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DOE OSTI2023

DeepONet-grid-UQ: A trustworthy deep operator framework for predicting the power grid’s post-fault trajectories

This paper proposes a novel data-driven method for the reliable prediction of the power grid’s post-fault trajectories, i.e., the power grid’s dynamic response after a disturbance or fault. Here, the proposed method is based on the recently proposed concept of Deep Operator Networks (DeepONets). Unlike traditional neural networks that learn to approximate functions, DeepONets are designed to approximate nonlinear operators, i.e., mappings between infinite-dimensional spaces. Under this operator framework, we design a novel and efficient DeepONet that (i) takes as inputs the trajectories collected before and during the fault and (ii) outputs the predicted post-fault trajectories. In addition, we endow our method with the much-needed ability to balance efficiency with reliable/trustworthy predictions via uncertainty quantification. To this end, we propose and compare two novel methods that enable quantifying the predictive uncertainty. First, we propose a Bayesian DeepONet (B-DeepONet) that uses stochastic gradient Hamiltonian Monte-Carlo to sample from the posterior distribution of the DeepONet trainable parameters. Then, we design a Probabilistic DeepONet (Prob-DeepONet) that uses a probabilistic training strategy to enable quantifying uncertainty at virtually no extra computational cost. Finally, we validate the proposed methods’ predictive power and uncertainty quantification capability using the New York-New England power grid model.

24 POWER TRANSMISSION AND DISTRIBUTION↗

DOE OSTI2026

Machine learning approach for vibronically renormalized electronic band structures

Here, we present a machine learning (ML) method for efficient computation of vibrational thermal expectation values of physical properties from first principles. Our approach is based on the nonperturbative frozen phonon formulation in which stochastic Monte Carlo algorithm is employed to sample configurations of nuclei in a supercell at finite temperatures based on a first-principles phonon model. A deep-learning neural network is trained to accurately predict physical properties associated with sampled phonon configurations, thus bypassing the time-consuming ab initio calculations. To incorporate the point-group symmetry of the electronic system into the ML model, group-theoretical methods are used to develop a symmetry-invariant descriptor for phonon configurations in the supercell. We apply our ML approach to compute the temperature dependent electronic energy gap of silicon based on density functional theory (DFT). We show that, with less than a hundred DFT calculations for training the neural network model, an order of magnitude larger number of sampling can be achieved for the computation of the vibrational thermal expectation values. Our work highlights the promising potential of ML techniques for finite temperature first-principles electronic structure methods.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

NASA NTRS1990

Randomly Sampled-Data Control Systems

The purpose is to solve the Linear Quadratic Regulator (LQR) problem with random time sampling. Such a sampling scheme may arise from imperfect instrumentation as in the case of sampling jitter. It can also model the stochastic information exchange among decentralized controllers to name just a few. A practical suboptimal controller is proposed with the nice property of mean square stability. The proposed controller is suboptimal in the sense that the control structure is limited to be linear. Because of i. i. d. assumption, this does not seem unreasonable. Once the control structure is fixed, the stochastic discrete optimal control problem is transformed into an equivalent deterministic optimal control problem with dynamics described by the matrix difference equation. The N-horizon control problem is solved using the Lagrange's multiplier method. The infinite horizon control problem is formulated as a classical minimization problem. Assuming existence of solution to the minimization problem, the total system is shown to be mean square stable under certain observability conditions. Computer simulations are performed to illustrate these conditions.

Han, Kuoruey↗

DOE OSTI2020

A Dynamic Amplitude-Correcting Gradient Estimation Technique to Align X-ray Focusing Optics

High-brightness X-rays, as produced at synchrotrons and X-ray free electron laser (XFEL) facilities, are used to characterize materials in a variety of scientific experiments. In most cases, effective use of the high-energy light requires precisely-aligned focusing optics; one example being a compound refractive lens (CRL). To align a CRL, the position and rotation must be optimized along four axes. In practice, this is a labor-intensive, time-consuming manual process that can monopolize scarce experimental time at the necessary X-ray facilities. Models of the expected Xray transmission function suggest that this task can be automated; however, the temporally-varying intensity at X-ray free electron laser facilities preclude the direct use of standard implementations of optimization solvers such as steepest descent algorithms. In this paper, we propose a novel technique to estimate the gradient of noisy functions with temporally-varying amplitudes. We construct this dynamicamplitude correction by systematically sampling a fixed central location within the standard finite difference stencil, accounting for the observed changes in time, and normalizing the difference quotients against those fluctuations. In addition to a rigorous error analysis of the sampling technique, we demonstrate its efficacy in stochastic descent optimization methods. Further, we demonstrate how this approach may be implemented to optimize X-ray focusing optics at synchrotrons or XFEL facilities

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

Adaptive Uncertainty Quantification for Stochastic Hyperbolic Conservation Laws

Here, we propose a predictor-corrector adaptive method for the study of hyperbolic partial differential equations (PDEs) under uncertainty. Constructed around the framework of stochastic finite volume (SFV) methods, our approach circumvents sampling schemes or simulation ensembles while also preserving fundamental properties, in particular hyperbolicity of the resulting systems and conservation of the discrete solutions. Furthermore, we augment the existing SFV theory with a priori convergence results for statistical quantities, in particular push-forward densities, which we demonstrate through numerical experiments. By linking refinement indicators to regions of the physical and stochastic spaces, we drive anisotropic refinements of the discretizations, introducing new degrees of freedom where deemed profitable. To illustrate our proposed method, we consider a series of numerical examples for nonlinear hyperbolic PDEs based on Burgers’ and Euler’s equations.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1999

Efficient Heuristic Hypothesis Ranking

This paper considers the problem of learning the ranking of a set of stochastic alternatives based upon incomplete information (i.e., a limited number of samples).

machine↗

DOE OSTI2021

Quantum-assisted associative adversarial network: applying quantum annealing in deep learning

Abstract Generative models have the capacity to model and generate new examples from a dataset and have an increasingly diverse set of applications driven by commercial and academic interest. In this work, we present an algorithm for learning a latent variable generative model via generative adversarial learning where the canonical uniform noise input is replaced by samples from a graphical model. This graphical model is learned by a Boltzmann machine which learns low-dimensional feature representation of data extracted by the discriminator. A quantum processor can be used to sample from the model to train the Boltzmann machine. This novel hybrid quantum-classical algorithm joins a growing family of algorithms that use a quantum processor sampling subroutine in deep learning, and provides a scalable framework to test the advantages of quantum-assisted learning. For the latent space model, fully connected, symmetric bipartite and Chimera graph topologies are compared on a reduced stochastically binarized MNIST dataset, for both classical and quantum sampling methods. The quantum-assisted associative adversarial network successfully learns a generative model of the MNIST dataset for all topologies. Evaluated using the Fréchet inception distance and inception score, the quantum and classical versions of the algorithm are found to have equivalent performance for learning an implicit generative model of the MNIST dataset. Classical sampling is used to demonstrate the algorithm on the LSUN bedrooms dataset, indicating scalability to larger and color datasets. Though the quantum processor used here is a quantum annealer, the algorithm is general enough such that any quantum processor, such as gate model quantum computers, may be substituted as a sampler.

Wilson, Max (ORCID:0000000207983391)↗

DOE OSTI2020

UQpy: A general purpose Python package and development environment for uncertainty quantification

In this paper, we present the UQpy software toolbox, an open-source Python package for general uncertainty quantification (UQ) in mathematical and physical systems. The software serves as both a user-ready toolbox that includes many of the latest methods for UQ in computational modeling and a convenient development environment for Python programmers advancing the field of UQ. The paper presents an introduction to the software's architecture and existing capabilities, divided in the code in a set of modules centered around different UQ tasks such as sampling methods, generation of random processes and random fields, probabilistic inverse modeling, reliability analysis, surrogate modeling, and active learning. The paper also highlights the importance of the RunModel module, which is used to drive simulations in the uncertainty analyses performed in UQpy. This module conveniently allows the user to define computational models directly in Python, or to run simulations from a third-party software in serial or in parallel. To illustrate the various capabilities, two examples are tracked throughout the paper and analyzed repeatedly for various UQ tasks. The first is a Python model solving a nonlinear structural dynamics problem, used to illustrate UQpy's capabilities in sampling and forward propagation of high dimensional random vectors (stochastic processes), and probabilistic inference. The second model is a third-party Abaqus finite element model solving the thermomechanical response of a beam structure. This example is used to illustrate UQpy's capabilities in variance reduction sampling techniques, reliability analysis, surrogate modeling and active learning techniques.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

Stochastic AC optimal power flow: A data-driven approach

There is an emerging need for efficient solutions to stochastic AC Optimal Power Flow (AC-OPF) to ensure optimal and reliable grid operations in the presence of increasing demand and generation uncertainty. Herein this paper presents a highly scalable data-driven algorithm for stochastic AC-OPF that has extremely low sample requirement. The novelty behind the algorithm’s performance involves an iterative scenario design approach that merges information regarding constraint violations in the system with data-driven sparse regression. Compared to conventional methods with random scenario sampling, our approach is able to provide feasible operating points for realistic systems with much lower sample requirements. Furthermore, multiple sub-tasks in our approach can be easily paralleled and based on historical data to enhance its performance and application. We demonstrate the computational improvements of our approach through simulations on different test cases in the IEEE PES PGLib-OPF benchmark library.

42 ENGINEERING↗

DOE OSTI2025

Efficient Training of Deep Neural Operator Networks via Randomized Sampling

Neural operators (NOs) employ deep neural networks to learn the mappings between infinitedimensional function spaces. Deep operator network (DeepONet), a popular NO architecture, has demonstrated success in the real-time prediction of complex dynamics across various scientific and engineering applications. In this work, we introduce a random sampling technique to be adopted during the training of DeepONet, aimed at improving the generalization ability of the model, while significantly reducing the computational time. The proposed approach targets the trunk network of the DeepONet model that outputs the basis functions corresponding to the spatiotemporal locations of the bounded domain on which the physical system is defined. While constructing the loss function, DeepONet training traditionally considers a uniform grid of spatiotemporal points at which all the output functions are evaluated for each iteration. This approach leads to a larger batch size, resulting in poor generalization and increased memory demands, due to the limitations of the stochastic gradient descent (SGD) optimizer. The proposed random sampling over the inputs of the trunk net mitigates these challenges, improving generalization and reducing the memory requirements during training, resulting in significant computational gains. We validate our hypothesis through three benchmark examples, demonstrating substantial reductions in training time while achieving comparable or lower overall test errors relative to the traditional training approach. Our results indicate that incorporating randomization in the trunk network inputs during training enhances the efficiency and robustness of DeepONet, offering a promising avenue for improving the framework’s performance in modeling complex physical systems.

Karumuri, Sharmila [Department of Civil & Systems ↗

DOE OSTI2021

Towards a Robust Sampling Approach: A Computational Review and Design

As pointed out in several other works, the estimation of the reliability of the electrical grid can not be conducted without the estimation of the stochastic phenomena of electicity demand and electricity production by variable renewable sources. Therefore, sampling procedures have become integral in the design of engineering structures and analysis. Commonly referred to as Monte Carlo uncertainty analysis or integration, the general objective of these procedures is to establish specifics about the uncertainty of an output characteristic of such an engineering system, given uncertainty about its input characteristics. These sampling procedures are applied in a context in which establishing such specifics cannot be performed through other means. Variance-reduction techniques are designed to lessen the variability among estimators to estimate statistics of those output uncertainties. Importance-sampling techniques, on the other hand, are designed to reduce the number of samples needed to estimate a particular statistic—e.g., a tail probability. The combination of these approaches can reduce the computational burden considerably for a particular estimator and statistic. Importance sampling—geared and designed as it is toward improving a particular estimator—suffers, unfortunately, from the unintended consequence of reducing the performance of other estimators in terms of their variance. The objective of this paper is to offer an alternative sampling procedure where this variance does not grow unacceptably large for a suite of estimators. Moreover, it is anticipated that, with additional knowledge of how an engineering output characteristic responds to its input characteristics, tuning parameters of the input’s sampling procedure can be set to improve the output characteristic’s estimation. This work proves the effectiveness of the suggested alternative approach. Such positive outcome will lead to a decrease of the computational burden of performing stochastic optimization of integrated energy systems (e.g., components dispatch, and portfolio composition). In particular, capturing the contribution to the overall system cost of rare and unlikely events and patterns of the electricity demand and production will become less computationally expensive. This is due to the fact that the approach demonstrated here will allow the sampling of those rare occurrences more frequently without misrepresenting their probabilistic impacts.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗

NASA NTRS2004

Fossil Signatures Using Elemental Abundance Distributions and Bayesian Probabilistic Classification

Elemental abundances (C6, N7, O8, Na11, Mg12, Al3, P15, S16, Cl17, K19, Ca20, Ti22, Mn25, Fe26, and Ni28) were obtained for a set of terrestrial fossils and the rock matrix surrounding them. Principal Component Analysis extracted five factors accounting for the 92.5% of the data variance, i.e. information content, of the elemental abundance data. Hierarchical Cluster Analysis provided unsupervised sample classification distinguishing fossil from matrix samples on the basis of either raw abundances or PCA input that agreed strongly with visual classification. A stochastic, non-linear Artificial Neural Network produced a Bayesian probability of correct sample classification. The results provide a quantitative probabilistic methodology for discriminating terrestrial fossils from the surrounding rock matrix using chemical information. To demonstrate the applicability of these techniques to the assessment of meteoritic samples or in situ extraterrestrial exploration, we present preliminary data on samples of the Orgueil meteorite. In both systems an elemental signature produces target classification decisions remarkably consistent with morphological classification by a human expert using only structural (visual) information. We discuss the possibility of implementing a complexity analysis metric capable of automating certain image analysis and pattern recognition abilities of the human eye using low magnification optical microscopy images and discuss the extension of this technique across multiple scales.

Hoover, Richard B.↗

DOE OSTI2020

Benders Cut Classification via Support Vector Machines for Solving Two-Stage Stochastic Programs

In this work, we consider Benders decomposition for solving two-stage stochastic programs with complete recourse based on finite samples of the uncertain parameters. We define the Benders cuts binding at the final optimal solution or the ones significantly improving bounds over iterations as valuable cuts. We propose a learning-enhanced Benders decomposition (LearnBD) algorithm, which adds a cut classification step in each iteration to selectively generate cuts that are more likely to be valuable cuts. The LearnBD algorithm includes two phases: (i) sampling cuts and collecting information from training problems and (ii) solving testing problems with a support vector machine (SVM) cut classifier. We run the LearnBD algorithm on instances of capacitated facility location and multicommodity network design under uncertain demand. Our results show that SVM cut classifier works effectively for identifying valuable cuts, and the LearnBD algorithm reduces the total solving time of all instances for different problems with various sizes and complexities.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

Global stochastic optimization of stellarator coil configurations

In the construction of a stellarator, the manufacturing and assembling of the coil system is a dominant cost. These coils need to satisfy strict engineering tolerances, and if those are not met the project could be cancelled as in the case of the National Compact Stellarator Experiment (NCSX) project. Therefore, our goal is to find coil configurations that increase construction tolerances without compromising the performance of the magnetic field. In this paper, we develop a gradient-based stochastic optimization model which seeks robust stellarator coil configurations in high dimensions. In particular, we design a two-step method: first, we perform an approximate global search by a sample efficient trust-region Bayesian optimization; second, we refine the minima found in step one with a stochastic local optimizer. To this end, we introduce two stochastic local optimizers: BFGS applied to the sample average approximation; and Adam, equipped with a control variate for variance reduction. Numerical simulations performed on a W7-X-like coil configuration demonstrate that our global optimization approach finds a variety of promising local solutions at less than 0.1% of the cost of previous work, which considered solely local stochastic optimization.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2025

Demonstration of Optimal Benchmark Selection Website and Validation of the q c Coverage Metric Using HEU-SOL-THERM-013-003 Experiment

In the work documented in this interim report, the experiment selection toolkit web site was demonstrated and q C coverage metric methodology was validated for IEU-MET-FAST-002-001, MIX-COMP-THERM 004-004, and HEU-SOL-THERM-013-003 experiments. 𝑞 𝐶 is an information-theoretic measure based on mutual information that quantifies the ability of candidate benchmark experiments to reduce the bias and uncertainty of a target criticality safety application. The metric and an accompanying open-source Python toolkit with a web-based interface were tested against a benchmark set of 425 experiments drawn from the International Criticality Safety Benchmark Evaluation Project Handbook. The interface is hosted at https://edim.covdef.com. It accepts sensitivity data files produced by the TSUNAMI-IP module of the SCALE code system and supports both (i) deterministic analysis using the ENDF/B-VII.0 covariance library and (ii) stochastic analysis based on user-supplied keff samples. Demonstrations on representative applications across a range of material composition, spectrum, and form show that q C -guided benchmark selection achieves greater uncertainty reduction with fewer experiments and yields more stable posterior bias and uncertainty estimates than traditional similarity coefficient ( c k )–based selection, while also capturing valuable low-ck experiments that one-to-one metrics overlook.

Abdel-khalik, Hany S. [Indiana Univ.-Purdue Univ. ↗

DOE OSTI2022

Learning Stochastic Parametric Differentiable Predictive Control Policies

We present a scalable unsupervised learning-based method for obtaining explicit control policies for model predictive control problems for stochastic linear systems with additive uncertainties subject to nonlinear chance constraints. We call the proposed method stochastic parametric differentiable predictive control (SP-DPC), which extends the recently proposed deterministic DPC policy optimization algorithm. We formulate the SP-DPC as a deterministic approximation to the stochastic parametric constrained optimal control problem via independent sampling of the problem's parameters and uncertainties. This formulation allows us to directly compute the policy gradients via automatic differentiation of the problem's value function, evaluated over sampled parameters and uncertainties. In particular, the computed expectation of the problem's value function is backpropagated through the finite-time closed-loop system rollouts parametrized by a known nominal system dynamics model and neural control policy. We also provide theoretical probabilistic guarantees on closed-loop stability and chance constraints satisfaction for systems controlled by learned neural policies. We demonstrate the computational efficiency and scalability of the proposed policy optimization algorithm in three numerical examples, including systems with a large number of states or subject to nonlinear constraints.

Drgona, Jan↗

NASA NTRS2021

Multi-Pass Sequential Mini-Batch Stochastic Gradient Descent Algorithms for Noise Covariance Estimation in Adaptive Kalman Filtering

Estimation of unknown noise covariances in a Kalman filter is a problem of significant practical interest in a wide array of applications. Although this problem has a long history, reliable algorithms for their estimation were scant, and necessary and sufficient conditions for identifiability of the covariances were in dispute until recently. Necessary and sufficient conditions for covariance estimation and a batch estimation algorithm were presented in our previous study. This paper presents stochastic gradient descent algorithms for noise covariance estimation in adaptive Kalman filters that are an order of magnitude faster than the batch method for similar or better root mean square error. More significantly, these algorithms are applicable to non-stationary systems where the noise covariances can occasionally jump up or down by an unknown magnitude. The computational efficiency of the new algorithms stems from adaptive thresholds for convergence, recursive fading memory estimation of the sample cross-correlations of the innovations, and accelerated stochastic gradient descent algorithms. The comparative evaluation of the proposed methods on a number of test cases demonstrates their computational efficiency and accuracy.

Adaptive Kalman filtering↗

DOE OSTI2026

Stochastic Thermo-Hydro Modeling and Neural Network Surrogate Development for Thermal Resource Assessment of the Galleries-to-Calories Geobattery

The Galleries-to-Calories Geobattery concept explores the use of abandoned coal mine workings for large-scale thermal energy transport and storage. The system involves injecting waste heat from a supercomputing facility into flooded mine galleries, where groundwater flow can store and transport thermal energy for potential recovery in downgradient district heating and cooling applications. To evaluate the feasibility and performance of the Geobattery under geological and operational uncertainty, we developed a suite of stochastic thermo-hydrological (TH) simulations using Monte Carlo sampling of key uncertain parameters (e.g., permeability, porosity, thermal conductivity, specific heat capacity) and operating conditions (e.g., injection rate, injection temperature). Results identified injection rate and temperature as the most influential parameters governing thermal front propagation, while the geometry of the room-and-pillar structure played a critical role in directing the extent and orientation of thermal advancement. Optimal combinations of material properties for maximizing heat recovery were also determined. To address the high computational cost of coupled-process stochastic modeling, we trained a neural network surrogate model on 24,000 physics-based realizations, achieving an R² > 0.99 and MAE < 0.1 for temperature predictions at monitoring locations. This surrogate enabled an additional 100,000 realizations for global sensitivity analysis and probabilistic thermal resource assessment. The integrated stochastic physics–surrogate modeling framework offers a computationally efficient tool for quantifying uncertainty, identifying key drivers, and informing early-stage design decisions for Geobattery systems.

15 - GEOTHERMAL ENERGY↗