Student-t Processes as Alternatives to Gaussian Processes

We investigate the Student-t process as an alternative to the Gaussian process as a nonparametric prior over functions. We derive closed form expressions for the marginal likelihood and predictive distribution of a Student-t process, by integrating away an inverse Wishart process prior over the covariance kernel of a Gaussian process model. We show surprising equivalences between different hierarchical Gaussian process models leading to Student-t processes, and derive a new sampling scheme for the inverse Wishart process, which helps elucidate these equivalences. Overall, we show that a Student-t process can retain the attractive properties of a Gaussian process -- a nonparametric representation, analytic marginal and predictive distributions, and easy model selection through covariance kernels -- but has enhanced flexibility, and predictive covariances that, unlike a Gaussian process, explicitly depend on the values of training observations. We verify empirically that a Student-t process is especially useful in situations where there are changes in covariance structure, or in applications like Bayesian optimization, where accurate predictive covariances are critical for good performance. These advantages come at no additional computational cost over Gaussian processes.Comment: 13 pages, 6 figures, 1 table. To appear in "The Seventeenth International Conference on Artificial Intelligence and Statistics (AISTATS), 2014.

Machine Learning for Fluid Mechanics

The field of fluid mechanics is rapidly advancing, driven by unprecedented volumes of data from field measurements, experiments and large-scale simulations at multiple spatiotemporal scales. Machine learning offers a wealth of techniques to extract information from data that could be translated into knowledge about the underlying fluid mechanics. Moreover, machine learning algorithms can augment domain knowledge and automate tasks related to flow control and optimization. This article presents an overview of past history, current developments, and emerging opportunities of machine learning for fluid mechanics. It outlines fundamental machine learning methodologies and discusses their uses for understanding, modeling, optimizing, and controlling fluid flows. The strengths and limitations of these methods are addressed from the perspective of scientific inquiry that considers data as an inherent part of modeling, experimentation, and simulation. Machine learning provides a powerful information processing framework that can enrich, and possibly even transform, current lines of fluid mechanics research and industrial applications.Comment: To appear in the Annual Reviews of Fluid Mechanics, 202

Optimal Control of Energy Efficient Buildings

The building sector consumes a large part of the energy used in the United States and is responsible for nearly 40% of greenhouse gas emissions. Therefore, it is economically and environmentally important to reduce the building energy consumption to realize massive energy savings. Commercial buildings are complex, multi-physics, and highly stochastic dynamic systems. Recent work has focused on integrating modern modeling, simulation, and control techniques to solving this challenging problem. The overall focus of this thesis is directed toward designing an energy efficient building by controlling room temperature. One approach is based on a distributed parameter model represented by a three dimensional (3D) heat equation in a room with heater/cooler located at ceiling. The finite element method is implemented as part of a novel solution to this problem. A reduced order model of only few states is derived using Proper Orthogonal Decomposition (POD). A Linear Quadratic Regulator (LQR) is computed based on the reduced model, and applied to the full order model to control room temperature. Also, a receding horizon constrained linear quadratic Gaussian (LQG) controller is developed by minimizing energy cost of heating and cooling while satisfying hard and probabilistic temperature constraints. A stochastic receding horizon controller (RHC) is employed to solve the optimization problem with the so-called chance constraints governed by probability temperature levels. Furthermore, a constrained stochastic linear quadratic control (SLQC) approach was developed for such purposes. The cost function to be minimized is quadratic, and two different cases are considered. The first case assumes the disturbance is Gaussian and the problem is formulated to minimize the expected cost subject to a linear constraint and a probabilistic constraint. The second case assumes the disturbance is norm-bounded with distribution unknown and the problem is formulated as a min-max problem. By using SLQC, both problems are reduced to semidefinite optimization problems, where the optimal control may be computed efficiently. Later, some discussions on solving more requirements by SLQC are provided. Simulation and numerical results are given to demonstrate the validity of the proposed techniques shown in this thesis