Final Technical Report: Probabilistic Learning, Dimension Reduction, and Spectral Representations for High-Dimensional Uncertainty Quantification
Final Technical Report
Ghanem, Roger [University of Southern California] ↗
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Results for “High-dimensional uncertainty quantification”
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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Randomized physics-informed machine learning for uncertainty quantification in high-dimensional inverse problems
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97 MATHEMATICS AND COMPUTING↗
Hierarchical Gaussian Random Field Sampling for Multilevel Markov Chain Monte Carlo: Coupling Stochastic Partial Differential Equation and the Karhunen–Loève Decomposition
This work introduces structure preserving hierarchical decompositions for sampling Gaussian random fields (GRFs) within the context of multilevel Bayesian inference in high-dimensional space. Existing scalable hierarchical sampling methods, such as those based on stochastic partial differential equations (SPDEs), often reduce the dimensionality of the sample space at the cost of accuracy of inference. Other approaches, such that those based on Karhunen-Loève (KL) expansions, offer sample space dimensionality reduction but sacrifice GRF representation accuracy and ergodicity of the Markov chain Monte Carlo (MCMC) sampler and are computationally expensive for high-dimensional problems. The proposed method integrates the dimensionality reduction capabilities of KL expansions with the scalability of SPDE-based sampling, thereby providing a robust, unified framework for high-dimensional uncertainty quantification (UQ) that is scalable and accurate, preserves ergodicity, and offers dimensionality reduction of the sample space. The hierarchy in our multilevel algorithm is derived from the geometric multigrid hierarchy. By constructing a hierarchical decomposition that maintains the covariance structure across the levels in the hierarchy, the approach enables efficient coarse-to-fine sampling while ensuring that all samples are drawn from the desired distribution. The effectiveness of the proposed method is demonstrated on a benchmark subsurface flow problem, demonstrating its effectiveness in improving computational efficiency and statistical accuracy. Furthermore, our proposed technique is more efficient and accurate and displays better convergence properties than existing methods for high-dimensional Bayesian inference problems.
Gaussian random fields↗
Solving key challenges in collider physics with foundation models
Foundation models are neural networks that are capable of simultaneously solving many problems. Large language foundation models like ChatGPT have revolutionized many aspects of daily life, but their impact for science is not yet clear. In this paper, we use a new foundation model for hadronic jets to solve three key challenges in collider physics. In particular, we show how experiments can (1) save significant computing power when developing reconstruction algorithms, (2) perform a complete uncertainty quantification for high-dimensional measurements, and (3) search for new physics with model agnostic methods using low-level inputs. In each case, there are significant computational or methodological challenges with current methods that limit the science potential of deep learning algorithms. By solving each problem, we take jet foundation models beyond proof-of-principle studies and into the toolkit of practitioners.
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
Gradient-informed Hamiltonian Monte Carlo for multicomponent CALPHAD model optimization and uncertainty quantification
CALPHAD model parameter optimization is inherently challenging due to non-smooth objective functions, high-dimensional parameter spaces, and the need for uncertainty quantification (UQ). Traditional weighted nonlinear least squares approaches are computationally efficient but local, whereas black-box global optimizers and ensemble Markov Chain Monte Carlo (MCMC) methods provide broader exploration at substantial computational cost. The objective of this work is to combine the global exploration capability of gradient-informed Hamiltonian Monte Carlo – specifically the No-U-Turn Sampler (NUTS) – with local deterministic refinement using BFGS to efficiently optimize multicomponent CALPHAD models with minimal manual intervention. Analytic gradients are computed via the Jansson derivative framework. The methodology is demonstrated on the Cr—Fe binary system and extended to the Cr—Fe—Ni ternary system with 32 degrees of freedom. For Cr—Fe, NUTS achieves comparable or superior optimality relative to ensemble MCMC while requiring over an order-of-magnitude fewer likelihood evaluations. Parameter uncertainties are quantified through NUTS sampling and propagated to thermodynamic observables using local expansion, demonstrating a novel modular approach that combines binary and ternary parameter subsets without requiring global relaxation. These results establish gradient-informed exploration as a scalable strategy for multicomponent CALPHAD optimization and provide a practical route towards efficient higher-order database development with quantified uncertainty.
36 MATERIALS SCIENCE↗
Uncertainty quantification for misspecified machine learned interatomic potentials
The use of high-dimensional regression techniques from machine learning has significantly improved the quantitative accuracy of interatomic potentials. Atomic simulations can now plausibly target quantitative predictions in a variety of settings, which has brought renewed interest in robust means to quantify uncertainties. In many practical settings where model complexity is constrained (e.g., due to performance considerations), misspecification — the inability of any one choice of model parameters to exactly match all training data — is a key contributor to errors that is often disregarded. Here, we employ a recent misspecification-aware regression technique to quantify parameter uncertainties, which is then propagated to a broad range of phase and defect properties in tungsten. The propagation is performed through both brute-force resampling and implicit Taylor expansion. The propagated misspecification uncertainties robustly quantify and bound errors on a broad range of material properties. We demonstrate application to recent foundational machine learning interatomic potentials, accurately predicting and bounding errors in MACE-MPA-0 energy predictions across the diverse materials project database.
36 MATERIALS SCIENCE↗
Multidimensional Distributional Neural Network Output Demonstrated in Super‐Resolution of Surface Wind Speed
Accurate quantification of uncertainty in neural network predictions remains a central challenge for scientific applications involving high-dimensional, correlated data. While existing methods capture either aleatoric or epistemic uncertainty, few offer closed-form, multidimensional distributions that preserve spatial correlation while remaining computationally tractable. In this work, we present a framework for training neural networks with a multidimensional Gaussian loss, generating a closed-form predictive distribution over outputs informed by non-identically distributed training data. Our approach captures aleatoric uncertainty by iteratively estimating the means and covariance matrices, and is demonstrated on a super-resolution example out-of-training-sample. We leverage a Fourier representation of the covariance matrix to stabilize network training and preserve spatial correlation. We introduce a novel regularization strategy—referred to as information sharing—that interpolates between image-specific and global covariance estimates, enabling convergence of the super-resolution downscaling network trained on image-specific distributional loss functions. This framework allows for efficient sampling, explicit correlation modeling, and extensions to more complex distribution families all without disrupting prediction performance. We demonstrate the method on a surface wind speed downscaling task and discuss its broader applicability to uncertainty-aware prediction in scientific models.
17 WIND ENERGY↗
Data and scripts from: “Denoising autoencoder for reconstructing sensor observation data and predicting evapotranspiration: noisy and missing values repair and uncertainty quantification”
This data package includes data and scripts from the manuscript “Denoising autoencoder for reconstructing sensor observation data and predicting evapotranspiration: noisy and missing values repair and uncertainty quantification”.The study addressed common challenges faced in environmental sensing and modeling, including uncertain input data, missing sensor observations, and high-dimensional datasets with interrelated but redundant variables. Point-scaled meteorological and soil sensor observations were perturbed with noises and missing values, and denoising autoencoder (DAE) neural networks were developed to reconstruct the perturbed data and further predict evapotranspiration. This study concluded that (1) the reconstruction quality of each variable depends on its cross-correlation and alignment to the underlying data structure, (2) uncertainties from the models were overall stronger than those from the data corruption, and (3) there was a tradeoff between reducing bias and reducing variance when evaluating the uncertainty of the machine learning models.This package includes:(1) Four ipython scripts (.ipynb): “DAE_train.ipynb” trains and evaluates DAE neural networks, “DAE_predict.ipynb” makes predictions from the trained DAE models, “ET_train.ipynb” trains and evaluates ET prediction neural networks, and “ET_predict.ipynb” makes predictions from trained ET models.(2) One python file (.py): “methods.py” includes all user-defined functions and python codes used in the ipython scripts.(3) A “sub_models” folder that includes five trained DAE neural networks (in pytorch format, .pt), which could be used to ingest input data before being fed to the downstream ET models in ‘ET_train.ipynb” or ‘ET_predict.ipynb’.(4) Two data files (.csv). Daily meteorological, vegetation, and soil data is in “df_data.csv”, where “df_meta.csv” contains the location and time information of “df_data.csv”. Each row (index) in “df_meta.csv” corresponds to each row in “df_data.csv”. These data files are formatted to follow the data structure requirements and be directly used in the ipython scripts, and they have been shuffled chronologically to train machine learning models. The meteorological and soil data was collected using point sensors between 2019-2023 at(4.a) Three shrub-dominated field sites in East River, Colorado (named “ph1”, “ph2” and “sg5” in “df_meta.csv”, where “ph1” and “ph2” were located at PumpHouse Hillslopes, and “sg5” was at Snodgrass Mountain meadow) and(4.b) One outdoor, mesoscale, and herbaceous-dominated experiment in Berkeley, California (named “tb” in “df_meta.csv”, short for Smartsoils Testbed at Lawrence Berkeley National Lab).- See "df_data_dd.csv" and "df_meta_dd.csv" for variable descriptions and the Methods section for additional data processing steps. See "flmd.csv" and "README.txt" for brief file descriptions.- All ipython scripts and python files are written in and require PYTHON language software.
54 ENVIRONMENTAL SCIENCES↗
Reliable and Efficient Machine Learning (Final Technical Report)
Modern scientific experiments generate massive amounts of data at a pace much faster than humans can manually analyze. While machine learning has revolutionized commercial data analysis (such as recommending movies or recognizing faces), applying these tools to complex scientific discovery is challenging because scientific answers must be precise, interpretable, and adhere to physical laws. The research under this project aims to develop new mathematical tools and computer algorithms specifically designed for scientific applications. Major progress has been made in automatically cleaning and deconstructing messy experimental data, analyzing the visual information of physical phenomena, determining the underlying physical variables, and providing rig orous mathematical analysis of interesting algorithms and concepts widely used in machine learning. This project addressed the critical gap between our ability to generate massive scientific data and our ability to extract interpretable information from it. We established mathematical foundations for Scientific Machine Learning (SciML) aimed at effective data analytics and automated discovery. Our work focused on three core objectives: (1) developing reliable feature extraction methods for dynamic high-dimensional data, (2) establishing mathematical foundations for discovering dynamics via neural networks, and (3) creating rigorous optimization techniques for these models. Key outcomes come from two fronts. On the practical side, they include the development of algorithms that significantly enhance the extraction of signals from field data, as well as the capability to handle situations that exhibit smooth variations or physical stretching due to temperature changes. They also include the creation of an automated framework for discovering fundamental state variables from raw experimental data, demonstrating the ability to identify intrinsic physical dimensions without prior knowledge of the governing laws. On the theoretical front, the research results in theoretical advances in Optimal Transport, a widely used notion in SciML, specifically regarding functions with fixed-size nodal sets, provide sharp bounds relevant to uncertainty quantification. Meanwhile, the outcomes also include the establishment of convergence theories for nonlocal gradient descent methods, enabling robust optimization with noisy data in high-dimensional settings commonly encountered in scientific modeling. The project also helps creating opportunities to train the next generation of researchers, equipping them with the necessary technical skills for today’s workplace and preparing them for future advances.
97 MATHEMATICS AND COMPUTING↗
Improving the Quasi‐Biennial Oscillation via a Surrogate‐Accelerated Multi‐Objective Optimization
Accurate simulation of the quasi-biennial oscillation (QBO) is challenging due to uncertainties in representing convectively generated gravity waves. We develop an end-to-end uncertainty quantification workflow that calibrates these gravity wave processes in E3SM for a realistic QBO. Central to our approach is a domain knowledge-informed, compressed representation of high-dimensional spatio-temporal wind fields. By employing a parsimonious statistical model that learns the fundamental frequency from complex observations, we extract interpretable and physically meaningful quantities capturing key attributes. Building on this, we train a probabilistic surrogate model that approximates the fundamental characteristics of the QBO as functions of critical physics parameters governing gravity wave generation. Leveraging the Karhunen–Loève decomposition, our surrogate efficiently represents these characteristics as a set of orthogonal features, capturing cross-correlations among multiple physics quantities evaluated at different pressure levels and enabling rapid surrogate-based inference at a fraction of the computational cost of full-scale simulations. Finally, we analyze the inverse problem using a multi-objective approach. Our study reveals a tension between amplitude and period that constrains the QBO representation, precluding a single optimal solution. To navigate this, we quantify the bi-criteria trade-off and generate a set of Pareto optimal parameter values that balance the conflicting objectives. This integrated workflow improves the fidelity of QBO simulations and offers a versatile template for uncertainty quantification in complex geophysical models.
54 ENVIRONMENTAL SCIENCES↗
Intrusive Uncertainty Quantification and Optimal Experiment Design in the Open-Source Pyomo Ecosystem
This contribution describes ParmEst and Pyomo.DoE, two pillars of the open-source Python-based Pyomo ecosystem for computational optimization with (partial differential) algebraic equation mathematical models. Specifically, ParmEst facilitates intrusive frequentist parameter estimation (PE) and uncertainty quantification (UQ) through built-in features, such as covariance matrix estimation, bootstrapping, and likelihood ratio tests. Complementary, Pyomo.DoE enables optimal experiment design by maximizing various metrics of the Fisher information matrix, such as A-optimality (trace), D-optimality (determinant), E-optimality (minimum eigenvalue), and ME-optimality (condition number). ParmEst and Pyomo.DoE can solve high-dimensional optimization problems by leveraging the model structure and exact derivative information. Finally, we will discuss future opportunities to integrate PE and UQ capabilities with optimization under uncertainty, including robust optimization with non-convex models via PyROS.
97 MATHEMATICS AND COMPUTING↗
Multi-head physics-informed neural networks for learning functional priors and uncertainty quantification
In numerous applications, the integration of prior knowledge and historical information is essential, particularly for tasks requiring the solution of ordinary or partial differential equations (ODEs/PDEs) in data-sparse or noisy environments. For instance, achieving accurate solutions to time-dependent PDEs with limited initial condition measurements necessitates an effective strategy for embedding prior knowledge. Hard-parameter sharing architectures in neural networks (NNs) have demonstrated success in both traditional and scientific machine learning domains, facilitating the learning of informative representations. Here, in this study, we introduce a novel, yet efficient, method to enhance physics-informed neural networks (PINNs) by incorporating a multi-head structure that enables the learning of functional priors from both empirical data and governing physical laws. This prior information can then be used to address data sparsity and high-level noise in solving ODE/PDE problems with uncertainty quantification (UQ). The approach, termed Multi-Head PINN (MH-PINN), consists of a shared body NN and multiple head NNs, each corresponding to an individual PINN instance. Our framework for functional prior learning is carried out in two stages: (1) training the MH-PINNs to develop a shared body NN alongside multiple head NNs, and (2) employing these trained head NNs to estimate a prior distribution through a normalizing flow-based density estimator. The learned functional prior can then be applied as a regularization mechanism in deterministic contexts or as an informative prior within a Bayesian inference framework, aiding in the resolution of subsequent ODE/PDE tasks. We evaluate the efficacy of MH-PINNs across five benchmark problems, including a high-dimensional parametric PDE, all characterized by data sparsity or substantial noise levels. Our findings reveal that MH-PINNs deliver accurate solutions and robust UQ, demonstrating adaptability across a range of complex and challenging scenarios.
Bayesian inference↗
Examples of Mission-driven Data Science from Jefferson Lab and ACES
This presentation details mission-driven data science initiatives at Jefferson Lab and the Joint Institute for Advanced Computing on Environmental Studies (ACES). JLab, a U.S. Department of Energy Office of Science national laboratory, operates the Continuous Electron Beam Accelerator Facility (CEBAF), and is the lead institute for the new High Performance Data Facility (HPDF) Hub. The Joint Institute for ACES brings together interdisciplinary teams in health informatics, climate modeling, computer science, and physics to address environmental challenges, including flood modeling. The Hampton Roads region, particularly Norfolk and Virginia Beach, faces increasing flood risks, motivating the need for rapid, reliable, and risk-aware decision support. ACES’s flooding work has a focus on uncertainty quantification (UQ) and machine learning (ML) for coastal flood management. The work is motivated by the increasing vulnerability of communities such as Norfolk and Virginia Beach, Virginia, to frequent coastal flooding events, and the need for rapid, reliable decision support. The research develops computationally efficient ML surrogate models to forecast water levels and flooding risk. A central theme is the quantification and calibration of predictive uncertainty, especially for out-of-distribution (OOD) scenarios, using techniques such as Monte Carlo Dropout, Deep Ensembles, Gaussian Processes, and Deep Quantile Regression (DQR). The study demonstrates that distance-aware UQ is critical for reliable scientific AI, particularly in high-dimensional, safety-critical, and real-time applications.
McSpadden, Diana [Thomas Jefferson National Accele↗
Accelerating uncertainty quantification in incremental dynamic analysis using dimension reduction-based surrogate modeling
We propose a surrogate modeling framework based on dimension reduction to facilitate the quantification of seismic risk of structural systems in performance-based earthquake engineering. The framework adopts incremental dynamic analysis (IDA) for addressing hazard variability, and promotes significant computational efficiency improvement for propagating epistemic uncertainties associated with the structural models. It utilizes both linear and nonlinear dimension reduction approaches, equipped with inverse mappings, to learn a functional between the input parameter space (e.g., the epistemic uncertainties of the structure) to the high-dimensional output space created through the IDA implementation across different ground motions and seismic intensity levels. Polynomial chaos expansion is adopted as the surrogate model to learn this functional in the reduced space. A nine-story steel moment-resisting frame with uncertain structural properties is used as a testbed. Furthermore, we select the seismic fragility curves as a measure of the structure’s seismic performance, since it provides an estimate of the probability of entering specified damage states for given levels of ground shaking.
42 ENGINEERING↗
Neural chaos: A spectral stochastic neural operator
Building surrogate models for operators with uncertainty quantification capabilities is essential for many engineering applications where randomness–such as variability in material properties, boundary conditions, and initial conditions–is unavoidable. Polynomial Chaos Expansion (PCE) is widely recognized as a go-to method for constructing stochastic surrogates in both intrusive and non-intrusive ways, and it has recently been used in the context of operator learning. However, its application becomes challenging for complex or high-dimensional processes, as achieving accuracy requires higher-order polynomials, which can increase computational demand and/or the risk of overfitting. Furthermore, PCE requires specialized treatments to manage random variables that are not independent, and these treatments may be problem-dependent or may fail with increasing complexity. Here, in this work, we adopt the same formalism as the spectral expansion used in PCE; however, we replace the classical polynomial basis functions with neural network (NN) basis functions to leverage their expressivity. To achieve this, we propose an algorithm that identifies NN-parameterized basis functions in a purely data-driven manner, without any prior assumptions about the joint distribution of the random variables involved, whether independent or dependent, or about their marginal distributions. The proposed algorithm identifies each NN-parameterized basis function sequentially, ensuring they are orthogonal with respect to the data distribution. The basis functions are constructed directly on the joint stochastic variables without requiring a tensor product structure or assuming independence of the random variables. This approach may offer greater flexibility for complex stochastic models, while simplifying implementation compared to the tensor product structures typically used in PCE to handle random vectors. This is particularly advantageous given the current state of open-source packages, where building and training neural networks can be done with just a few lines of code and extensive community support. We demonstrate the effectiveness of the proposed scheme through several numerical examples of varying complexity and provide comparisons with classical PCE.
Polynomial chaos expansion↗
A copula-based rank histogram ensemble filter
Serial ensemble filters implement triangular probability transport maps to reduce high-dimensional inference problems to sequences of state-by-state univariate inference problems. The univariate inference problems are solved by sampling posterior probability densities obtained by combining constructed prior densities with observational likelihoods according to Bayes' rule. Many serial filters in the literature focus on representing the marginal posterior densities of each state. However, rigorously capturing the conditional dependencies between the different univariate inferences is crucial to correctly sampling multidimensional posteriors. This work proposes a new serial ensemble filter, called the copula rank histogram filter (CoRHF), that seeks to capture the conditional dependency structure between variables via empirical copula estimates; these estimates are used to rigorously implement the triangular (state-by-state univariate) Bayesian inference. The success of the CoRHF is demonstrated on two-dimensional examples and the Lorenz'63 problem. A practical extension to the high-dimensional setting is developed by localizing the empirical copula estimation, and is demonstrated on the Lorenz'96 problem.
97 MATHEMATICS AND COMPUTING↗
Discriminative versus generative approaches to simulation-based inference
Most of the fundamental, emergent, and phenomenological parameters of particle and nuclear physics are determined through parametric template fits. Simulations are used to populate histograms which are then matched to data. This approach is inherently lossy, since histograms are binned and low-dimensional. Deep learning has enabled unbinned and high-dimensional parameter estimation through neural likelihood(-ratio) estimation. We compare two approaches for neural simulation-based inference (NSBI): one based on discriminative learning (classification) and one based on generative modeling. These two approaches are directly evaluated on the same datasets, with a similar level of hyperparameter optimization in both cases. In addition to a Gaussian dataset, we study NSBI using a Higgs boson dataset from the FAIR Universe Challenge. We find that both the direct likelihood and likelihood ratio estimation are able to effectively extract parameters with reasonable uncertainties. For the numerical examples and within the set of hyperparameters studied, we found that the likelihood ratio method is more accurate and/or precise. Both methods have a significant spread from the network training and would require ensembling or other mitigation strategies in practice.
high energy physics↗
Improving Trustworthiness of Data-Driven Power Grid Contingency Analysis With Bayesian Residual Graph Neural Networks
The evolving energy landscape requires novel tools to efficiently perform contingency analysis and reliability assessment of power grids, potentially in real-time. The high computational cost of traditional power flow solvers limits their applicability in practice. Machine learning (ML) surrogates such as deep neural networks (NNs) accelerate power flow solvers computations, enabling high-order contingency analysis and real-time decision-making by learning highly nonlinear functions and integrating grid topology via graph architectures. However, (graph) NNs lack predictive power away from training data and do not provide predictive confidence estimates. Here, we present a Bayesian residual graph NN that integrates knowledge from low-fidelity data via residual training and embeds granular quantification of uncertainties, improving trustworthiness critical for high-consequence decision-making. Applying Bayesian concepts to NNs is challenging due to the high-dimensionality of both the parameter space, complicating derivation of a meaningful prior, and the output space in large grid systems, requiring enhanced techniques to assess the predicted high-dimensional uncertainties. Our contributions include: (1) Deriving a prior for fully connected and graph NNs that leverages low-fidelity data to guide mean predictions and appropriately control prior predictive uncertainty. (2) Integrating this prior within an ensembling with anchoring scheme for efficient approximate posterior inference. (3) Deriving enhanced metrics to assess accuracy of both the mean and uncertainty predictions in high dimensions, appropriately accounting for correlations propagated through graph layers. The resulting Bayesian residual graph NN is tested on a contingency analysis task for 14-bus and 118-bus grids.
24 - POWER TRANSMISSION AND DISTRIBUTION↗