GRADIENT-BASED STOCHASTIC OPTIMIZATION METHODS IN BAYESIAN EXPERIMENTAL DESIGN

Optimal experimental design (OED) seeks experiments expected to yield the most useful data for some purpose. In practical circumstances where experiments are time-consuming or resource-intensive, OED can yield enormous savings. We pursue OED for nonlinear systems from a Bayesian perspective, with the goal of choosing experiments that are optimal for parameter inference. Our objective in this context is the expected information gain in model parameters, which in general can only be estimated using Monte Carlo methods. Maximizing this objective thus becomes a stochastic optimization problem. This paper develops gradient-based stochastic optimization methods for the design of experiments on a continuous parameter space. Given a Monte Carlo estimator of expected information gain, we use infinitesimal perturbation analysis to derive gradients of this estimator.We are then able to formulate two gradient-based stochastic optimization approaches: (i) Robbins-Monro stochastic approximation, and (ii) sample average approximation combined with a deterministic quasi-Newton method. A polynomial chaos approximation of the forward model accelerates objective and gradient evaluations in both cases.We discuss the implementation of these optimization methods, then conduct an empirical comparison of their performance. To demonstrate design in a nonlinear setting with partial differential equation forward models, we use the problem of sensor placement for source inversion. Numerical results yield useful guidelines on the choice of algorithm and sample sizes, assess the impact of estimator bias, and quantify tradeoffs of computational cost versus solution quality and robustness.United States. Air Force Office of Scientific Research (Computational Mathematics Program)National Science Foundation (U.S.) (Award ECCS-1128147

Fault Prognosis of Turbofan Engines: Eventual Failure Prediction and Remaining Useful Life Estimation

In the era of industrial big data, prognostics and health management is essential to improve the prediction of future failures to minimize inventory, maintenance, and human costs. Used for the 2021 PHM Data Challenge, the new Commercial Modular Aero-Propulsion System Simulation dataset from NASA is an open-source benchmark containing simulated turbofan engine units flown under realistic flight conditions. Deep learning approaches implemented previously for this application attempt to predict the remaining useful life of the engine units, but have not utilized labeled failure mode information, impeding practical usage and explainability. To address these limitations, a new prognostics approach is formulated with a customized loss function to simultaneously predict the current health state, the eventual failing component(s), and the remaining useful life. The proposed method incorporates principal component analysis to orthogonalize statistical time-domain features, which are inputs into supervised regressors such as random forests, extreme random forests, XGBoost, and artificial neural networks. The highest performing algorithm, ANN-Flux, achieves AUROC and AUPR scores exceeding 0.95 for each classification. In addition, ANN-Flux reduces the remaining useful life RMSE by 38% for the same test split of the dataset compared to past work, with significantly less computational cost.Comment: Preprint with 10 pages, 5 figures. Submitted to International Journal of Prognostics and Health Management (IJPHM

Accelerated Bayesian experimental design for chemical kinetic models

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2010.Cataloged from PDF version of thesis.Includes bibliographical references (p. 129-136).The optimal selection of experimental conditions is essential in maximizing the value of data for inference and prediction, particularly in situations where experiments are time-consuming and expensive to conduct. A general Bayesian framework for optimal experimental design with nonlinear simulation-based models is proposed. The formulation accounts for uncertainty in model parameters, observables, and experimental conditions. Straightforward Monte Carlo evaluation of the objective function - which reflects expected information gain (Kullback-Leibler divergence) from prior to posterior - is intractable when the likelihood is computationally intensive. Instead, polynomial chaos expansions are introduced to capture the dependence of observables on model parameters and on design conditions. Under suitable regularity conditions, these expansions converge exponentially fast. Since both the parameter space and the design space can be high-dimensional, dimension-adaptive sparse quadrature is used to construct the polynomial expansions. Stochastic optimization methods will be used in the future to maximize the expected utility. While this approach is broadly applicable, it is demonstrated on a chemical kinetic system with strong nonlinearities. In particular, the Arrhenius rate parameters in a combustion reaction mechanism are estimated from observations of autoignition. Results show multiple order-of-magnitude speedups in both experimental design and parameter inference.by Xun Huan.S.M

FP-IRL: Fokker-Planck-based Inverse Reinforcement Learning -- A Physics-Constrained Approach to Markov Decision Processes

Inverse Reinforcement Learning (IRL) is a compelling technique for revealing the rationale underlying the behavior of autonomous agents. IRL seeks to estimate the unknown reward function of a Markov decision process (MDP) from observed agent trajectories. However, IRL needs a transition function, and most algorithms assume it is known or can be estimated in advance from data. It therefore becomes even more challenging when such transition dynamics is not known a-priori, since it enters the estimation of the policy in addition to determining the system's evolution. When the dynamics of these agents in the state-action space is described by stochastic differential equations (SDE) in It^{o} calculus, these transitions can be inferred from the mean-field theory described by the Fokker-Planck (FP) equation. We conjecture there exists an isomorphism between the time-discrete FP and MDP that extends beyond the minimization of free energy (in FP) and maximization of the reward (in MDP). We identify specific manifestations of this isomorphism and use them to create a novel physics-aware IRL algorithm, FP-IRL, which can simultaneously infer the transition and reward functions using only observed trajectories. We employ variational system identification to infer the potential function in FP, which consequently allows the evaluation of reward, transition, and policy by leveraging the conjecture. We demonstrate the effectiveness of FP-IRL by applying it to a synthetic benchmark and a biological problem of cancer cell dynamics, where the transition function is inaccessible

Expert Elicitation and Data Noise Learning for Material Flow Analysis using Bayesian Inference

Bayesian inference allows the transparent communication of uncertainty in material flow analyses (MFAs), and a systematic update of uncertainty as new data become available. However, the method is undermined by the difficultly of defining proper priors for the MFA parameters and quantifying the noise in the collected data. We start to address these issues by first deriving and implementing an expert elicitation procedure suitable for generating MFA parameter priors. Second, we propose to learn the data noise concurrent with the parametric uncertainty. These methods are demonstrated using a case study on the 2012 U.S. steel flow. Eight experts are interviewed to elicit distributions on steel flow uncertainty from raw materials to intermediate goods. The experts' distributions are combined and weighted according to the expertise demonstrated in response to seeding questions. These aggregated distributions form our model parameters' prior. A sensible, weakly-informative prior is also adopted for learning the data noise. Bayesian inference is then performed to update the parametric and data noise uncertainty given MFA data collected from the United States Geological Survey (USGS) and the World Steel Association (WSA). The results show a reduction in MFA parametric uncertainty when incorporating the collected data. Only a modest reduction in data noise uncertainty was observed; however, greater reductions were achieved when using data from multiple years in the inference. These methods generate transparent MFA and data noise uncertainties learned from data rather than pre-assumed data noise levels, providing a more robust basis for decision-making that affects the system.Comment: 23 pages of main paper and 10 pages of supporting informatio

Compressive sensing adaptation for polynomial chaos expansions

Basis adaptation in Homogeneous Chaos spaces rely on a suitable rotation of the underlying Gaussian germ. Several rotations have been proposed in the literature resulting in adaptations with different convergence properties. In this paper we present a new adaptation mechanism that builds on compressive sensing algorithms, resulting in a reduced polynomial chaos approximation with optimal sparsity. The developed adaptation algorithm consists of a two-step optimization procedure that computes the optimal coefficients and the input projection matrix of a low dimensional chaos expansion with respect to an optimally rotated basis. We demonstrate the attractive features of our algorithm through several numerical examples including the application on Large-Eddy Simulation (LES) calculations of turbulent combustion in a HIFiRE scramjet engine.Comment: Submitted to Journal of Computational Physic