Well Posedness and Convergence Analysis of the Ensemble Kalman Inversion

The ensemble Kalman inversion is widely used in practice to estimate unknown parameters from noisy measurement data. Its low computational costs, straightforward implementation, and non-intrusive nature makes the method appealing in various areas of application. We present a complete analysis of the ensemble Kalman inversion with perturbed observations for a fixed ensemble size when applied to linear inverse problems. The well-posedness and convergence results are based on the continuous time scaling limits of the method. The resulting coupled system of stochastic differential equations allows to derive estimates on the long-time behaviour and provides insights into the convergence properties of the ensemble Kalman inversion. We view the method as a derivative free optimization method for the least-squares misfit functional, which opens up the perspective to use the method in various areas of applications such as imaging, groundwater flow problems, biological problems as well as in the context of the training of neural networks

Value iteration with deep neural networks for optimal control of input-affine nonlinear systems

This paper proposes a new algorithm with deep neural networks to solve optimal control problems for continuous-time input nonlinear systems based on a value iteration algorithm. The proposed algorithm applies the networks to approximating the value functions and control inputs in the iterations. Consequently, the partial differential equations of the original algorithm reduce to the optimization problems for the parameters of the networks. Although the conventional algorithm can obtain the optimal control with iterative computations, each of the computations needs to be completed precisely, and it is hard to achieve sufficient precision in practice. Instead, the proposed method provides a practical method using deep neural networks and overcomes the difficulty based on a property of the networks, under which our convergence analysis shows that the proposed algorithm can achieve the minimum of the value function and the corresponding optimal controller. The effectiveness of the proposed method even with reasonable computational resources is demonstrated in two numerical simulations

Generalized Policy Iteration for Optimal Control in Continuous Time

This paper proposes the Deep Generalized Policy Iteration (DGPI) algorithm to find the infinite horizon optimal control policy for general nonlinear continuous-time systems with known dynamics. Unlike existing adaptive dynamic programming algorithms for continuous time systems, DGPI does not require the admissibility of initialized policy, and input-affine nature of controlled systems for convergence. Our algorithm employs the actor-critic architecture to approximate both policy and value functions with the purpose of iteratively solving the Hamilton-Jacobi-Bellman equation. Both the policy and value functions are approximated by deep neural networks. Given any arbitrary initial policy, the proposed DGPI algorithm can eventually converge to an admissible, and subsequently an optimal policy for an arbitrary nonlinear system. We also relax the update termination conditions of both the policy evaluation and improvement processes, which leads to a faster convergence speed than conventional Policy Iteration (PI) methods, for the same architecture of function approximators. We further prove the convergence and optimality of the algorithm with thorough Lyapunov analysis, and demonstrate its generality and efficacy using two detailed numerical examples

Neural Parametric Fokker-Planck Equations

In this paper, we develop and analyze numerical methods for high dimensional Fokker-Planck equations by leveraging generative models from deep learning. Our starting point is a formulation of the Fokker-Planck equation as a system of ordinary differential equations (ODEs) on finite-dimensional parameter space with the parameters inherited from generative models such as normalizing flows. We call such ODEs neural parametric Fokker-Planck equation. The fact that the Fokker-Planck equation can be viewed as the $L^2$-Wasserstein gradient flow of Kullback-Leibler (KL) divergence allows us to derive the ODEs as the constrained $L^2$-Wasserstein gradient flow of KL divergence on the set of probability densities generated by neural networks. For numerical computation, we design a variational semi-implicit scheme for the time discretization of the proposed ODE. Such an algorithm is sampling-based, which can readily handle Fokker-Planck equations in higher dimensional spaces. Moreover, we also establish bounds for the asymptotic convergence analysis of the neural parametric Fokker-Planck equation as well as its error analysis for both the continuous and discrete (forward-Euler time discretization) versions. Several numerical examples are provided to illustrate the performance of the proposed algorithms and analysis