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Results for “Finite expression method”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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14 records
All sourcesNASA NTRSDOE OSTI
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

Identifying stochastic dynamics via finite expression methods

Modeling stochastic differential equations (SDEs) is crucial for understanding complex dynamical systems in various scientific fields. Recent methods often employ neural network-based models, which typically represent SDEs through a combination of deterministic and stochastic terms. However, these models usually lack interpretability and have difficulty in generalizing beyond their training domain. Here, this paper introduces the Finite Expression Method (FEX), a symbolic learning approach designed to derive interpretable mathematical representations of the deterministic component of SDEs. For the stochastic component, we integrate FEX with advanced generative modeling techniques to provide a comprehensive representation of SDEs. The numerical experiments on linear, nonlinear, and multidimensional SDEs demonstrate that FEX generalizes well beyond the training domain and delivers more accurate long-term predictions compared to neural network-based methods. The symbolic expressions identified by FEX not only improve prediction accuracy but also offer valuable scientific insights into the underlying dynamics of the systems.

Complex dynamical systems↗
DOE OSTI2026

Coulomb corrections for the nonflip and spin-flip electromagnetic 𝑝 ↑⁢ 𝐴 amplitudes

It is demonstrated that, within the eikonal approach, the Coulomb corrections to the elastic electromagnetic nonflip and spin-flip proton-nucleus amplitudes are identical when the two amplitudes share the same exponential form factors. This result allows Coulomb corrections to be computed numerically, and with high precision, for both electromagnetic and hadronic elastic 𝑝 ↑⁢ 𝐴 amplitudes in the massless-photon limit, including the effects of soft magnetic photon exchange. The method relies on analytical expressions and numerical integrations over a finite impact-parameter range with nonsingular integrands, providing a practical and systematically controlled framework for phenomenological applications.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2025

Additional considerations in analytical solution for time-dependent heat conduction in a three-dimensional multilayer sphere

This work presents an analytical method to solve the heat conduction equation in three dimensions for problems consisting of multilayer concentric spheres. The method can be used to treat time-varying heat conduction problems where the heat source that drives the transient is time-invariant. Equally applicable to all Poisson-type problems with concentric spherical geometry, the method consists of representing the solution as a summation of weighted eigenfunctions. The weights for each eigenfunction are computed algebraically. Previous work has already established the core constituents of the methodology. The current work augments the existing methods by including consideration of nonzero interface resistance between layers and explicit discussion on the boundary condition homogenization required to treat inhomogeneous problems. Also, two demonstration problems are presented. One demonstration problem is based on the method of manufactured solutions and therefore allows for comparison with exact expressions for the solution temperature distribution. The second, more complex, demonstration problem relies on the finite element method for comparisons. The expected convergence behavior is observed for both demonstration problems.

97 - MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Exact-Two-Component Complete Active Space Method with Variational Treatment of Magnetic Field and Spin–Orbit Coupling: Application to X-ray Magnetic Circular Dichroism Spectroscopy

We introduce an exact-two-component complete active space self-consistent-field (X2C-CASSCF) method formulated under the restricted-magnetic-balance condition. This framework allows for the nonperturbative treatment of static magnetic fields using gauge-including atomic orbitals (GIAOs). The GIAO-X2C-CASSCF methodology effectively captures all microstates within the same 2J + 1-degenerate manifold and their splitting in a static magnetic field, which are not accessible through single-reference-based methods. We also present mathematical recursive expressions for evaluating one-electron relativistic integrals by using GIAOs in the presence of a finite magnetic field. Benchmark studies include oxygen and nitrogen K-edge X-ray magnetic circular dichroism spectroscopy (XMCD) for closed-shell organic compounds, as well as L-edge XMCD spectroscopy for the high-spin open-shell transition metal ion Mn 2+ and the tetrahedral Mn(II)O 4 6– complex.

Chemical calculations↗
DOE OSTI2024

Solution of the Schrödinger equation for quasi-one-dimensional materials using helical waves

We formulate and implement a spectral method for solving the Schrödinger equation, as it applies to quasi-one-dimensional materials and structures. This allows for computation of the electronic structure of important technological materials such as nanotubes (of arbitrary chirality), nanowires, nanoribbons, chiral nanoassemblies, nanosprings and nanocoils, in an accurate, efficient and systematic manner. Our work is motivated by the observation that one of the most successful methods for carrying out electronic structure calculations of bulk/crystalline systems — the plane-wave method — is a spectral method based on eigenfunction expansion. Our scheme avoids computationally onerous approximations involving periodic supercells often employed in conventional plane-wave calculations of quasi-one-dimensional materials, and also overcomes several limitations of other discretization strategies, e.g., those based on finite differences and atomic orbitals. The basis functions in our method — called helical waves (or twisted waves) — are eigenfunctions of the Laplacian with symmetry adapted boundary conditions, and are expressible in terms of plane waves and Bessel functions in helical coordinates. We describe the setup of fast transforms to carry out discretization of the governing equations using our basis set, and the use of matrix-free iterative diagonalization to obtain the electronic eigenstates. Miscellaneous computational details, including the choice of eigensolvers, use of a preconditioning scheme, evaluation of oscillatory radial integrals and the imposition of a kinetic energy cutoff are discussed. We have implemented these strategies into a computational package called HelicES (Helical Electronic Structure). We demonstrate the utility of our method in carrying out systematic electronic structure calculations of various quasi-one-dimensional materials through numerous examples involving nanotubes, nanoribbons and nanowires. We also explore the convergence properties of our method, and assess its accuracy and computational efficiency by comparison against reference finite difference, transfer matrix method and plane-wave results. We anticipate that our method will find applications in computational nanomechanics and multiscale modeling, for carrying out transport calculations of interest to the field of semiconductor devices, and for the discovery of novel chiral phases of matter that are of relevance to the burgeoning quantum hardware industry.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2024

Convergent laser beam shapes: Unveiling the dynamics of Laser-induced elastic waves in composite materials

Overcoming the low signal-to-noise ratio (SNR) in laser ultrasonic testing of composite materials remains a significant challenge. Current efforts focus on enhancing SNR by inserting more energy into the material through temporal and/or spatial modulation of the laser beam. However, potential SNR improvements through wave convergence and wave energy manipulation have been overlooked. This paper addresses this gap by demonstrating the convergence of different wave types to a designated point and by showing the feasibility of directing absorbed laser energy into a specific wave type through spatial modulation of the laser beam. To achieve this, mathematical expressions for the convergent laser beams are derived. Various laser beam profiles are then introduced to the thermoelastic equations and solved using the finite element method. The sample under investigation is a transversely isotropic unidirectional carbon fiber reinforced plastic, characterized by anisotropic thermal expansion coefficients and thermal conductivities. Results reveal pronounced convergence of the intended wave type at the center due to laser beam shaping. This study showcases the ability to direct absorbed laser energy toward a specific wave type through spatial modulation of the laser beam and highlights the role of material anisotropy in energy focusing.

composite materials↗
DOE OSTI2026

Feynman diagrams for matter wave interferometry

We introduce a new theoretical framework based on Feynman diagrams to compute phase shifts in matter wave interferometry. The method allows for analytic computation of higher order quantum corrections, beyond the traditional semi-classical approximation. These additional terms depend on the finite size of the initial matter wavefunction and/or have higher order dependence on ℏ. We apply the method to compute the response of matter wave interferometers to power law potentials and potentials with an arbitrary spatial dependence. The analytic expressions are validated by comparing to numerical simulations, and estimates are provided for the scale of the quantum corrections to the phase shift response to the gravitational field of the earth, anharmonic trapping potentials, and gravitational fields from local proof masses. We also find that for certain experimentally feasible parameters, these corrections are large enough to be measured and could lead to systematic errors if they are not mitigated. We find that to first order in a spatially dependent potential, quantum corrections vanish when the initial matter wavepacket has spherical symmetry and the potential satisfies Laplace's equation. We anticipate these quantum corrections will be especially important for trapped matter wave interferometers and for free-space matter wave interferometers in the presence of proof masses. These interferometers are becoming increasingly sensitive tools for mobile inertial sensing, gravity surveying, tests of gravity and its interplay with quantum mechanics, and searches for dark energy.

Glick, Jonah [Northwestern U.; Fermilab] (ORCID:00↗
DOE OSTI2026

Beyond Contrast Transfer: Spectral SNR as a Finite-Dose Metric for STEM Phase Retrieval

The contrast transfer function (CTF) is widely used to evaluate phase retrieval methods in scanning transmission electron microscopy (STEM), including center-of-mass imaging, parallax imaging, direct ptychography, and iterative ptychography. However, the CTF reflects only the maximum usable signal, neglecting the effects of finite electron fluence and the Poisson-limited nature of detection. As a result, it can significantly overestimate practical performance, especially in low-dose regimes. Here, we employ the spectral signal-to-noise ratio (SSNR), as a finite-dose statistical framework to evaluate the recoverable signal as a function of spatial frequency. Using numerical reconstructions of white-noise objects, we show that center-of-mass, parallax, and direct ptychography exhibit dose-independent SSNRs, with close-form analytic expressions. In contrast, iterative ptychography exhibits a surprising dose dependence: at low fluence, its SSNR converges to that of direct ptychography; at high fluence, it saturates at a value consistent with the maximum detective quantum efficiency predicted by recent quantum Fisher information bounds. The results highlight the limitations of CTF-based evaluation and motivate SSNR as a more accurate, finite-dose metric for assessing STEM phase retrieval methods.

STEM phase retrieval↗
DOE OSTI2025

Analytical expression of a finite, long, conical canted-cosine-theta coil for particle collider interaction regions

Magnets in the accelerator interaction region (IR) present significant challenges because of high field requirements and limited available space. Conical-shaped magnets offer advantages in these environments by allowing closer placement to the interaction point while maintaining clearance from synchrotron radiation. Interestingly, numerical studies have shown that conical canted-cosine-theta (CCT) designs produce a constant field distribution along the axial direction in the IR quadrupoles for the Electron-Ion Collider (EIC) at Brookhaven National Laboratory. However, the field harmonics generated by conical CCT windings are not yet fully understood. This paper presents an analytical approach to describe the magnetic field produced by a conical surface current and proposes a method for designing conical CCT magnets for accelerator applications. First, we begin with a surface current sheet having a general cosine-theta distribution in spherical coordinates and solve the vector potential using the Green’s function. The magnetic fields generated by the conical current sheet are expressed using associated Legendre polynomials. These results are then related to circular field harmonics and integral field harmonics for designing a coil that produces a pure multipole field. Next, a single layer of the conical CCT winding path is produced based on the cosine-theta current distribution. Finally, the magnetic field quality of dipole and quadrupole conical CCT coils with multiple layers is verified using the Biot-Savart law.

Yang, Ye↗
DOE OSTI2026

Commuting embeddings for parallel strategies in non-local games

Non-local games provide a versatile framework for probing quantum correlations and for benchmarking the power of entanglement. In finite dimensions, the standard method for playing several games in parallel requires a tensor product of the local Hilbert spaces, which scales additively in the number of qubits. In this work, we show that this additive cost can be reduced by exploiting algebraic embeddings. We introduce two forms of compressions. First, when a referee selects one game from a finite collection of games at random, the game quantum strategy can be implemented using a maximally entangled state of dimension equal to the largest individual game, thereby eliminating the need for repeated state preparations. Second, we establish conditions under which several games can be played simultaneously in parallel on fewer qubits than the tensor product baseline. These conditions are expressed in terms of commuting embeddings of the game algebras. Moreover, we provide a constructive framework for building such embeddings. Using tools from Lie theory, we show that aligning the various game algebras into a common Cartan decomposition enables such a qubit reduction. Beyond the theoretical contribution, our framework casts NLGs as algebraic primitives for distributed and resource-constrained quantum computations and suggested NLGs as a comparable device-independent dimension witness.

Commuting embeddings↗
DOE OSTI2026

A Unified Design Theory for Multi-Port Polyphase Transformers Enabling Scalable Power-Multiplexed EV Fleet Charging Systems

This paper presents a unified analytical design theory for multi-port polyphase transformers, targeting scalable and isolated high-power Electric Vehicle (EV) fleet charging systems with power multiplexing capability. As fleet electrification accelerates, conventional one-to-one charger architectures face significant challenges in infrastructure cost, peak power demand, and low utilization of installed power electronics. Power-multiplexed charging architectures, which dynamically distribute power from a shared pool of converter modules across multiple vehicles, have emerged as a promising solution. However, such architectures require scalable, isolated multi-port power interfaces capable of routing energy among multiple inputs and outputs, whose design remains complex and dependent on iterative modeling. To address this gap, the proposed theory provides closed-form expressions for self-inductance, leakage inductance, and mutual coupling terms for arbitrary multi-phase, multi-port transformer structures. The formulation enables direct synthesis of isolated multi-input and multi-output resonant converter systems without reliance on geometry-specific finite-element analysis or extensive parameter extraction. This capability is particularly critical for power-multiplexed systems, where modular converter structures must interface with multiple vehicles while maintaining galvanic isolation and flexible power allocation. The effectiveness of the proposed framework is demonstrated through the design of a 360 kW multi-phase system operating over a 700–900 VDC input and 400–1250 VDC output range. PLECS simulation results confirm accurate prediction of system behavior and validate the applicability of the approach to multi-port, power-multiplexed charging scenarios. The proposed method significantly reduces design complexity while enabling scalable, cost-effective, and fully utilized EV fleet charging infrastructure.

Asa, Erdem [ORNL] (ORCID:0000000190884812)↗
DOE OSTI2025

Conformal Hierarchical Simulation-Based Inference with Local Validity

Trustworthy and interpretable uncertainty quantification is a long-standing challenge in artificial intelligence. Simulation-based inference (SBI) comprises a broad swath of approaches for estimating latent parameters with uncertainties. Although flexible neural density estimators in SBI can be remark- ably expressive capturing highly structured, high-dimensional posteriors their credible regions can be badly mis-calibrated and are often only accompanied by heuristic coverage checks. We present the first SBI framework that delivers finite-sample local valid coverage guarantees that hold in the neighborhood of each observation. Our framework can couple any off-the-shelf hierarchical SBI engine with a confor- mal Bayesian post-processing step that operates on the posterior predictive density. A kernel-weighted conformity score adapts the conformal quantile to the local geometry of the data, yielding prediction sets that are simultaneously (i) marginally calibrated, (ii) locally valid, and (iii) hierarchical, handling global and observation-specific parameters in a single pass. Through experiments on synthetic data and benchmarks from neuroscience and physics, we show that our approach attains 1 − α coverage, where prior SBI methods under- or over-cover. Our approach also maintains a competitive, credible set size with minimal computational overhead. Finally, our approach can be used to make predictions on real data and give valid credible regions modulo weight-initialization-based model mis-specification.

Trivedi, Shubhendu [Fermilab]↗
DOE OSTI2024

NSTXU Diagnostic Disruption Dynamic Loading Represented by Response Spectra

This article presents the results of transient dynamic simulations of loads due to disruption eddy currents on the NSTXU vacuum vessel. Dynamic loading at diagnostic mounting locations is expressed as response spectra derived from the time history results of the dynamic structural simulations of a variety of disruption scenarios. The disruption simulations draw on a history of the project assessments of worst case disruptions for specific components. Major efforts to assess disruption loading have included the vacuum vessel which is the major structural support for the machine, as well as the passive plates (PPs), high harmonic fast wave (HHFW) antenna, and centerstack casing. Each one of these efforts included transient electromagnetic simulations producing time-dependent eddy current Lorentz loads (and in some cases halo loads) which then were applied to time-dependent structural dynamic analyses intended to obtain the proper dynamic amplification factors. In some instances, the EM model and structural model were identical allowing direct transfer of EM forces to the structural model. In other cases, the EM and structural model were not identical and the vector potential (VP) transfer method was used. The results files from these analyses were available (or re-run) to post process in ANSYS Classic time history postprocessor. In conclusion, the ANSYS command is used to create response spectra from time history data at desired points on the vessel.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗