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DOE OSTI2023

Analysis of Neural Networks as Random Dynamical Systems

In this report we present our findings and outcomes of the NNRDS (analysis of Neural Networks as Random Dynamical Systems) project. The work is largely motivated by the analogy of a large class of neural networks (NNs) with a discretized ordinary differential equation (ODE) schemes. Namely, residual NNs, or ResNets, can be viewed as a discretization of neural ODEs (NODEs) where the NN depth plays the role of the time evolution. We employ several legacy tools from ODE theory, such as stiffness, nonlocality, autonomicity, to enable regularization of ResNets thus improving their generalization capabilities. Furthermore, armed with NN analysis tools borrowed from the ODE theory, we are able to efficiently augment NN predictions with uncertainty overcoming wellknown dimensionality challenges and adding a degree of trust towards NN predictions. Finally, we have developed a Python library QUiNN (Quantification of Uncertainties in Neural Networks) that incorporates improved-architecture ResNets, besides classical feed-forward NNs, and contains wrappers to PyTorch NN models enabling several major classes of uncertainty quantification methods for NNs. Besides synthetic problems, we demonstrate the methods on datasets from climate modeling and materials science.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

When ancient numerical demons meet physics-informed machine learning: adjoint-based gradients for implicit differentiable modeling

Recent advances in differentiable modeling, a genre of physics-informed machine learning that trains neural networks (NNs) together with process-based equations, have shown promise in enhancing hydrological models' accuracy, interpretability, and knowledge-discovery potential. Current differentiable models are efficient for NN-based parameter regionalization, but the simple explicit numerical schemes paired with sequential calculations (operator splitting) can incur numerical errors whose impacts on models' representation power and learned parameters are not clear. Implicit schemes, however, cannot rely on automatic differentiation to calculate gradients due to potential issues of gradient vanishing and memory demand. Here we propose a “discretize-then-optimize” adjoint method to enable differentiable implicit numerical schemes for the first time for large-scale hydrological modeling. The adjoint model demonstrates comprehensively improved performance, with Kling–Gupta efficiency coefficients, peak-flow and low-flow metrics, and evapotranspiration that moderately surpass the already-competitive explicit model. Therefore, the previous sequential-calculation approach had a detrimental impact on the model's ability to represent hydrological dynamics. Furthermore, with a structural update that describes capillary rise, the adjoint model can better describe baseflow in arid regions and also produce low flows that outperform even pure machine learning methods such as long short-term memory networks. The adjoint model rectified some parameter distortions but did not alter spatial parameter distributions, demonstrating the robustness of regionalized parameterization. Despite higher computational expenses and modest improvements, the adjoint model's success removes the barrier for complex implicit schemes to enrich differentiable modeling in hydrology.

58 GEOSCIENCES↗

DOE OSTI2025

Fast and Scalable FFT-Based GPU-Accelerated Algorithms for Block-Triangular Toeplitz Matrices with Application to Linear Inverse Problems Governed by Autonomous Dynamical Systems

In this work, we present an efficient and scalable algorithm for performing matrix-vector multiplications (matvecs) for block Toeplitz matrices. Such matrices, which are shift-invariant with respect to their blocks, arise in the context of solving inverse problems governed by autonomous systems, and time-invariant systems in particular. In this article, we consider inverse problems that infer unknown parameters from observational data of a linear time-invariant dynamical system given in the form of partial differential equations (PDEs). Matrix-free Newton-conjugate-gradient methods are often the gold standard for solving these inverse problems, but they require numerous actions of the Hessian on a vector. Matrix-free adjoint-based Hessian matvecs require solution of a pair of linearized forward/adjoint PDE solves per Hessian action, which may be prohibitive for large-scale inverse problems. Time invariance of the forward PDE problem leads to a block Toeplitz structure of the discretized parameter-to-observable (p2o) map defining the mapping from inputs (parameters) to outputs (observables) of the PDEs. This block Toeplitz structure enables us to exploit two key properties: (1) compact storage of the p2o map and its adjoint, and (2) efficient fast Fourier transform–based Hessian matvecs. The proposed algorithm is mapped onto large multi-GPU clusters and achieves more than 80% of peak bandwidth on NVIDIA A100 GPUs. Excellent weak scaling is shown for up to 48 A100 GPUs. For the targeted problems, the implementation executes Hessian matvecs within fractions of a second, which is orders of magnitude faster than can be achieved by conventional matrix-free Hessian matvecs via forward/adjoint PDE solves.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

Collocation methods for nonlinear differential equations on low-rank manifolds

We introduce new methods for integrating nonlinear differential equations on low-rank manifolds. These methods rely on interpolatory projections onto the tangent space, enabling low-rank time integration of vector fields that can be evaluated entry-wise. A key advantage of our approach is that it does not require the vector field to exhibit low-rank structure, thereby overcoming significant limitations of traditional dynamical low-rank methods based on orthogonal projection. To construct the interpolatory projectors, we develop a sparse tensor sampling algorithm based on the discrete empirical interpolation method (DEIM) that parameterizes tensor train manifolds and their tangent spaces with cross interpolation. Using these projectors, we propose two time integration schemes on low-rank tensor train manifolds. The first scheme integrates the solution at selected interpolation indices and constructs the solution with cross interpolation. The second scheme generalizes the well-known orthogonal projector-splitting integrator to interpolatory projectors. We demonstrate the proposed methods with applications to several tensor differential equations arising from the discretization of partial differential equations.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

A high-order finite difference method for moving immersed domain boundaries and material interfaces

Here, we present a high-order sharp treatment of immersed moving domain boundaries and material interfaces, and apply it to the advection-diffusion equation in two and three dimensions. The spatial discretization combines dimension-split finite difference schemes with an immersed boundary treatment based on a weighted least-squares reconstruction of the solution, providing stable discretizations with up to sixth order accuracy for diffusion terms and third order accuracy for advection terms. The temporal discretization relies on a novel strategy for maintaining high-order temporal accuracy in problems with moving boundaries that minimizes implementation complexity and allows arbitrary explicit or diagonally-implicit Runge-Kutta schemes. The approach is broadly compatible with popular PDE-specialized Runge-Kutta time integrators, including low-storage, strong stability preserving, and diagonally implicit schemes. Through numerical experiments we demonstrate that the full discretization maintains high-order spatial and temporal accuracy in the presence of complex 3D geometries and for a range of boundary conditions, including Dirichlet, Neumann, and flux conditions with large jumps in coefficients.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Maximum bound principle preserving integrating factor Runge–Kutta methods for semilinear parabolic equations

A large class of semilinear parabolic equations satisfy the maximum bound principle (MBP) in the sense that the time-dependent solution preserves for any time a uniform pointwise bound imposed by its initial and boundary conditions. Here, the MBP plays a crucial role in understanding the physical meaning and the well-posedness of the mathematical model. Investigation on numerical algorithms with preservation of the MBP has attracted increasingly attentions in recent years, especially for the temporal discretizations, since the violation of MBP may lead to nonphysical solutions or even blow-ups of the algorithms. In this paper, we study high-order MBP-preserving time integration schemes by means of the integrating factor Runge-Kutta (IFRK) method. Beginning with the space-discrete system of semilinear parabolic equations, we present the IFRK method in general form and derive the sufficient conditions for the method to preserve the MBP. In particular, we show that the classic four-stage, fourth-order IFRK scheme is MBP preserving for some typical semilinear systems although not strong stability preserving, which can be instantly applied to the Allen-Cahn type of equations. To our best knowledge, this is the first time to present a fourth-order linear numerical method preserving the MBP. In addition, convergence of these numerical schemes is proved theoretically and verified numerically, as well as their efficiency by simulations of 2D and 3D long-time evolutional behaviors. Numerical experiments are also carried out for a model which is not a typical gradient flow as the Allen-Cahn type of equations.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

Route Optimization for Energy Efficient Airport Shuttle Operations - A Case Study from Dallas Fort Worth International Airport

Air travel and requisite surface traffic supporting passenger arrival/departure constitutes a significant portion of travel and emissions in cities with large airports. An airport trip can segment into three parts namely: i) travel from a location in the city to the airport; ii) travel from a parking lot or rental car center to the terminal (i.e., within the airport premises), and iii) travel inside the terminal. Depending on the airport access mode all or a part of these legs comprise a traveler’s journey to the airport. The priority of airport ground transport management teams is to provide passengers with a seamless travel experience within the airport, so it is understandable that within airport shuttle routes might not be optimized for minimizing energy consumption. Solutions that meet the dual objective of reducing energy consumption from airport shuttle operations without compromising on passenger travel experience are key to improving system efficiency. There is currently a dearth of research and tools that can inform airports in making such decisions. Addressing this need, this research effort puts forth an optimization model that generates optimal shuttle routes for a given set of constraints, and a discrete-event simulator that evaluates the optimal solutions in a stochastic environment to understand the tradeoffs between passenger wait times, and within airport shuttle energy consumption. The proposed set of tools are tested in the context of optimizing airport shuttles routes within the Dallas Fort Worth International Airport (DFW). In addition to shuttle spatial positioning, and passenger demand information, high-fidelity vehicle data was collected using data loggers installed on DFW shuttles. Results show that 20% energy reduction in shuttle operations is possible with a modest two-minute increase in average passenger wait times. The tools developed in this research effort are designed to be generalizable and can help optimize shuttle operations planning at any major airport.

air travel↗

DOE OSTI2024

Efficient formulation of multitime generalized quantum master equations: Taming the cost of simulating 2D spectra

Modern 4-wave mixing spectroscopies are expensive to obtain experimentally and computationally. In certain cases, the unfavorable scaling of quantum dynamics problems can be improved using a generalized quantum master equation (GQME) approach. However, the inclusion of multiple (light–matter) interactions complicates the equation of motion and leads to seemingly unavoidable cubic scaling in time. In this paper, we present a formulation that greatly simplifies and reduces the computational cost of previous work that extended the GQME framework to treat arbitrary numbers of quantum measurements. Specifically, we remove the time derivatives of quantum correlation functions from the modified Mori–Nakajima–Zwanzig framework by switching to a discrete-convolution implementation inspired by the transfer tensor approach. We then demonstrate the method’s capabilities by simulating 2D electronic spectra for the excitation-energy-transfer dimer model. In our method, the resolution of data can be arbitrarily coarsened, especially along the t2 axis, which mirrors how the data are obtained experimentally. Even in a modest case, this demands O(103) fewer data points. We are further able to decompose the spectra into one-, two-, and three-time correlations, showing how and when the system enters a Markovian regime where further measurements are unnecessary to predict future spectra and the scaling becomes quadratic. This offers the ability to generate long-time spectra using only short-time data, enabling access to timescales previously beyond the reach of standard methodologies.

Chemistry↗

DOE OSTI2023

Peridynamic elastic waves in two-dimensional unbounded domains: Construction of nonlocal Dirichlet-type absorbing boundary conditions

The focus of this paper is on application of peridynamics (PD) to propagation of elastic waves in unbounded domains. We construct absorbing boundary conditions (ABCs) derived from a semi-analytical solution of the PD governing equation at the exterior region. This solution is made up of a finite series of plane waves, as fundamental solutions (modes), which satisfy the PD dispersion relations. The modes are adjusted to transmit the energy from the interior region (near field) to the exterior region (far field). The corresponding unknown coefficients of the series are found in terms of the displacement field at a layer of points adjacent to the absorbing boundary. This is accomplished through a collocation procedure at subregions (clouds) around each absorbing point. The proposed ABCs offer appealing advantages, which facilitate their application to PD. They are of Dirichlet-type, hence their implementation is relatively simple as no derivatives of the field variables are required. They are constructed in the time and space domains and thus application of Fourier and Laplace transforms, cumbersome for nonlocal models, is not required. At the discrete level, the modes satisfy the same numerical dispersion relations of the near field, which makes the far-field solution compatible with that of the near field. We scrutinize the performance of the proposed ABCs through several examples. So our investigation shows that the proposed ABCs perform stably in time with an appropriate level of accuracy even in problems characterized by highly-dispersive propagating waves, including crack propagation in semi-unbounded brittle solids.

42 ENGINEERING↗

DOE OSTI2020

Discrete ordinates analysis of the forced-flight variance reduction technique in Monte Carlo neutral particle transport simulations

This paper presents mathematical formulations and methods to predict the effect of forced-flight variance reduction on Monte Carlo tally variance and calculation time. This includes deducing biasing operators that are then used to construct a history-score probability density function (HSPDF), which represents all possible Monte Carlo random walks and gives the probability of a Monte Carlo history scoring in a tally from a particular phase-space position. The history-score moment equations (HSMEs), the statistical moments of the HSPDF, are then derived to calculate the statistical behavior of the Monte Carlo tally when forced-flight variance reduction is applied. In addition, the future-time equation (FTE) is derived to predict the Monte Carlo computational time as a result of applying forced-flight variance reduction. The solutions of the HSMEs and FTE can be used to predict Monte Carlo computational cost. Furthermore, this work also describes a discrete ordinates method to solve the forced-flight HSMEs and FTE. Several 1-D and 2-D test problems verify that the derivations are performed and implemented correctly.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2023

Real-time control of connected vehicles in signalized corridors using pseudospectral convex optimization

Recent advances in Connected and Automated Vehicle (CAV) technologies have opened up new opportunities to enable safe, efficient, and sustainable transportation systems. However, developing reliable and rapid speed control algorithms in highly dynamic environments with complex inter-vehicle interactions and nonlinear vehicle dynamics is still a daunting task. In this paper, we develop a novel speed control method for CAVs to produce optimal speed profiles that minimize the fuel consumption and avoid idling at signalized intersections. To this end, an optimal control problem is formulated using the information of the upcoming traffic signal to adapt vehicles' speeds to avoid frequent stop-and-go driving patterns. Here, by applying the pseudospectral discretization method and the sequential convex programming method, the computational efficiency is greatly improved, enabling potential real-time on-vehicle applications. In addition, the algorithm is implemented under a model predictive control framework to ensure online control with instant response for collision avoidance and robust vehicle coordination. The proposed algorithm is verified through numerical simulations of three different traffic scenarios. The convergence and accuracy of the proposed approach are demonstrated by comparing with a popular nonlinear solver. Furthermore, the benefit of the proposed method in both traffic mobility and fuel efficiency is validated using the speed profile determined from a traffic following model in a simulation software as the baseline.

42 ENGINEERING↗

DOE OSTI2023

Compression of tokamak boundary plasma simulation data using a maximum volume algorithm for matrix skeleton decomposition

This report demonstrates satisfactory data compression of SOLPS-ITER simulation output ranging from 2D fields, 1D profiles, and 0D scalar variables with a novel matrix decomposition approach. The singular value decomposition (SVD) scales poorly for large matrix sizes and is unsuited to the application on high dimensional data common to fusion plasma physics simulation. In this work, we employ the columns-submatrix-rows (CUR) matrix factorization technique in order to compute a low-rank approximation up to two orders of magnitude faster than the SVD, but within a nominal L2-norm relative error of ε = 10 –2 . In addition, the CUR approach maintains the original format of the data, in its extracted columns and rows, allowing for interpretable data storage at the original resolution of the simulation. We utilize an iterative algorithm to compute the CUR decomposition of simulation output by maximizing the volume, or linearly independent information content, of a low-rank submatrix contained within the data. Experiments over $\textit{n} × \textit{n}$ randomized test matrices with embedded rank-deficient features show that this maximum volume implementation of CUR matrix approximation has reduced asymptotic computational complexity on the order of n compared to the SVD, which scales approximately as $n^3$. These results show that the CUR technique can be used to effectively select time step snapshots (columns) of over 140 SOLPS-ITER output variables and the associated discretized coordinate timeseries (rows) allowing for reconstruction of the complete simulation dynamics.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2022

Primitive quantum gates for dihedral gauge theories

We describe the simulation of dihedral gauge theories on digital quantum computers. The nonabelian discrete gauge group $D_N$-- the dihedral group -- serves as an approximation to $U(1)\times\mathbb{Z}_2$ lattice gauge theory. In order to carry out such a lattice simulation, we detail the construction of efficient quantum circuits to realize basic primitives including the nonabelian Fourier transform over $D_N$, the trace operation, and the group multiplication and inversion operations. For each case the required quantum resources scale linearly or as low-degree polynomials in $n=\log N$. We experimentally benchmark our gates on the Rigetti Aspen-9 quantum processor for the case of $D_4$. The fidelity of all $D_4$ gates was found to exceed $80\%$.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2021

Comparison of NMC estimates with trans-stilbene, EJ-309, and He-3 detection systems

Neutron multiplicity counting (NMC) is a technique for the assay of fissile material. In this work, three detection systems are utilized for active interrogation assay of shells of the Rocky Flats shells (highly enriched uranium, 93% 235U) stacked from 13.25-54.92 kg assemblies. The singles ($R_1$) and doubles ($R_2$) rates are calculated with each system to estimate two sample parameters: $M_L$- the leakage multiplication and F - the sample fission rate. The estimated mass, m, is found by dividing F by a constant activity per unit mass. Since we are interrogating HEU, the α ratio of (α; n) to fission neutrons is taken to be zero. The system of equations to calculate these quantities was originally derived. The Neutron Multiplicity 3 He Array Detector (NOMAD) consists of 15 3 He tubes inside polyethylene and is the traditional, capture-based detection system for NMC. The polyethylene moderates incoming neutrons for thermal capture in individual tubes. The low gamma background, discrete capture signals, and high efficiency of the NOMAD are beneficial for NMC. However, the time to slow down neutrons to thermal energies leads to a system die away time on the scale of microseconds. Comparatively, the Rossi-alpha Measurements – Rapid Organic (n, γ) Discrimination Detector (RAMRODD) and the Organic Scintillator Array (OSCAR) are scatter-based detection systems. RAMRODD consists of 8, 5.08 by 5.08 cm EJ-309 liquid scintillators in 4 pairs, evenly-spaced with 90 degrees separation about the center of each assembly. OSCAR consists of a single array of 12, 5.08 by 5.08 cm trans-stilbene crystals aligned with the center of the assembly. Scatter-based systems detect fast neutrons without any moderation, leading to system die away times on the scale of tens of nanoseconds. This allows much shorter timing gate widths compared to thermal systems, thus increasing counting statistics of true correlated fission events. However, scatter-based systems are susceptible to neutron cross-talk when an incident neutron scatters off one detector and interacts in an adjacent detector, causing two seemingly correlated detection signals. The equations developed adjust for cross-talk to conduct NMC with scatter-based systems.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗

DOE OSTI2025

Geometric Interpretation of the Cluster Location Problem Part I: Theory

We present a new framing of the seismic location problem using principles drawn from differential geometry. Our interpretation relies upon the common assumption that travel times observed across a network are continuous, differentiable functions of source location. In consequence, travel‐time functions constitute a differentiable map between the source region and a Riemannian manifold. The manifold is said to be the image of the source region embedded in a generally high‐dimension travel‐time vector space. A cluster of events in the source region has an image of discrete points on the manifold, that, except in the simplest cases, cannot be viewed directly. However, it is possible to project the image of a cluster into a tangent space of the manifold for direct visualization. The projection operator can be computed directly from the data without a velocity model, but produces a distorted rendering of the cluster geometry. With a model we can predict the distortions and correct them to estimate cluster geometry. We develop these points with the simplest possible example, one for which direct visualization of the manifold is possible, using the example as an introduction to the relevant concepts from differential geometry in a familiar setting. The tangent space, a local linearization of the manifold, plays a key role. We develop a metric to estimate the limits of linearization, that is, to determine when the curvature of the manifold invalidates the linear assumption. We also examine the interplay of model error, inadequate network geometry, and pick error. We then generalize our results from the simple case to the general case of 3D source regions observed by general networks. Although we do suggest a new “project and correct” method for location, we do not develop it into a practical algorithm. In conclusion, our intention rather is to highlight new analytical methods grounded in differential geometry.

East Pacific Ocean Islands↗

DOE OSTI2021

Port and optimize the CEED software stack to Aurora/Frontier EA (ECP Milestone Report)

The goal of this milestone was to port the CEED software stack, including Nek, MFEM and libCEED to the Frontier and Aurora early access hardware, work on optimizing the performance on AMD and Intel GPUs, and demonstrate impact in CEED-enabled ECP applications. As part of this milestone, we also performed a number of other activities, including continued optimizations for CEED applications at scale on Summit, performance evaluation on non-ECP hardware, including the Fugaku’s A64FX chip and the NVIDIA A100 GPU architecture, exploring the use of just-in-time compilations in applications at scale and potential of mixed precision optimizations in the CEED discretization and solver algorithms, and more. During the milestone period, the CEED team released new versions of 6 of its packages, including major releases of MFEM, NekRS and OCCA. We also organized the fifth CEED Annual meeting (CEED5AM) which included nearly 100 researchers from national labs, universities and industry.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2026

Hofmann Stability Charts Revisited for PIP-II: From Classical Theory to Assumption-Free and ML-Driven Maps

The Hofmann stability chart remains a standard for visualizing parametric resonances in space-charge–dominated linacs, but its use typically relies on non-oscillatory Vlasov dispersion relations with simplifying assumptions (continuous focusing, KV phase space, linear optics, limited transverse–longitudinal coupling). We revisit the chart for the PIP-II linac along three tracks. (1) We reproduce the conventional maps in the (νz/νx, νx/ν0x) plane for relevant εz/εx, providing a validated reference. (2) We remove key assumptions by deriving stability surfaces directly from multi-particle tracking with realistic lattice discreteness, RF defocusing, solenoid/quad optics, and bunched-beam dynamics; local tunes and early-time growth rates are estimated from envelope oscillations and projected to the same coordinates. These assumption-reduced maps recover the canonical stopbands while revealing shifts and broadenings driven by tune modulation, non-KV distributions, and transverse–longitudinal coupling at PIP-II intensities. (3) We train a compact machine-learning surrogate that emulates the growth surface from zero-current optics, tune depression, emittance ratio, bunching factor, and selected lattice descriptors, enabling rapid scans and online working-point selection. We compare the three representations on representative PIP-II sections and discuss implications for commissioning guard bands, resonance avoidance, and routine operations.

Pathak, Abhishek [Fermilab] (ORCID:000000021704208↗

DOE OSTI2022

Low power implementation of high frequency SiPM readout for Cherenkov and scintillation detectors in TOF-PET

State-of-the-art (SoA) electronic readout for silicon photomultiplier (SiPM)-based scintillation detectors that demonstrate experimental limits in achievable coincidence time resolution (CTR) leverage low noise, high frequency signal processing to facilitate a single photon time response that is near the limit of the SiPMs architecture. This readout strategy can optimally exploit fast luminescence and prompt photon populations, and promising measurements show detector concepts employing this readout can greatly advance PET detector CTR, relative to SoA in clinical systems. However, the technique employs power hungry components which make the electronics chain impractical for channel-dense time-of-flight (TOF)-PET detectors. We have developed and tested a low noise and high frequency readout circuit which is performant at low power and consists of discrete elements with small footprints, making it feasible for integration into TOF-PET detector prototypes. A 3 × 3 mm 2 Broadcom SiPM with this readout chain exhibited sub-100 ps single photon time resolution at 10 mW of power consumption, with a relatively minor performance degradation to 120 ± 2 ps FWHM at 5 mW. CTR measurements with 3 × 3 × 20 mm 3 LYSO and fast LGSO scintillators demonstrated 127 ± 3 ps and 113 ± 2 ps FWHM at optimal power operation and 133 ± 2 ps and 121 ± 3 ps CTR at 5 mW. BGO crystals 3 × 3 × 20 mm 3 in size show 271 ± 5 ps FWHM CTR (1174 ± 14 ps full-width-at-tenth-maximum (FWTM)) at optimal power dissipation and 289 ± 8 ps (1296 ± 33 ps FWTM) at 5 mW. The compact and low power readout topology that achieves this performance thereby offers a platform to greatly advance PET system CTR and also opportunities to provide high performance TOF-PET at reduced material cost.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗