, leaving many unbound, due to limitations of its iso-orbital indicator. To correct this, in this work we propose a new local scaling, LSIC-α, based on the iso-orbital indicator ασ (which additionally identifies regions of overlapping density tails). A two-parameter scaling function of ασ is fitted to a subset of the nonbonded appropriate norms for the SCAN and r2SCAN meta- GGAs, and tested on many properties of main-group atoms, molecules, and molecular complexes. LSIC-α greatly improves the interaction energies of weakly bonded systems in the S22 data set while retaining LSIC’s accuracy for other properties. This work shows that the errors of LSDA (and presumably of higher-level DFAs) can be largely but not entirely repaired by a proper “do no harm” self-interaction correction.

Approximation↗
DOE OSTI2025

How the choice of exchange–correlation functional affects DFT-based simulations of the hydrated electron

Hydrated electrons are anionic species that are formed when an excess electron is introduced into liquid water. Building an understanding of how hydrated electrons behave in solution has been a long-standing effort of simulation methods, of which density functional theory (DFT) has come to the fore in recent years. The ability of DFT to model the reactive chemistry of hydrated electrons is an attractive advantage over semi-classical methodologies; however, relatively few density functional approximations (DFAs) have been used for the hydrated electron simulations presented in the literature. Here, we simulate hydrated electron systems using a series of exchange–correlation (XC) functionals spanning Jacob’s ladder. We calculate a variety of experimental and other observables of the hydrated electron and compare the XC functional dependence for each quantity. We find that the formation of a stable localized hydrated electron is not necessarily limited to hybrid XC functionals and that some hybrid functionals produce delocalized hydrated electrons or electrons that react with the surrounding water at an unphysically fast rate. Here, we further characterize how different DFAs impact the solvent structure and predicted spectroscopy of the hydrated electron, considering several methods for calculating the hydrated electron’s absorption spectrum for the best comparison between structures generated using different density functionals. None of the dozen or so DFAs that we investigated are able to correctly predict the hydrated electron’s spectroscopy, vertical detachment energy, or molar solvation volume.

Ab-initio molecular dynamics↗
DOE OSTI2024

Delocalization error poisons the density-functional many-body expansion

The many-body expansion is a fragment-based approach to large-scale quantum chemistry that partitions a single monolithic calculation into manageable subsystems. This technique is increasingly being used as a basis for fitting classical force fields to electronic structure data, especially for water and aqueous ions, and for machine learning. Here, we show that the many-body expansion based on semilocal density functional theory affords wild oscillations and runaway error accumulation for ion–water interactions, typified by F − (H 2 O) N with N ≳ 15. We attribute these oscillations to self-interaction error in the density-functional approximation. The effect is minor or negligible in small water clusters, explaining why it has not been noticed previously, but grows to catastrophic proportion in clusters that are only moderately larger. This behavior can be counteracted with hybrid functionals but only if the fraction of exact exchange is ≳50%, whereas modern meta-generalized gradient approximations including ωB97X-V, SCAN, and SCAN0 are insufficient to eliminate divergent behavior. Other mitigation strategies including counterpoise correction, density correction (i.e., exchange–correlation functionals evaluated atop Hartree–Fock densities), and dielectric continuum boundary conditions do little to curtail the problematic oscillations. In contrast, energy-based screening to cull unimportant subsystems can successfully forestall divergent behavior. These results suggest that extreme caution is warranted when the many-body expansion is combined with density functional theory.

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

Hydrogen Bond Benchmark: Focal‐Point Analysis and Assessment of DFT Functionals

We performed a hierarchical, convergent ab initio benchmark study and systematically analyzed the performance of density functional approximations for describing hydrogen bonds in small neutral, cationic, and anionic complexes, as well as in larger systems involving amide, urea, deltamide, and squaramide moieties. Focal point analyses (FPA), extrapolating to the ab initio limit, were carried out using correlated wave function methods up to CCSDT(Q) for the small complexes and CCSD(T) for the larger systems, together with correlation-consistent Gaussian basis sets up to the complete basis set limit. Optimized geometries and vibrational frequencies were obtained at the CCSD(T) level. The resulting FPA hydrogen-bond energies converge within a few tenths of a kcal mol −1 . These reference data were used to evaluate 60 density functionals (including 12 dispersion-corrected), spanning the local-density approximation (LDA), generalized gradient approximations (GGAs), meta-GGAs, hybrids, meta-hybrids, double-hybrids, and range-separated hybrids. Overall, the meta-hybrid M06-2X provides the best performance for both hydrogen bond energies and geometries, while the dispersion-corrected GGAs BLYP-D3(BJ) and BLYP-D4 also yield accurate hydrogen-bond data and can serve as cost-effective options for studying large and complex systems.

coupled cluster theory↗
DOE OSTI2026

Assessing the Limitations of Self-Interaction-Corrected Functionals for Describing the Hydrated Electron

Simulating the hydrated electron using density functional theory is challenging due to the prevalence of self-interaction error in standard functionals. Hybrid functionals like PBEh(40) can reasonably describe the chemistry of an excess electron in water and partially mitigate self-interaction error by incorporating exact Hartree–Fock exchange, but they are computationally expensive making them impractical for large-scale and long-time ab initio molecular dynamics simulations. Explicit self-interaction correction schemes that are applied on an orbital-by-orbital basis offer a potential alternative when the correction is limited to the singly occupied molecular orbital obtained with a generalized gradient approximation functional. Here, we examine whether the Perdew–Zunger self-interaction correction scheme applied to the revPBE functional can provide a computationally efficient and physically sensible alternative to PBEh(40) for the hydrated electron. We find that functionals incorporating a self-interaction correction scheme should be viewed with caution when applied to the hydrated electron and its reactivity. Furthermore, we show that it is critical to consider extensive sampling and diverse chemical environments when validating their performance.

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

Direct NeTS sampling of nuclear graphite $S(α, β, T)$ in Serpent

For advanced reactor applications, Neural Thermal Scattering (NeTS) modules were developed to predict the thermal scattering law (TSL or $S(α, β, T)$) of a nuclear graphite neutron moderator. NeTS are multi-layer, feedforward artificial neural networks, which act as universal function approximators designed for TSL datasets. In this case, a 4-layer neural network with 164 neurons per layer is trained using FLASSH evaluated data in PyTorch and serialized as a torchscript dictionary to predict $S(α, β, T)$ on-the-fly. Relative, absolute and maximum percent deviations of NeTS from File 7 data generated using the FLASSH code are on the order of 0.01%, 0.1% and 1%, respectively, with low inference latencies of 0.000172 s per $S(α, β, T)$ at a given temperature. Capturing the full dimensionality of possible inelastic neutron-lattice interactions, NeTS functionality is embedded in the Serpent Monte Carlo code, where $S(α, β, T)_{NeTS}$ sampling is conducted on-the-fly and compared to ACE look-up-tables for predicting TREAT criticality. k-eff differences between sampling algorithms of 6 pcm are observed and are within the order of Monte Carlo uncertainty. Compared to discrete and continuous-energy ACE files (30 MB and 131 MB per temperature), the NeTS format is on the order of 200–300 kB for a continuous-temperature, interpolation-free representation of $S(α, β, T)$ and cross sections. NeTS-in-Serpent runtimes comparable with ACE look-up tables are achieved by scaling NeTS for high performance computing architectures with hybrid OpenMP + MPI parallelization. This work validates a novel, self-contained reactor physics framework for predictive cross sections, and demonstrates a general methodology for embedding modern machine learning libraries within existing neutronic analysis frameworks.

Nuclear Criticality Safety Program (NCSP)↗
DOE OSTI2025

Solving high-dimensional partial integral differential equations: The finite expression method

Partial integro-differential equations (PIDEs) have broad applications in the sciences, from electro-magnetism to options pricing. Here, in this paper, we introduce a new finite expression method (FEX) to solve PIDEs. This approach builds upon the original FEX and its inherent advantages with new advances: 1) A novel method of parameter grouping is proposed to reduce the number of coefficients in high-dimensional function approximation; 2) A Taylor series approximation method is implemented to significantly improve the computational efficiency and accuracy of the evaluation of the integral terms of PIDEs. The new FEX based method, denoted FEX-PG to indicate the addition of the parameter grouping (PG) step to the algorithm, provides both high accuracy and interpretable numerical solutions, with the outcome being an explicit equation that facilitates intuitive understanding of the underlying solution structures. These features are often absent in traditional methods, such as finite element methods (FEM) and finite difference methods, as well as in deep learning-based approaches. To benchmark our method against recent advances, we apply the new FEX-PG to solve benchmark PIDEs in the literature. In high-dimensional settings, FEX-PG exhibits strong and robust performance, achieving relative errors on the order of single precision machine epsilon, significantly outperforming existing approaches based on neural networks.

Combinatorial optimization↗
DOE OSTI2026

Thorium Monosilicide, ThSi: An Experimental and Theoretical Study

The present theoretical and experimental combination study investigates the ThSi molecule in detail. Computationally, we utilized high-level multireference and coupled-cluster levels of theory conjoined with large correlation consistent basis sets to study a series of electronic and spin–orbit states of ThSi. Here, we report potential energy curves (PECs), electron configurations at equilibrium distances, spectroscopic constants, energetics, and spin–orbit coupling effects for 16 electronic states of ThSi. The studied 16 electronic states are arranged tightly within 0.9 eV, highlighting the complexity of the electronic spectrum of ThSi. The ground electronic state of ThSi is a single-reference 1 1 Σ + state that derives from the 1σ 2 2σ 2 1π 4 electronic configuration. The Ω = 0 + spin–orbit ground state of ThSi is composed of 1 1 Σ + (47%) and 13Π (44%) electronic states. Our measured bond energy (D0) of ThSi, obtained using resonant two-photon ionization (R2PI) spectroscopy is 3.146(4) eV, where the assigned error limit is given in parentheses in units of the last quoted digits. The computed D0 of ThSi (Ω = 0 + ) at the CBS-C-CCSD(T)-δT(Q)-δDK-δSO level (3.181 eV) is in good agreement with the experimental value. Our derived enthalpy of formation for ThSi, Δ f H 0K o (ThSi(g)), is 971.8(6.0) kJ/mol. Finally, we have performed density functional theory (DFT) calculations for ThSi(1 1 Σ + ) using 16 exchange correlation functionals that span multiple rungs of “Jacob’s ladder” of density functional approximation (DFA) to assess the DFT errors on D 0 , r e , and ω e of ThSi with respect to experimental and ab initio coupled-cluster values.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗