Deep neural networks (DNNs) have achieved enormous success in a wide range of domains, such as computer vision, natural language processing and scientific areas. However, one key bottleneck of DNNs is that they are ignorant about the uncertainties in their predictions. They can produce wildly wrong predictions without realizing, and can even be confident about their mistakes. Such mistakes can cause misguided decisions-sometimes catastrophic in critical applications, ranging from self-driving cars to cyber security to automatic medical diagnosis. In this tutorial, we present recent advancements in uncertainty quantification for DNNs and their applications across various domains. We first provide an overview of the motivation behind uncertainty quantification, different sources of uncertainty, and evaluation metrics. Then, we delve into several representative uncertainty quantification methods for predictive models, including ensembles, Bayesian neural networks, conformal prediction, and others. We go on to discuss how uncertainty can be utilized for label-efficient learning, continual learning, robust decision-making, and experimental design. Furthermore, we showcase examples of uncertainty-aware DNNs in various domains, such as health, robotics, and scientific machine learning. Finally, we summarize open challenges and future directions in this area.

The curse of dimensionality is a longstanding challenge in Bayesian inference in high dimensions. In this work, we propose a {projected Stein variational gradient descent} (pSVGD) method to overcome this challenge by exploiting the fundamental property of intrinsic low dimensionality of the data informed subspace stemming from ill-posedness of such problems. We adaptively construct the subspace using a gradient information matrix of the log-likelihood, and apply pSVGD to the much lower-dimensional coefficients of the parameter projection. The method is demonstrated to be more accurate and efficient than SVGD. It is also shown to be more scalable with respect to the number of parameters, samples, data points, and processor cores via experiments with parameters dimensions ranging from the hundreds to the tens of thousands.

We propose a projected Stein variational Newton (pSVN) method for high-dimensional Bayesian inference. To address the curse of dimensionality, we exploit the intrinsic low-dimensional geometric structure of the posterior distribution in the high-dimensional parameter space via its Hessian (of the log posterior) operator and perform a parallel update of the parameter samples projected into a low-dimensional subspace by an SVN method. The subspace is adaptively constructed using the eigenvectors of the averaged Hessian at the current samples. We demonstrate fast convergence of the proposed method, complexity independent of the parameter and sample dimensions, and parallel scalability.