Scalable Approach to Uncertainty Quantification and Robust Design of Interconnected Dynamical Systems

Development of robust dynamical systems and networks such as autonomous aircraft systems capable of accomplishing complex missions faces challenges due to the dynamically evolving uncertainties coming from model uncertainties, necessity to operate in a hostile cluttered urban environment, and the distributed and dynamic nature of the communication and computation resources. Model-based robust design is difficult because of the complexity of the hybrid dynamic models including continuous vehicle dynamics, the discrete models of computations and communications, and the size of the problem. We will overview recent advances in methodology and tools to model, analyze, and design robust autonomous aerospace systems operating in uncertain environment, with stress on efficient uncertainty quantification and robust design using the case studies of the mission including model-based target tracking and search, and trajectory planning in uncertain urban environment. To show that the methodology is generally applicable to uncertain dynamical systems, we will also show examples of application of the new methods to efficient uncertainty quantification of energy usage in buildings, and stability assessment of interconnected power networks

Selectively decentralized reinforcement learning

Indiana University-Purdue University Indianapolis (IUPUI)The main contributions in this thesis include the selectively decentralized method in solving multi-agent reinforcement learning problems and the discretized Markov-decision-process (MDP) algorithm to compute the sub-optimal learning policy in completely unknown learning and control problems. These contributions tackle several challenges in multi-agent reinforcement learning: the unknown and dynamic nature of the learning environment, the difficulty in computing the closed-form solution of the learning problem, the slow learning performance in large-scale systems, and the questions of how/when/to whom the learning agents should communicate among themselves. Through this thesis, the selectively decentralized method, which evaluates all of the possible communicative strategies, not only increases the learning speed, achieves better learning goals but also could learn the communicative policy for each learning agent. Compared to the other state-of-the-art approaches, this thesis’s contributions offer two advantages. First, the selectively decentralized method could incorporate a wide range of well-known algorithms, including the discretized MDP, in single-agent reinforcement learning; meanwhile, the state-of-the-art approaches usually could be applied for one class of algorithms. Second, the discretized MDP algorithm could compute the sub-optimal learning policy when the environment is described in general nonlinear format; meanwhile, the other state-of-the-art approaches often assume that the environment is in limited format, particularly in feedback-linearization form. This thesis also discusses several alternative approaches for multi-agent learning, including Multidisciplinary Optimization. In addition, this thesis shows how the selectively decentralized method could successfully solve several real-worlds problems, particularly in mechanical and biological systems

Error estimates of penalty schemes for quasi-variational inequalities arising from impulse control problems

This paper proposes penalty schemes for a class of weakly coupled systems of Hamilton-Jacobi-Bellman quasi-variational inequalities (HJBQVIs) arising from stochastic hybrid control problems of regime-switching models with both continuous and impulse controls. We show that the solutions of the penalized equations converge monotonically to those of the HJBQVIs. We further establish that the schemes are half-order accurate for HJBQVIs with Lipschitz coefficients, and first-order accurate for equations with more regular coefficients. Moreover, we construct the action regions and optimal impulse controls based on the error estimates and the penalized solutions. The penalty schemes and convergence results are then extended to HJBQVIs with possibly negative impulse costs. We also demonstrate the convergence of monotone discretizations of the penalized equations, and establish that policy iteration applied to the discrete equation is monotonically convergent with an arbitrary initial guess in an infinite dimensional setting. Numerical examples for infinite-horizon optimal switching problems are presented to illustrate the effectiveness of the penalty schemes over the conventional direct control scheme.Comment: Accepted for publication in SIAM Journal on Control and Optimizatio

Piecewise Constant Policy Approximations to Hamilton-Jacobi-Bellman Equations

An advantageous feature of piecewise constant policy timestepping for Hamilton-Jacobi-Bellman (HJB) equations is that different linear approximation schemes, and indeed different meshes, can be used for the resulting linear equations for different control parameters. Standard convergence analysis suggests that monotone (i.e., linear) interpolation must be used to transfer data between meshes. Using the equivalence to a switching system and an adaptation of the usual arguments based on consistency, stability and monotonicity, we show that if limited, potentially higher order interpolation is used for the mesh transfer, convergence is guaranteed. We provide numerical tests for the mean-variance optimal investment problem and the uncertain volatility option pricing model, and compare the results to published test cases

A fast iterative PDE-based algorithm for feedback controls of nonsmooth mean-field control problems

A PDE-based accelerated gradient algorithm is proposed to seek optimal feedback controls of McKean-Vlasov dynamics subject to nonsmooth costs, whose coefficients involve mean-field interactions both on the state and action. It exploits a forward-backward splitting approach and iteratively refines the approximate controls based on the gradients of smooth costs, the proximal maps of nonsmooth costs, and dynamically updated momentum parameters. At each step, the state dynamics is realized via a particle approximation, and the required gradient is evaluated through a coupled system of nonlocal linear PDEs. The latter is solved by finite difference approximation or neural network-based residual approximation, depending on the state dimension. Exhaustive numerical experiments for low and high-dimensional mean-field control problems, including sparse stabilization of stochastic Cucker-Smale models, are presented, which reveal that our algorithm captures important structures of the optimal feedback control, and achieves a robust performance with respect to parameter perturbation.Comment: Add Sections 2.3 and 2.4 for theoretical convergence result

Deep Learning for Mean Field Games with non-separable Hamiltonians

This paper introduces a new method based on Deep Galerkin Methods (DGMs) for solving high-dimensional stochastic Mean Field Games (MFGs). We achieve this by using two neural networks to approximate the unknown solutions of the MFG system and forward-backward conditions. Our method is efficient, even with a small number of iterations, and is capable of handling up to 300 dimensions with a single layer, which makes it faster than other approaches. In contrast, methods based on Generative Adversarial Networks (GANs) cannot solve MFGs with non-separable Hamiltonians. We demonstrate the effectiveness of our approach by applying it to a traffic flow problem, which was previously solved using the Newton iteration method only in the deterministic case. We compare the results of our method to analytical solutions and previous approaches, showing its efficiency. We also prove the convergence of our neural network approximation with a single hidden layer using the universal approximation theorem