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All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

Compactly‐Supported Nonstationary Kernels for Computing Exact Gaussian Processes on Big Data

The Gaussian process (GP) is a widely used method for analyzing large-scale data sets, including spatio-temporal measurements of nonlinear processes that are now commonplace in the environmental sciences. Traditional implementations of GPs involve stationary kernels (also termed covariance functions) that limit their flexibility, and exact methods for inference that prevent application to data sets with more than about 10,000 points. Modern approaches to address stationarity assumptions generally fail to accommodate large data sets, while all attempts to address scalability focus on approximating the Gaussian likelihood, which can involve subjectivity and lead to inaccuracies. In this work, we explicitly derive an alternative kernel that can discover and encode both sparsity and nonstationarity. We embed the kernel within a fully Bayesian GP model and leverage high-performance computing resources to enable the analysis of massive data sets. We demonstrate the favorable performance of our novel kernel relative to existing exact and approximate GP methods across a variety of synthetic data examples. Furthermore, we conduct space–time prediction based on more than 1 million measurements of daily maximum temperature and verify that our results outperform state-of-the-art methods in the Earth sciences. More broadly, having access to exact GPs that use ultra-scalable, sparsity-discovering, nonstationary kernels allows GP methods to truly compete with a wide variety of machine learning methods.

Gaussian processes↗
DOE OSTI2023

A localized ensemble of approximate Gaussian processes for fast sequential emulation

More attention has been given to the computational cost associated with the fitting of an emulator. Substantially less attention is given to the computational cost of using that emulator for prediction. This is primarily because the cost of fitting an emulator is usually far greater than that of obtaining a single prediction, and predictions can often be obtained in parallel. In many settings, especially those requiring Markov Chain Monte Carlo, predictions may arrive sequentially and parallelization is not possible. In this case, using an emulator procedure which can produce accurate predictions efficiently can lead to substantial time savings in practice. In this paper, we propose a global model approximate Gaussian process framework via extension of a popular local approximate Gaussian process (laGP) framework. Our proposed emulator can be viewed as a treed Gaussian process where the leaf nodes are laGP models, and the tree structure is learned greedily as a function of the prediction stream. The suggested method (called leapGP) has interpretable tuning parameters which control the time‐memory trade‐off. One reasonable choice of settings leads to an emulator with a training cost and makes predictions rapidly with an asymptotic amortized cost of .

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Online Learning of Effective Turbine Wind Speed in Wind Farms

To develop better wind farm controllers that can meet more complex objectives, methods of modeling the wind turbine wakes at low computational expense are needed. Gaussian process (GP) regression offers a computationally inexpensive framework for learning complex functions from noisy measurements with very few datapoints. In this work, an online learning approach is presented to learn the rotor-averaged wind velocity at downstream wind turbines with GPs, using the available datastream of wind field measurements and wind turbine control set-points. This framework can readily be integrated into model-based controls methods because the model a) is updated online at low computational expense, b) assumes a mathematically favorable Gaussian form, and c) explicitly quantifies the stochastic nature of the wake field so that the trade-off between exploration and exploitation, and the uncertainty in the prediction, can be utilized. We show that a GP-learned model can match true values with errors within 0.5% on average, with as few as 5 training data points.

Gaussian process↗
DOE OSTI2024

Non-smooth Bayesian optimization in tuning scientific applications

Tuning algorithmic parameters to optimize the performance of large, complicated computational codes is an important problem involving finding the optima and identifying regimes defined by non-smooth boundaries in black-box functions. Within the Bayesian optimization framework, the Gaussian process surrogate model produces smooth mean functions, but functions in the tuning problem are often non-smooth, which is exacerbated by the fact that we usually have limited sequential samples from the black-box function. Here, motivated by these issues encountered in tuning, we propose a novel Gaussian process model called a clustered Gaussian process (cGP), where the components are dynamically updated by clustering. In our studies, the performance of cGP can be better than stationary GPs in nearly 90% of the experiments and better than non-stationary GPs in nearly 70% of the repeated experiments while requiring less computational cost. cGP provides a novel approach for dynamic GP, computes more efficiently than recursive partitioning, and discovers non-smoothness regimes. We provide extensive experiments including high-performance computing (HPC) and industrial simulation functions to show the effectiveness of our methods.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Recipes for when physics fails: recovering robust learning of physics informed neural networks

Abstract Physics-informed neural networks (PINNs) have been shown to be effective in solving partial differential equations by capturing the physics induced constraints as a part of the training loss function. This paper shows that a PINN can be sensitive to errors in training data and overfit itself in dynamically propagating these errors over the domain of the solution of the PDE. It also shows how physical regularizations based on continuity criteria and conservation laws fail to address this issue and rather introduce problems of their own causing the deep network to converge to a physics-obeying local minimum instead of the global minimum. We introduce Gaussian process (GP) based smoothing that recovers the performance of a PINN and promises a robust architecture against noise/errors in measurements. Additionally, we illustrate an inexpensive method of quantifying the evolution of uncertainty based on the variance estimation of GPs on boundary data. Robust PINN performance is also shown to be achievable by choice of sparse sets of inducing points based on sparsely induced GPs. We demonstrate the performance of our proposed methods and compare the results from existing benchmark models in literature for time-dependent Schrödinger and Burgers’ equations.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Evaluating Gaussian process metamodels and sequential designs for noisy level set estimation

Abstract We consider the problem of learning the level set for which a noisy black-box function exceeds a given threshold. To efficiently reconstruct the level set, we investigate Gaussian process (GP) metamodels. Our focus is on strongly stochastic simulators, in particular with heavy-tailed simulation noise and low signal-to-noise ratio. To guard against noise misspecification, we assess the performance of three variants: (i) GPs with Student- t observations; (ii) Student- t processes (TPs); and (iii) classification GPs modeling the sign of the response. In conjunction with these metamodels, we analyze several acquisition functions for guiding the sequential experimental designs, extending existing stepwise uncertainty reduction criteria to the stochastic contour-finding context. This also motivates our development of (approximate) updating formulas to efficiently compute such acquisition functions. Our schemes are benchmarked by using a variety of synthetic experiments in 1–6 dimensions. We also consider an application of level set estimation for determining the optimal exercise policy of Bermudan options in finance.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Sequential Learning of Active Subspaces

This study shows that in recent years, active subspace methods (ASMs) have become a popular means of performing subspace sensitivity analysis on black-box functions. Naively applied, however, ASMs require gradient evaluations of the target function. In the event of noisy, expensive, or stochastic simulators, evaluating gradients via finite differencing may be infeasible. In such cases, often a surrogate model is employed, on which finite differencing is performed. When the surrogate model is a Gaussian process (GP), we show that the ASM estimator is available in closed form, rendering the finite-difference approximation unnecessary. We use our closed-form solution to develop acquisition functions focused on sequential learning tailored to sensitivity analysis on top of ASMs. We also show that the traditional ASM estimator may be viewed as a method of moments estimator for a certain class of GPs. We demonstrate how uncertainty on GP hyperparameters may be propagated to uncertainty on the sensitivity analysis, allowing model-based confidence intervals on the active subspace. Our methodological developments are illustrated on several examples.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

A Bayesian Learning Approach to Wireless Outdoor Heatmap Construction using Deep Gaussian Process

We present a novel Bayesian learning approach to outdoor radio heatmap construction utilizing deep Gaussian process (GP). The proposed approach employs a two-layer hierarchy which consists of two cascaded Gaussian processes that are capable of modeling more complex input-output relations than standard single-layer Gaussian processes. Since deriving the exact model likelihood is challenging, a lower bound is optimized instead so that gradient descent-based methods can be performed to find out the optimal model parameters. Typically, inducing points are used in GPs to facilitate low-rank approximation of covariance (kernel) matrices for computation speedup. However, the inaccuracy induced by inducing points can accumulate when stacking multiple layers of GP which may hinder the performance of deep GP. Moreover, since inducing points need to be learned, having them at all layers of deep GP also incurs computational burden. To overcome the above challenges, in contrast to the canonical deep GP model, we use a modified architecture where a full standard GP resides in the first layer and inducing points are only introduced for the second layer. This modified architecture strikes a balance between model accuracy and training complexity. In the proposed model, the noise parameter of the first GP layer is also eliminated to improve the training efficiency as the noise parameter at the output of the second layer suffices to model the uncertainty in the output. The proposed approach is evaluated on real-world datasets, in the form of location-Received Signal Strength (RSS) pairs, collected from the Platform for Open Wireless Data-driven Experimental Research (POWDER) located at the campus of the University of Utah. Experiment results show that the proposed approach can achieve smaller prediction errors on various training and testing data configurations than DNN-based and GP-based methods.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
DOE OSTI2025

Data-Efficient Strategies for Probabilistic Voltage Envelopes under Network Contingencies

This work presents an efficient data-driven method to construct probabilistic voltage envelopes (PVE) using power flow learning in grids with network contingencies. First, a network-aware Gaussian process (GP) termed Vertex-Degree Kernel (VDK-GP), developed in prior work, is used to estimate voltage–power functions for a few network configurations. The paper introduces a novel multi-task vertex degree kernel (MT-VDK) that amalgamates the learned VDK-GPs to determine power flows for unseen networks, with a significant reduction in the computational complexity and hyperparameter requirements compared to alternate approaches. Simulations on the IEEE 30-Bus network demonstrate the retention and transfer of power flow knowledge in both N-1 and N-2 contingency scenarios. The MT-VDK-GP approach achieves over 50 % reduction in mean prediction error for novel N-1 contingency network configurations in low training data regimes (50–250 samples) over VDK-GP. Additionally, MT-VDK-GP outperforms a hyper-parameter based transfer learning approach in over 75 % of N-2 contingency network structures, even without historical N-2 outage data. Furthermore, the proposed method demonstrates the ability to achieve PVEs using sixteen times fewer power flow solutions compared to Monte-Carlo sampling-based methods.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2022

Learning-Based Demand Response in Grid Interactive Buildings via Gaussian Processes

This paper presents a predictive controller for a grid-interactive multi-zone building where the temperature dynamics are learned via Gaussian Process (GP) regression. We investigate the development of a learning-based predictive control with two main objectives: (i) continuously learn the temperature dynamics of the building based on data; and, (ii) use the learned dynamics to solve a multi-objective predictive control problem to guarantee occupants' comfort and energy efficiency during normal conditions and demand response events. We leverage the probabilistic non-parametric properties of GPs to estimate the (unknown) non-linear temperature dynamics of the building and to incorporate the uncertainty of those predictions in a multi-objective optimization problem. The GP-based predictive control is solved via a zero-order primal-dual projected-gradient algorithm. We evaluate numerically the performance of the proposed controller using a five-zone commercial building.

demand response↗
NASA NTRS2022

Searching for Quasi-periodic Oscillations in Astrophysical Transients Using Gaussian Processes

Analyses of quasi-periodic oscillations(QPOs)are important to understanding the dynamic behavior in manyastrophysical objects during transient events like gamma-ray bursts, solarflares, magnetarflares, and fast radiobursts. Astrophysicists often search for QPOs with frequency-domain methods such as(Lomb–Scargle)periodograms, which generally assume power-law models plus some excess around the QPO frequency. Time-series data can alternatively be investigated directly in the time domain using Gaussian process(GP)regression.While GP regression is computationally expensive in the general case, the properties of astrophysical data andmodels allow fast likelihood strategies. Heteroscedasticity and nonstationarity in data have been shown to causebias in periodogram-based analyses. GPs can take account of these properties. Using GPs, we model QPOs as astochastic process on top of a deterministicflare shape. Using Bayesian inference, we demonstrate how to infer GPhyperparameters and assign them physical meaning, such as the QPO frequency. We also perform model selectionbetween QPOs and alternative models such as red noise and show that this can be used to reliablyfind QPOs. Thismethod is easily applicable to a variety of different astrophysical data sets. We demonstrate the use of this methodon a range of short transients: a gamma-ray burst, a magnetarflare, a magnetar giantflare, and simulated solarflare data.

Moritz Hubner↗
DOE OSTI2023

Exact Gaussian processes for massive datasets via non-stationary sparsity-discovering kernels

Abstract A Gaussian Process (GP) is a prominent mathematical framework for stochastic function approximation in science and engineering applications. Its success is largely attributed to the GP’s analytical tractability, robustness, and natural inclusion of uncertainty quantification. Unfortunately, the use of exact GPs is prohibitively expensive for large datasets due to their unfavorable numerical complexity of $$O(N^3)$$ O ( N 3 ) in computation and $$O(N^2)$$ O ( N 2 ) in storage. All existing methods addressing this issue utilize some form of approximation—usually considering subsets of the full dataset or finding representative pseudo-points that render the covariance matrix well-structured and sparse. These approximate methods can lead to inaccuracies in function approximations and often limit the user’s flexibility in designing expressive kernels. Instead of inducing sparsity via data-point geometry and structure, we propose to take advantage of naturally-occurring sparsity by allowing the kernel to discover—instead of induce—sparse structure. The premise of this paper is that the data sets and physical processes modeled by GPs often exhibit natural or implicit sparsities, but commonly-used kernels do not allow us to exploit such sparsity. The core concept of exact, and at the same time sparse GPs relies on kernel definitions that provide enough flexibility to learn and encode not only non-zero but also zero covariances. This principle of ultra-flexible, compactly-supported, and non-stationary kernels, combined with HPC and constrained optimization, lets us scale exact GPs well beyond 5 million data points.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Scalable Gaussian Processes, GPyTorch Application Benchmarking, and Targeted Adaptive Design (TAD) on ThetaGPU

We aim at showcasing the scalability of Gaussian Process (GP). The naive GP implementation scales cubically with data size, which can be prohibitive, so GP has not heretofore been considered suitable for very large-scale problem settings. We take advantage of GPyTorch, a library for scalable GPs built on top of PyTorch that incorporates GPU acceleration. With GPyTorch, one can achieve nearly linear scaling with structured kernel interpolation (SKI) and constant-time predictive covariances computation with LanczOs Variance Estimates (LOVE) while preserving accuracy. We also take advantage of the computational power of ThetaGPU, a supercomputer of Argonne Leadership Computing Facility (ALCF). In addition, we implement a scalable, GPU-ready version of Targeted Adaptive Design (TAD), a GP-based data-driven algorithm that efficiently searches the control space of an advanced manufacturing experiment for settings capable of producing a required design within a specified tolerance, despite the poorly known mapping from control settings to design. We finally show our benchmarking for GPyTorch and TAD performance on CPU vs. ThetaGPU and discuss the results and implications.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

MOOSE ProbML: Parallelized probabilistic machine learning and uncertainty quantification for computational energy applications

Here, this paper presents the development and demonstration of massively parallel probabilistic machine learning (ML) and uncertainty quantification (UQ) capabilities within the Multiphysics Object-Oriented Simulation Environment (MOOSE), an open-source computational platform for parallel finite element and finite volume analyses. In addressing the computational expense and uncertainties inherent in complex multiphysics simulations, this paper integrates Gaussian process (GP) variants, active learning, Bayesian inverse UQ, adaptive forward UQ, Bayesian optimization, evolutionary optimization, and Markov chain Monte Carlo (MCMC) within MOOSE. It also elaborates on the interaction among key MOOSE systems — Sampler, MultiApp, Reporter, and Surrogate — in enabling these capabilities. The modularity offered by these systems enables development of a multitude of probabilistic ML and UQ algorithms in MOOSE. Example code demonstrations include parallel active learning and parallel Bayesian inference via active learning. The impact of these developments is illustrated through five applications relevant to computational energy applications: UQ of nuclear fuel fission product release, using parallel active learning Bayesian inference; very rare events analysis in nuclear microreactors using active learning; advanced manufacturing process modeling using multi-output GPs (MOGPs) and dimensionality reduction; fluid flow using deep GPs (DGPs); and tritium transport model parameter optimization for fusion energy, using batch Bayesian optimization. These capabilities are part of the MOOSE framework.

97 - MATHEMATICS AND COMPUTING↗
NASA NTRS1990

The Navstar GPS master control station's Kalman filter experience

The Navstar Global Positioning System (GPS) is a highly accurate space based navigation system providing all weather, 24 hour a day service to both military and civilian users. The system provides a Gaussian position solution with four satellites, each providing its ephemeris and clock offset with respect to GPS time. The GPS Master Clock Station (MCS) is charged with tracking each Navstar spacecraft and precisely defining the ephemeris and clock parameters for upload into the vehicle's navigation message. Briefly described here are the Navstar system and the Kalman filter estimation process used by MCS to determine, predict, and ensure quality control for each of the satellite's ephemeris and clock states. Routine performance is shown. Kalman filter reaction and response is discussed for anomalous clock behavior and trajectory perturbations. Particular attention is given to MCS efforts to improve orbital adjust modeling. The satellite out of service time due to orbital maneuvering has been reduced in the past year from four days to under twelve hours. The planning, reference trajectory model, and Kalman filter management improvements are explained.

Scardera, Michael P.↗
DOE OSTI2024

Codiscovering graphical structure and functional relationships within data: A Gaussian Process framework for connecting the dots

Most problems within and beyond the scientific domain can be framed into one of the following three levels of complexity of function approximation. Type 1: Approximate an unknown function given input/output data. Type 2: Consider a collection of variables and functions, some of which are unknown, indexed by the nodes and hyperedges of a hypergraph (a generalized graph where edges can connect more than two vertices). Given partial observations of the variables of the hypergraph (satisfying the functional dependencies imposed by its structure), approximate all the unobserved variables and unknown functions. Type 3: Expanding on Type 2, if the hypergraph structure itself is unknown, use partial observations of the variables of the hypergraph to discover its structure and approximate its unknown functions. These hypergraphs offer a natural platform for organizing, communicating, and processing computational knowledge. While most scientific problems can be framed as the data-driven discovery of unknown functions in a computational hypergraph whose structure is known (Type 2), many require the data-driven discovery of the structure (connectivity) of the hypergraph itself (Type 3). We introduce an interpretable Gaussian Process (GP) framework for such (Type 3) problems that does not require randomization of the data, access to or control over its sampling, or sparsity of the unknown functions in a known or learned basis. Its polynomial complexity, which contrasts sharply with the super-exponential complexity of causal inference methods, is enabled by the nonlinear ANOVA capabilities of GPs used as a sensing mechanism.

Science & Technology - Other Topics↗
NASA NTRS1990

Ridge Regression Signal Processing

The introduction of the Global Positioning System (GPS) into the National Airspace System (NAS) necessitates the development of Receiver Autonomous Integrity Monitoring (RAIM) techniques. In order to guarantee a certain level of integrity, a thorough understanding of modern estimation techniques applied to navigational problems is required. The extended Kalman filter (EKF) is derived and analyzed under poor geometry conditions. It was found that the performance of the EKF is difficult to predict, since the EKF is designed for a Gaussian environment. A novel approach is implemented which incorporates ridge regression to explain the behavior of an EKF in the presence of dynamics under poor geometry conditions. The basic principles of ridge regression theory are presented, followed by the derivation of a linearized recursive ridge estimator. Computer simulations are performed to confirm the underlying theory and to provide a comparative analysis of the EKF and the recursive ridge estimator.

Kuhl, Mark R.↗
DOE OSTI2021

Fast Emulation of Expensive Simulations using Approximate Gaussian Processes [Slides]

Nuclear Computational Low-Energy Initiative (NUCLEI) collaboration uses Density Functional Theory (DFT) simulations to predict the structure and binding energies of nuclei over a wide range of proton (Z) and neutron (N) numbers. The DFT simulations utilize a particular parameterization of a Skyrme energy density functional called UNEDF1 which depends on 12 free parameters that must be fit to data (M Kortelainen et al 2014). Fitting involves comparing (e.g.) predicted binding energies of nuclei to experimentally measured values. We use only binding energies as observables, but DFT with UNEDF1 will predict structure (shape) observables as well. In this work, assessing the capability of approximate GP emulators to balance emulator accuracy with computational speed to facilitate improved UNEDF1 calibration. Sparse GPs are straightforward to train and accurate. Calibration is not straightforward with MCMC (using MH or HMC/NUTS). We produced reusable software for continuing and building on this work as well as accessing and using Darwin cluster compute resources

97 MATHEMATICS AND COMPUTING↗