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DOE OSTI2025

Static Subspace Approximation for Random Phase Approximation Correlation Energies: Applications to Materials for Catalysis and Electrochemistry

Modeling complex materials using high-fidelity, ab initio methods at low cost is a fundamental goal for quantum chemical software packages. The GW approximation and random phase approximation (RPA) provide a unified description of both electronic structure and total energies using the same physics in a many-body perturbative approach that can be more accurate than generalized-gradient density functional theory (DFT) methods. However, GW/RPA implementations have historically been limited to either specific materials classes or application toward small chemical systems. Here, the static subspace approximation allows for reduced cost full-frequency GW/RPA calculations and has previously been benchmarked thoroughly for GW calculations. Here, we describe our approach to including partial occupations of electronic orbitals in full-frequency GW and RPA calculations for the study of electrocatalysts. We benchmarked RPA total energy calculations using the subspace approximation across a diverse test suite of materials for a variety of computational parameters. The benchmarking quantifies the impact of different extrapolation procedures for representing the static polarizability at infinite screened cutoff, and shows that using screened cutoffs above 20-25 Ryd result in diminishing accuracy returns for predicting RPA total energies. Additionally, for moderately sized electrocatalytic models, 2-3 times fewer computational resources are used to compute RPA total energies by representing the static polarizability with 20-30% of the static subspace basis, with an error of approximately 0.01 eV or better in RPA adsorption energy calculations. Finally, we show that for these electrochemical models RPA can shift DFT adsorption energy shifts by up to 0.5 eV and that GW can frequently shift DFT eigenvalues of surface and adsorbate states by approximately 0.5-1 eV.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2026

Taylor approximation variance reduction for approximation errors in PDE-constrained Bayesian inverse problems

In numerous applications, surrogate models are used as a replacement for accurate parameter-to-observable mappings when solving large-scale inverse problems governed by partial differential equations (PDEs). The surrogate model may be a computationally cheaper alternative to the accurate parameter-to-observable mappings and/or may ignore additional unknowns or sources of uncertainty. The Bayesian approximation error (BAE) approach provides a means to account for the induced uncertainties and approximation errors, i.e. the errors between the accurate parameter-to-observable mapping and the surrogate. The statistics of these errors are, however, in general unknown a priori, and are thus calculated using Monte Carlo sampling. Although the sampling is typically carried out offline, i.e. before considering the data, the process can still represent a computational bottleneck. In this work, we develop a scalable computational approach for reducing the costs associated with the sampling stage of the BAE approach. Specifically, we consider the Taylor expansion of the accurate and surrogate forward models with respect to the uncertain parameter fields either as a control variate for variance reduction or as a means to directly and efficiently approximate the mean and covariance of the approximation errors. We propose efficient methods for evaluating the expressions for the mean and covariance of the Taylor approximations based on linear(-ized) PDE solves. Furthermore, the proposed approach is independent of the dimension of the uncertain parameter, depending instead on the intrinsic dimension of the data, ensuring scalability to high-dimensional problems. The potential benefits of the proposed approach are demonstrated for two high-dimensional inverse problems governed by PDE examples, namely for the estimation of a distributed Robin boundary coefficient in a linear diffusion problem, and for a coefficient estimation problem governed by a nonlinear diffusion problem.

Bayesian approximation error↗

DOE OSTI2024

Static Subspace Approximation for Random Phase Approximation Correlation Energies: Implementation and Performance

Developing theoretical understanding of complex reactions and processes at interfaces requires using methods that go beyond semilocal density functional theory to accurately describe the interactions between solvent, reactants and substrates. Methods based on many-body perturbation theory, such as the random phase approximation (RPA), have previously been limited due to their computational complexity. However, this is now a surmountable barrier due to the advances in computational power available, in particular through modern GPU-based supercomputers. In this work, we describe the implementation of RPA calculations within BerkeleyGW and show its favorable computational performance on large complex systems relevant for catalysis and electrochemistry applications. Our implementation builds off of the static subspace approximation which, by employing a compressed representation of the frequency dependent polarizability, enables the evaluation of the RPA correlation energy with significant acceleration and systematically controllable accuracy. We find that the computational cost of calculating the RPA correlation energy scales only linearly with system size for systems containing up to 50 thousand bands, and is expected to scale quadratically thereafter. We also show excellent strong scaling results across several supercomputers, demonstrating the performance and portability of this implementation.

algorithmic development↗

DOE OSTI2026

Direct Nonlinear Approximation for Security Region Boundary of Integrated Energy Systems: A Polynomial Chaos Expansion Solution

The strong interdependence of electricity, gas, and heating systems can facilitate fault propagation within integrated energy systems (IESs), posing significant challenges to secure operation. This paper proposes a polynomial chaos expansion (PCE)-based approximation method to accurately characterize the IES security region boundary (IES–SRB). By integrating the Karush-Kuhn-Tucker conditions with PCE theory, the IES-SRB approximation problem is reformulated as a set of nonlinear equations concerning the approximation coefficients. Using the Galerkin projection method, these equations are further transformed into a system of projection equations that govern the polynomial approximation coefficients in the IES-SRB approximation. To reduce computational complexity while maintaining high approximation accuracy, a piecewise polynomial approximation method is proposed. Numerical studies on the E39-G20-H6 and E118-G96-H52 IES test systems demonstrate that the proposed method can accurately and effectively construct IES security regions.

Wu, Chenghao [Northeast Electric Power University]↗

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

Biased degenerate ground-state sampling of small Ising models with converged quantum approximate optimization algorithm

The quantum alternating operator ansatz, a generalization of the quantum approximate optimization algorithm (QAOA), is a quantum algorithm used for approximately solving combinatorial optimization problems. QAOA typically uses the transverse field mixer as the driving Hamiltonian. One of the interesting properties of the transverse field driving Hamiltonian is that it results in nonuniform sampling of degenerate ground states of optimization problems. In this study, we numerically examine the fair sampling properties of the transverse field mixer QAOA, and Grover mixer QAOA (GM-QAOA), which provides theoretical guarantees of fair sampling of degenerate optimal solutions, up to a large enough p such that the mean expectation value converges to an optimal approximation ratio of 1. This comparison is performed with high-quality heuristically computed, but not necessarily optimal, QAOA angles, which give strictly monotonically improving solution quality as p increases. These angles are computed using the Julia based numerical simulation software JuliQAOA. Fair sampling of degenerate ground states is quantified using the Shannon entropy of the ground-state amplitudes distribution. The fair sampling properties are reported on several quantum signature Hamiltonians from previous quantum annealing fair sampling studies. Small random fully connected spin glasses are shown, which exhibit exponential suppression of some degenerate ground states with transverse field mixer QAOA. The transverse field mixer QAOA simulations show that some problem instances clearly saturate the Shannon entropy of 0 with a maximally biased distribution that occurs when the learning converges to an approximation ratio of 1 while other problem instances never deviate from a maximum Shannon entropy (uniform distribution) at any p step. Published by the American Physical Society 2025

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2025

A Meta-Generalized Gradient Approximation for the Cavity-Dependent Exchange-Correlation Interaction in Strongly Coupled Light–Matter Systems

Strong light–matter coupling in optical cavities enables the manipulation of chemical and physical properties without altering molecular composition. Theoretical modeling of such phenomena requires exchange-correlation (XC) functionals that account for both electron–electron and electron–photon (ep) interactions within quantum electrodynamical density functional theory (QEDFT). In this work, we develop a meta-generalized gradient approximation (meta-GGA) specifically targeting the cavity-dependent XC interaction in strongly coupled light–matter systems. This novel approximation is built upon a new semilocal polarizability approximation, which draws from the jellium-with-a-gap model, and can be extended to a “global hybrid” variant that goes beyond the isotropic model from previous approximations. The polarizability model yields significantly improved dispersion coefficients and benchmark calculations with the cavity-dependent XC functional demonstrate improved agreement with QED Hartree–Fock (QED-HF) reference energies. Application to the regioselectivity of brominated nitrobenzene intermediates reveals the functional’s capacity to capture cavity-induced energetic shifts. In conclusion, our results advance the Jacob’s ladder of functionals for QEDFT and provide a practical tool for modeling polaritonic chemistry.

Approximation↗

DOE OSTI2025

Embedded random phase approximation for magnetic systems: H 2 dissociative adsorption on Fe(110)

The random phase approximation (RPA), a method for treating electron correlation, has been shown to be superior to standard density functional theory (DFT) approximations in numerous cases. However, the RPA’s computational cost is substantially higher than that of DFT, particularly restricting its application to extended surfaces. The recently introduced embedded RPA (emb-RPA) approach [Wei et al., J. Chem. Phys. 159(19), 194108 (2023)] reduces this computational cost by approximately two orders of magnitude. While previous applications of emb-RPA focused on non-spin-polarized systems, here we extend the approach to ferromagnetic ones. Unlike other embedded correlated wavefunction methods, such as embedded complete active space self-consistent field theory, emb-RPA is advantageous for spin-polarized systems because the RPA is compatible with unrestricted DFT solutions, which are eigenfunctions of the spin angular momentum operator S z but not the total spin-squared operator S 2 . By applying emb-RPA with specific magnetization constraints, we achieved a speedup of two to three orders of magnitude (one order when accounting for the one-time embedding potential optimization cost) with only small errors (∼50 meV) compared to full periodic RPA. Moreover, emb-RPA significantly reduces the over-binding errors of DFT approximations. In conclusion, we anticipate that the acceleration enabled by the spin-polarized emb-RPA approach will broaden the applicability of RPA to magnetic materials.

Density functional theory↗

DOE OSTI2024

Adaptive Power Flow Approximations With Second-Order Sensitivity Insights

The power flow equations are fundamental to power system planning, analysis, and control. However, the inherent non-linearity and non-convexity of these equations present formidable obstacles in problem-solving processes. To mitigate these challenges, recent research has proposed adaptive power flow linearizations that aim to achieve accuracy over wide operating ranges. The accuracy of these approximations inherently depends on the curvature of the power flow equations within these ranges, which necessitates considering second-order sensitivities. In this paper, we leverage second-order sensitivities to both analyze and improve power flow approximations. We evaluate the curvature across broad operational ranges and subsequently utilize this information to inform the computation of various sample-based power flow approximation techniques. Additionally, we leverage second-order sensitivities to guide the development of rational approximations that yield linear constraints in optimization problems. In conclusion, this approach is extended to enhance accuracy beyond the limitations of linear functions across varied operational scenarios.

24 POWER TRANSMISSION AND DISTRIBUTION↗

DOE OSTI2024

Approximate CFTs and random tensor models

Abstract A key issue in both the field of quantum chaos and quantum gravity is an effective description of chaotic conformal field theories (CFTs), that is CFTs that have a quantum ergodic limit. We develop a framework incorporating the constraints of conformal symmetry and locality, allowing the definition of ensembles of ‘CFT data’. These ensembles take on the same role as the ensembles of random Hamiltonians in more conventional quantum ergodic phases of many-body quantum systems. To describe individual members of the ensembles, we introduce the notion of approximate CFT, defined as a collection of ‘CFT data’ satisfying the usual CFT constraints approximately, i.e. up to small deviations. We show that they generically exist by providing concrete examples. Ensembles of approximate CFTs are very natural in holography, as every member of the ensemble is indistinguishable from a true CFT for low-energy probes that only have access to information from semi-classical gravity. To specify these ensembles, we impose successively higher moments of the CFT constraints. Lastly, we propose a theory of pure gravity in AdS 3 as a random matrix/tensor model implementing approximate CFT constraints. This tensor model is the maximum ignorance ensemble compatible with conformal symmetry, crossing invariance, and a primary gap to the black-hole threshold. The resulting theory is a random matrix/tensor model governed by the Virasoro 6j-symbol.

Physics↗

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 OSTI2024

Symmetry-Based Structured Matrices for Efficient Approximately Equivariant Networks

There has been much recent interest in designing symmetry-aware neural networks (NNs) exhibiting relaxed equivariance. Such NNs aim to interpolate between being exactly equivariant and being fully flexible, affording consistent performance benefits. In a separate line of work, certain structured parameter matrices -- those with displacement structure, characterized by low displacement rank (LDR) -- have been used to design small-footprint NNs. Displacement structure enables fast function and gradient evaluation, but permits accurate approximations via compression primarily to classical convolutional neural networks (CNNs). In this work, we propose a general framework -- based on a novel construction of symmetry-based structured matrices -- to build approximately equivariant NNs with significantly reduced parameter counts. Our framework integrates the two aforementioned lines of work via the use of so-called Group Matrices (GMs), a forgotten precursor to the modern notion of regular representations of finite groups. GMs allow the design of structured matrices -- resembling LDR matrices -- which generalize the linear operations of a classical CNN from cyclic groups to general finite groups and their homogeneous spaces. We show that GMs can be employed to extend all the elementary operations of CNNs to general discrete groups. Further, the theory of structured matrices based on GMs provides a generalization of LDR theory focussed on matrices with cyclic structure, providing a tool for implementing approximate equivariance for discrete groups. We test GM-based architectures on a variety of tasks in the presence of relaxed symmetry. We report that our framework consistently performs competitively compared to approximately equivariant NNs, and other structured matrix-based compression frameworks, sometimes with a one or two orders of magnitude lower parameter count.

Samudre, Ashwin↗

DOE OSTI2025

Simplifying activations with linear approximations in neural networks

A key step in Neural Networks is activation. Among the different types of activation functions, sigmoid, tanh, and others involve the usage of exponents for calculation. From a hardware perspective, exponential implementation implies the usage of Taylor series or repeated methods involving many addition, multiplication, and division steps, and as a result are power-hungry and consume many clock cycles. We implement a piecewise linear approximation of the sigmoid function as a replacement for standard sigmoid activation libraries. This approach provides a practical alternative by leveraging piecewise segmentation, which simplifies hardware implementation and improves computational efficiency. In this paper, we detail piecewise functions that can be implemented using linear approximations and their implications for overall model accuracy and performance gain. Our results show that for the DenseNet, ResNet, and GoogLeNet architectures, the piecewise linear approximation of the sigmoid function provides faster execution times compared to the standard TensorFlow sigmoid implementation while maintaining comparable accuracy. Specifically, for MNIST with DenseNet, accuracy reaches 99.91% (Piecewise) vs. 99.97% (Base) with up to 1.31x speedup in execution time. For CIFAR-10 with DenseNet, accuracy improves to 98.97% (Piecewise) vs. 99.40% (Base) while achieving 1.24x faster execution. Similarly, for CIFAR-100 with DenseNet, the accuracy is 97.93% (Piecewise) vs. 98.39% (Base), with a 1.18x execution time reduction. These results confirm the proposed method’s capability to efficiently process large-scale datasets and computationally demanding tasks, offering a practical means to accelerate deep learning models, including LSTMs, without compromising accuracy.

Activation function↗

DOE OSTI2026

Deep Neural Networks are Adaptive to Function Regularity and Data Distribution in Approximation and Estimation

Deep learning has exhibited remarkable results across diverse areas. To understand its success, substantial research has been directed towards its theoretical foundations. Nev- ertheless, the majority of these studies examine how well deep neural networks can model functions with uniform regularities. In this paper, we explore a different angle: how deep neural networks can adapt to varying degrees of smoothness in functions and nonuni- form data distributions across different locations and scales. More precisely, we focus on a broad class of functions defined by nonlinear tree-based approximation methods. This class encompasses a range of function types, such as functions with uniform regularities and discontinuous functions. We develop nonparametric approximation and estimation theories for this class using deep ReLU networks. Our results show that deep neural networks are adaptive to the nonuniform smoothness of functions and nonuniform data distributions at different locations and scales. We apply our results to several function classes, and derive the corresponding approximation and generalization errors. The validity of our results is demonstrated through numerical experiments.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

Approximation of ice phenology of Maine lakes using Aqua MODIS surface temperature data

Studies of lake ice phenology have historically relied on limited in situ data. Relatively few observations exist for ice out and fewer still for ice in, both of which are necessary to determine the temporal extent of ice cover. Satellite data provide an opportunity to better document patterns of ice phenology across landscapes and relate them to the climatological drivers behind changing ice phenology. We developed a model, the Cumulative Sum Method (CSM), that uses daytime and nighttime surface temperature observations from the Moderate Resolution Imaging Spectroradiometer (MODIS) sensor on board the Earth-observing Aqua satellite to approximate ice in (the onset of ice cover) and ice out from training datasets of 13 and 58 Maine lakes, respectively, during the 2002/2003 through 2017/2018 ice seasons. Ice in was signaled by reaching a threshold of cumulative negative degrees following the first day of the season below 0°C. Ice out was signaled by reaching a threshold of cumulative positive degrees following the first day of the year above 0°C. The comparison of observed and remotely sensed ice-in dates showed relative agreement with a correlation coefficient of 0.71 and a mean absolute error (MAE) of 9.8 days. Ice-out approximations had a correlation coefficient of 0.67 and an MAE of 8.8 days. Lakes smaller in surface area and nearer the Atlantic coast had the greatest error in approximation. Application of the CSM to 20 additional lakes in Maine produced a comparable ice-out MAE of 8.9 days. Ice-out model performance was weaker for the warmest years; there was a larger MAE of 12.0 days when the model was applied to the years 2019–2023 for the original 58 lakes. The development of this model, which utilizes daily satellite data, demonstrates the promise of remote sensing for quantifying ice phenology over short, temporal scales, and wider geographic regions than can be observed in situ, and allows exploration of the influence of surface temperature patterns on the process and timing of ice in and ice out.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2025

Performance evaluations of signed and unsigned noisy approximate quantum Fourier arithmetic

The Quantum Fourier Transform (QFT) grants competitive advantages, especially in resource usage and circuit approximation, for performing arithmetic operations on quantum computers, and offers a potential route toward a numerical quantum-computational paradigm. In this paper, we utilize efficient techniques to implement QFT-based integer addition and multiplications. These operations are fundamental to various quantum applications including Shor’s algorithm, weighted-sum optimization problems in data processing and machine learning, and quantum algorithms requiring inner products. We carry out performance evaluations of these implementations based on IBM’s superconducting-qubit architecture using different compatible noise models. We isolate the sensitivity of the component quantum circuits on both one-/two-qubit gate error rates, and the number of the arithmetic operands’ superposed integer states. We analyze performance and identify the most effective approximation depths for unsigned quantum addition and quantum multiplication within the given context. We then perform a similar analysis of signed addition and compare to the unsigned results. We observe significant dependency of the optimal approximation depth on the degree of machine noise and the number of superposed states in certain performance regimes. Finally, we elaborate on the algorithmic challenges—relevant to signed, unsigned, modular and non-modular versions—that could also be applied to current implementations of QFT-based subtraction, division, exponentiation, and their potential tensor extensions. Here, we analyze the performance trends in our results and speculate on possible future developments within this computational paradigm.

Computational models↗

DOE OSTI2025

Backpropagation-based learning with local derivative approximation and memory replay in biologically plausible neural systems

When learning, the brain modifies individual synaptic connections to reach a desired behavior. Animal and human brains have been shown to be incredibly capable of learning complex and varied functions across a wide variety of tasks. In recent years, artificial neural networks, inspired by human and animal brains, have shown great capabilities in learning a wide variety of difficult tasks. However, artificial neural networks primarily teach themselves through the use of backpropagation, a learning method which has no clear analogue within the brain. Additionally, Artificial Neural Networks primarily use continuous activation functions, which differ significantly from the spiking neuronal behavior present in the brain. In this paper, we discuss and demonstrate a biologically plausible learning method that approximates backpropagation through two techniques on Spiking Neural Networks. First, we show that the local temporal derivatives that are necessary for backpropagation can be approximately recovered through reconstruction using spike timings. Second, we show that through learning during a sleep phase, inspired by neuroscience research into memory replay, the localized parallel feedback path can learn to approximate the derivative through the forward path weight matrix, thus solving the weight transport problem. Lastly, we demonstrate that the combination of these two methods can approach or exceed the accuracy of backpropagation-based methods for a variety of neuromorphic vision tasks while maintaining biological plausibility.

42 ENGINEERING↗

DOE OSTI2024

A windowed mean trajectory approximation for condensed phase dynamics

We propose a trajectory-based quasi-classical method for approximating dynamics in condensed phase systems. Building upon the previously developed optimized mean trajectory approximation that has been used to compute linear and nonlinear spectra, we borrow some ideas from filtering trajectory methods to obtain a novel semiclassical method for the dynamical propagation of density matrices. This new approximation is tested rigorously against standard multistate electronic models, spin-boson models, and models of the Fenna–Matthews–Olson complex. For dissipative systems, the current method is significantly better or as good as many other semiclassical methods available, especially at low temperatures and for off-diagonal density matrix elements, whereas for scattering models, the current method bears similar limitations as mean-field propagation schemes. All results are tested against the numerically exact hierarchical equations of motion method. In conclusion, the new method shows excellent agreement across various parameter regimes with numerically exact results, highlighting the robustness and accuracy of our approach.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗