1,593 research outputs found
Uncertainty Quantification in Machine Learning for Engineering Design and Health Prognostics: A Tutorial
On top of machine learning models, uncertainty quantification (UQ) functions
as an essential layer of safety assurance that could lead to more principled
decision making by enabling sound risk assessment and management. The safety
and reliability improvement of ML models empowered by UQ has the potential to
significantly facilitate the broad adoption of ML solutions in high-stakes
decision settings, such as healthcare, manufacturing, and aviation, to name a
few. In this tutorial, we aim to provide a holistic lens on emerging UQ methods
for ML models with a particular focus on neural networks and the applications
of these UQ methods in tackling engineering design as well as prognostics and
health management problems. Toward this goal, we start with a comprehensive
classification of uncertainty types, sources, and causes pertaining to UQ of ML
models. Next, we provide a tutorial-style description of several
state-of-the-art UQ methods: Gaussian process regression, Bayesian neural
network, neural network ensemble, and deterministic UQ methods focusing on
spectral-normalized neural Gaussian process. Established upon the mathematical
formulations, we subsequently examine the soundness of these UQ methods
quantitatively and qualitatively (by a toy regression example) to examine their
strengths and shortcomings from different dimensions. Then, we review
quantitative metrics commonly used to assess the quality of predictive
uncertainty in classification and regression problems. Afterward, we discuss
the increasingly important role of UQ of ML models in solving challenging
problems in engineering design and health prognostics. Two case studies with
source codes available on GitHub are used to demonstrate these UQ methods and
compare their performance in the life prediction of lithium-ion batteries at
the early stage and the remaining useful life prediction of turbofan engines
Probabilistic Inference for Model Based Control
Robotic systems are essential for enhancing productivity, automation, and performing hazardous tasks. Addressing the unpredictability of physical systems, this thesis advances robotic planning and control under uncertainty, introducing learning-based methods for managing uncertain parameters and adapting to changing environments in real-time.
Our first contribution is a framework using Bayesian statistics for likelihood-free inference of model parameters. This allows employing complex simulators for designing efficient, robust controllers. The method, integrating the unscented transform with a variant of information theoretical model predictive control, shows better performance in trajectory evaluation compared to Monte Carlo sampling, easing the computational load in various control and robotics tasks.
Next, we reframe robotic planning and control as a Bayesian inference problem, focusing on the posterior distribution of actions and model parameters. An implicit variational inference algorithm, performing Stein Variational Gradient Descent, estimates distributions over model parameters and control inputs in real-time. This Bayesian approach effectively handles complex multi-modal posterior distributions, vital for dynamic and realistic robot navigation.
Finally, we tackle diversity in high-dimensional spaces. Our approach mitigates underestimation of uncertainty in posterior distributions, which leads to locally optimal solutions. Using the theory of rough paths, we develop an algorithm for parallel trajectory optimisation, enhancing solution diversity and avoiding mode collapse. This method extends our variational inference approach for trajectory estimation, employing diversity-enhancing kernels and leveraging path signature representation of trajectories. Empirical tests, ranging from 2-D navigation to robotic manipulators in cluttered environments, affirm our method's efficiency, outperforming existing alternatives
Reinforcement learning in large state action spaces
Reinforcement learning (RL) is a promising framework for training intelligent agents which learn to optimize long term utility by directly interacting with the environment. Creating RL methods which scale to large state-action spaces is a critical problem towards ensuring real world deployment of RL systems. However, several challenges limit the applicability of RL to large scale settings. These include difficulties with exploration, low sample efficiency, computational intractability, task constraints like decentralization and lack of guarantees about important properties like performance, generalization and robustness in potentially unseen scenarios.
This thesis is motivated towards bridging the aforementioned gap. We propose several principled algorithms and frameworks for studying and addressing the above challenges RL. The proposed methods cover a wide range of RL settings (single and multi-agent systems (MAS) with all the variations in the latter, prediction and control, model-based and model-free methods, value-based and policy-based methods). In this work we propose the first results on several different problems: e.g. tensorization of the Bellman equation which allows exponential sample efficiency gains (Chapter 4), provable suboptimality arising from structural constraints in MAS(Chapter 3), combinatorial generalization results in cooperative MAS(Chapter 5), generalization results on observation shifts(Chapter 7), learning deterministic policies in a probabilistic RL framework(Chapter 6). Our algorithms exhibit provably enhanced performance and sample efficiency along with better scalability. Additionally, we also shed light on generalization aspects of the agents under different frameworks. These properties have been been driven by the use of several advanced tools (e.g. statistical machine learning, state abstraction, variational inference, tensor theory).
In summary, the contributions in this thesis significantly advance progress towards making RL agents ready for large scale, real world applications
Singularity Formation in the High-Dimensional Euler Equations and Sampling of High-Dimensional Distributions by Deep Generative Networks
High dimensionality brings both opportunities and challenges to the study of applied mathematics. This thesis consists of two parts. The first part explores the singularity formation of the axisymmetric incompressible Euler equations with no swirl in ââż, which is closely related to the Millennium Prize Problem on the global singularity of the Navier-Stokes equations. In this part, the high dimensionality contributes to the singularity formation in finite time by enhancing the strength of the vortex stretching term. The second part focuses on sampling from a high-dimensional distribution using deep generative networks, which has wide applications in the Bayesian inverse problem and the image synthesis task. The high dimensionality in this part becomes a significant challenge to the numerical algorithms, known as the curse of dimensionality.
In the first part of this thesis, we consider the singularity formation in two scenarios. In the first scenario, for the axisymmetric Euler equations with no swirl, we consider the case when the initial condition for the angular vorticity is Cα Hölder continuous. We provide convincing numerical examples where the solutions develop potential self-similar blow-up in finite time when the Hölder exponent α < α*, and this upper bound α* can asymptotically approach 1 - 2/n. This result supports a conjecture from Drivas and Elgindi [37], and generalizes it to the high-dimensional case. This potential blow-up is insensitive to the perturbation of initial data. Based on assumptions summarized from numerical experiments, we study a limiting case of the Euler equations, and obtain α* = 1 - 2/n which agrees with the numerical result. For the general case, we propose a relatively simple one-dimensional model and numerically verify its approximation to the Euler equations. This one-dimensional model might suggest a possible way to show this finite-time blow-up scenario analytically. Compared to the first proved blow-up result of the 3D axisymmetric Euler equations with no swirl and Hölder continuous initial data by Elgindi in [40], our potential blow-up scenario has completely different scaling behavior and regularity of the initial condition. In the second scenario, we consider using smooth initial data, but modify the Euler equations by adding a factor Δ as the coefficient of the convection terms to weaken the convection effect. The new model is called the weak convection model. We provide convincing numerical examples of the weak convection model where the solutions develop potential self-similar blow-up in finite time when the convection strength Δ < Δ*, and this upper bound Δ* should be close to 1 - 2/n. This result is closely related to the infinite-dimensional case of an open question [37] stated by Drivas and Elgindi. Our numerical observations also inspire us to approximate the weak convection model with a one-dimensional model. We give a rigorous proof that the one-dimensional model will develop finite-time blow-up if Δ < 1 - 2/n, and study the approximation quality of the one-dimensional model to the weak convection model numerically, which could be beneficial to a rigorous proof of the potential finite-time blow-up.
In the second part of the thesis, we propose the Multiscale Invertible Generative Network (MsIGN) to sample from high-dimensional distributions by exploring the low-dimensional structure in the target distribution. The MsIGN models a transport map from a known reference distribution to the target distribution, and thus is very efficient in generating uncorrelated samples compared to MCMC-type methods. The MsIGN captures multiple modes in the target distribution by generating new samples hierarchically from a coarse scale to a fine scale with the help of a novel prior conditioning layer. The hierarchical structure of the MsIGN also allows training in a coarse-to-fine scale manner. The Jeffreys divergence is used as the objective function in training to avoid mode collapse. Importance sampling based on the prior conditioning layer is leveraged to estimate the Jeffreys divergence, which is intractable in previous deep generative networks. Numerically, when applied to two Bayesian inverse problems, the MsIGN clearly captures multiple modes in the high-dimensional posterior and approximates the posterior accurately, demonstrating its superior performance compared with previous methods. We also provide an ablation study to show the necessity of our proposed network architecture and training algorithm for the good numerical performance. Moreover, we also apply the MsIGN to the image synthesis task, where it achieves superior performance in terms of bits-per-dimension value over other flow-based generative models and yields very good interpretability of its neurons in intermediate layers.</p
Approximate Methods for Marginal Likelihood Estimation
We consider the estimation of the marginal likelihood in Bayesian statistics, a essential and
important task known to be computationally expensive when the dimension of the parameter space
is large. We propose a general algorithm with numerous extensions that can be widely applied to a
variety of problem settings and excels particularly when dealing with near log-concave posteriors.
Our method hinges on a novel idea that uses MCMC samples to partition the parameter space
and forms local approximations over these partition sets as a means of estimating the marginal
likelihood. In this dissertation, we provide both the motivation and the groundwork for developing
what we call the Hybrid estimator. Our numerical experiments show the versatility and accuracy of
the proposed estimator, even as the parameter space becomes increasingly high-dimensional and
complicated
Mean-field Variational Inference via Wasserstein Gradient Flow
Variational inference, such as the mean-field (MF) approximation, requires
certain conjugacy structures for efficient computation. These can impose
unnecessary restrictions on the viable prior distribution family and further
constraints on the variational approximation family. In this work, we introduce
a general computational framework to implement MF variational inference for
Bayesian models, with or without latent variables, using the Wasserstein
gradient flow (WGF), a modern mathematical technique for realizing a gradient
flow over the space of probability measures. Theoretically, we analyze the
algorithmic convergence of the proposed approaches, providing an explicit
expression for the contraction factor. We also strengthen existing results on
MF variational posterior concentration from a polynomial to an exponential
contraction, by utilizing the fixed point equation of the time-discretized WGF.
Computationally, we propose a new constraint-free function approximation method
using neural networks to numerically realize our algorithm. This method is
shown to be more precise and efficient than traditional particle approximation
methods based on Langevin dynamics
A Computational Framework for Efficient Reliability Analysis of Complex Networks
With the growing scale and complexity of modern infrastructure networks comes the challenge of developing efficient and dependable methods for analysing their reliability. Special attention must be given to potential network interdependencies as disregarding these can lead to catastrophic failures. Furthermore, it is of paramount importance to properly treat all uncertainties. The survival signature is a recent development built to effectively analyse complex networks that far exceeds standard techniques in several important areas. Its most distinguishing feature is the complete separation of system structure from probabilistic information. Because of this, it is possible to take into account a variety of component failure phenomena such as dependencies, common causes of failure, and imprecise probabilities without reevaluating the network structure.
This cumulative dissertation presents several key improvements to the survival signature ecosystem focused on the structural evaluation of the system as well as the modelling of component failures.
A new method is presented in which (inter)-dependencies between components and networks are modelled using vine copulas. Furthermore, aleatory and epistemic uncertainties are included by applying probability boxes and imprecise copulas. By leveraging the large number of available copula families it is possible to account for varying dependent effects. The graph-based design of vine copulas synergizes well with the typical descriptions of network topologies. The proposed method is tested on a challenging scenario using the IEEE reliability test system, demonstrating its usefulness and emphasizing the ability to represent complicated scenarios with a range of dependent failure modes.
The numerical effort required to analytically compute the survival signature is prohibitive for large complex systems. This work presents two methods for the approximation of the survival signature. In the first approach system configurations of low interest are excluded using percolation theory, while the remaining parts of the signature are estimated by Monte Carlo simulation. The method is able to accurately approximate the survival signature with very small errors while drastically reducing computational demand. Several simple test systems, as well as two real-world situations, are used to show the accuracy and performance.
However, with increasing network size and complexity this technique also reaches its limits. A second method is presented where the numerical demand is further reduced. Here, instead of approximating the whole survival signature only a few strategically selected values are computed using Monte Carlo simulation and used to build a surrogate model based on normalized radial basis functions. The uncertainty resulting from the approximation of the data points is then propagated through an interval predictor model which estimates bounds for the remaining survival signature values. This imprecise model provides bounds on the survival signature and therefore the network reliability. Because a few data points are sufficient to build the interval predictor model it allows for even larger systems to be analysed.
With the rising complexity of not just the system but also the individual components themselves comes the need for the components to be modelled as subsystems in a system-of-systems approach. A study is presented, where a previously developed framework for resilience decision-making is adapted to multidimensional scenarios in which the subsystems are represented as survival signatures. The survival signature of the subsystems can be computed ahead of the resilience analysis due to the inherent separation of structural information. This enables efficient analysis in which the failure rates of subsystems for various resilience-enhancing endowments are calculated directly from the survival function without reevaluating the system structure.
In addition to the advancements in the field of survival signature, this work also presents a new framework for uncertainty quantification developed as a package in the Julia programming language called UncertaintyQuantification.jl. Julia is a modern high-level dynamic programming language that is ideal for applications such as data analysis and scientific computing. UncertaintyQuantification.jl was built from the ground up to be generalised and versatile while remaining simple to use. The framework is in constant development and its goal is to become a toolbox encompassing state-of-the-art algorithms from all fields of uncertainty quantification and to serve as a valuable tool for both research and industry. UncertaintyQuantification.jl currently includes simulation-based reliability analysis utilising a wide range of sampling schemes, local and global sensitivity analysis, and surrogate modelling methodologies
Deep Statistical Models with Application to Environmental Data
When analyzing environmental data, constructing a realistic statistical model is important, not only to fully characterize the physical phenomena, but also to provide valid and useful predictions. Gaussian process models are amongst the most popular tools used for this purpose. However, many assumptions are usually made when using Gaussian processes, such as stationarity of the covariance function. There are several approaches to construct nonstationary spatial and spatio-temporal Gaussian processes, including the deformation approach. In the deformation approach, the geographical domain is warped into a new domain, on which the Gaussian process is modeled to be stationary. One of the main challenges with this approach is how to construct a deformation function that is complicated enough to adequately capture the nonstationarity in the process, but simple enough to facilitate statistical inference and prediction. In this thesis, by using ideas from deep learning, we construct deformation functions that are compositions of simple warping units. In particular, deformation functions that are composed of aligning functions and warping functions are introduced to model nonstationary and asymmetric multivariate spatial processes, while spatial and temporal warping functions are used to model nonstationary spatio-temporal processes. Similarly to the traditional deformation approach, familiar stationary models are used on the warped domain. It is shown that this new approach to model nonstationarity is computationally efficient, and that it can lead to predictions that are superior to those from stationary models. We show the utility of these models on both simulated data and real-world environmental data: ocean temperatures and surface-ice elevation. The developed warped nonstationary processes can also be used for emulation. We show that a warped, gradient-enhanced Gaussian process surrogate model can be embedded in algorithms such as importance sampling and delayed-acceptance Markov chain Monte Carlo. Our surrogate models can provide more accurate emulation than other traditional surrogate models, and can help speed up Bayesian inference in problems with exponential-family likelihoods with intractable normalizing constants, for example when analyzing satellite images using the Potts model
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