Modeling real-world phenomena to any degree of accuracy is a challenge that the scientific research community has navigated since its foundation. Lack of information and limited computational and observational resources necessitate modeling assumptions which, when invalid, lead to model-form error (MFE). The work reported herein explored a novel method to represent model-form uncertainty (MFU) that combines Bayesian statistics with the emerging field of universal differential equations (UDEs). The fundamental principle behind UDEs is simple: use known equational forms that govern a dynamical system when you have them; then incorporate data-driven approaches – in this case neural networks (NNs) – embedded within the governing equations to learn the interacting terms that were underrepresented. Utilizing epidemiology as our motivating exemplar, this report will highlight the challenges of modeling novel infectious diseases while introducing ways to incorporate NN approximations to MFE. Prior to embarking on a Bayesian calibration, we first explored methods to augment the standard (non-Bayesian) UDE training procedure to account for uncertainty and increase robustness of training. In addition, it is often the case that uncertainty in observations is significant; this may be due to randomness or lack of precision in the measurement process. This uncertainty typically manifests as “noisy” observations which deviate from a true underlying signal. To account for such variability, the NN approximation to MFE is endowed with a probabilistic representation and is updated using available observational data in a Bayesian framework. By representing the MFU explicitly and deploying an embedded, data-driven model, this approach enables an agile, expressive, and interpretable method for representing MFU. In this report we will provide evidence that Bayesian UDEs show promise as a novel framework for any science-based, data-driven MFU representation; while emphasizing that significant advances must be made in the calibration of Bayesian NNs to ensure a robust calibration procedure.
We present a surrogate modeling framework for conservatively estimating measures of risk from limited realizations of an expensive physical experiment or computational simulation. Risk measures combine objective probabilities with the subjective values of a decision maker to quantify anticipated outcomes. Given a set of samples, we construct a surrogate model that produces estimates of risk measures that are always greater than their empirical approximations obtained from the training data. These surrogate models limit over-confidence in reliability and safety assessments and produce estimates of risk measures that converge much faster to the true value than purely sample-based estimates. We first detail the construction of conservative surrogate models that can be tailored to a stakeholder's risk preferences and then present an approach, based on stochastic orders, for constructing surrogate models that are conservative with respect to families of risk measures. Our surrogate models include biases that permit them to conservatively estimate the target risk measures. We provide theoretical results that show that these biases decay at the same rate as the L2 error in the surrogate model. Numerical demonstrations confirm that risk-adapted surrogate models do indeed overestimate the target risk measures while converging at the expected rate.
High-quality factor resonant cavities are challenging structures to model in electromagnetics owing to their large sensitivity to minute parameter changes. Therefore, uncertainty quantification (UQ) strategies are pivotal to understanding key parameters affecting the cavity response. We discuss here some of these strategies focusing on shielding effectiveness (SE) properties of a canonical slotted cylindrical cavity that will be used to develop credibility evidence in support of predictions made using computational simulations for this application.
Constructing accurate statistical models of critical system responses typically requires an enormous amount of data from physical experiments or numerical simulations. Unfortunately, data generation is often expensive and time consuming. To streamline the data generation process, optimal experimental design determines the 'best' allocation of experiments with respect to a criterion that measures the ability to estimate some important aspect of an assumed statistical model. While optimal design has a vast literature, few researchers have developed design paradigms targeting tail statistics, such as quantiles. In this project, we tailored and extended traditional design paradigms to target distribution tails. Our approach included (i) the development of new optimality criteria to shape the distribution of prediction variances, (ii) the development of novel risk-adapted surrogate models that provably overestimate certain statistics including the probability of exceeding a threshold, and (iii) the asymptotic analysis of regression approaches that target tail statistics such as superquantile regression. To accompany our theoretical contributions, we released implementations of our methods for surrogate modeling and design of experiments in two complementary open source software packages, the ROL/OED Toolkit and PyApprox.
We present a surrogate modeling framework for conservatively estimating measures of risk from limited realizations of an expensive physical experiment or computational simulation. We adopt a probabilistic description of risk that assigns probabilities to consequences associated with an event and use risk measures, which combine objective evidence with the subjective values of decision makers, to quantify anticipated outcomes. Given a set of samples, we construct a surrogate model that produces estimates of risk measures that are always greater than their empirical estimates obtained from the training data. These surrogate models not only limit over-confidence in reliability and safety assessments, but produce estimates of risk measures that converge much faster to the true value than purely sample-based estimates. We first detail the construction of conservative surrogate models that can be tailored to the specific risk preferences of the stakeholder and then present an approach, based upon stochastic orders, for constructing surrogate models that are conservative with respect to families of risk measures. The surrogate models introduce a bias that allows them to conservatively estimate the target risk measures. We provide theoretical results that show that this bias decays at the same rate as the L2 error in the surrogate model. Our numerical examples confirm that risk-aware surrogate models do indeed over-estimate the target risk measures while converging at the expected rate.
This report describes the credibility activities undertaken in support of Gemma code development in FY20, which include Verification & Validation (V&V), Uncertainty Quantification (UQ), and Credibility process application. The main goal of these activities is to establish capabilities and process frameworks that can be more broadly applied to new and more advanced problems as the Gemma code development effort matures. This will provide Gemma developers and analysts with the tools needed to generate credibility evidence in support of Gemma predictions for future use cases. The FY20 Gemma V&V/UQ/Credibility activities described in this report include experimental uncertainty analysis, the development and use of methods for optimal design of computer experiments, and the development of a framework for validation. These initial activities supported the development of broader credibility planning for Gemma that continued into FY21.
In Bayesian model calibration, evaluation of the likelihood function usually involves finding the inverse and determinant of a covariance matrix. When Markov Chain Monte Carlo (MCMC) methods are used to sample from the posterior, hundreds of thousands of likelihood evaluations may be required. In this paper, we demonstrate that the structure of the covariance matrix can be exploited, leading to substantial time savings in practice. We also derive two simple equations for approximating the inverse of the covariance matrix in this setting, which can be computed in near-quadratic time. The practical implications of these strategies are demonstrated using a simple numerical case study and the "quack"R package. For a covariance matrix with 1000 rows, application of these strategies for a million likelihood evaluations leads to a speedup of roughly 4000 compared to the naive implementation.
In this article, we examine the coupling into an electrically short azimuthal slot on a cylindrical cavity operating at fundamental cavity modal frequencies. We first develop a matched bound formulation through which we can gather information for maximum achievable levels of interior cavity fields. Actual field levels are below this matched bound; therefore, we also develop an unmatched formulation for frequencies below the slot resonance to achieve a better insight on the physics of this coupling. Good agreement is observed between the unmatched formulation, full-wave simulations, and experimental data, providing a validation of our analytical models. We then extend the unmatched formulation to treat an array of slots, found again in good agreement with full-wave simulations. These analytical models can be used to investigate ways to mitigate electromagnetic interference and electromagnetic compatibility effects within cavities.
Understanding the effect of a high-altitude electromagnetic pulse (HEMP) on the equipment in the United States electrical power grid is important to national security. A present challenge to this understanding is evaluating the vulnerability of transformers to a HEMP. Evaluating vulnerability by direct testing is cost-prohibitive, due to the wide variation in transformers, their high cost, and the large number of tests required to establish vulnerability with confidence. Alternatively, material and component testing can be performed to quantify a model for transformer failure, and the model can be used to assess vulnerability of a wide variety of transformers. This project develops a model of the probability of equipment failure due to effects of a HEMP. Potential failure modes are cataloged, and a model structure is presented which can be quantified by the results of small-scale coupon tests.
In the presence of model discrepancy, the calibration of physics-based models for physical parameter inference is a challenging problem. Lack of identifiability between calibration parameters and model discrepancy requires additional identifiability constraints to be placed on the model discrepancy to obtain unique physical parameter estimates. If these assumptions are violated, the inference for the calibration parameters can be systematically biased. In many applications, such as in dynamic material property experiments, many of the calibration inputs refer to measurement uncertainties. In this setting, we develop a metric for identifying overfitting of these measurement uncertainties, propose a prior capable of reducing this overfitting, and show how this leads to a diagnostic tool for validation of physical parameter inference. The approach is demonstrated for a benchmark example and applied for a material property application to perform inference on the equation of state parameters of tantalum.