Qi Gong

Tenavi Nakamura-Zimmerer, Qi Gong, Wei Kang

Recent research has shown that supervised learning can be an effective tool for designing optimal feedback controllers for high-dimensional nonlinear dynamic systems. But the behavior of these neural network (NN) controllers is still not well understood. In this paper we use numerical simulations to demonstrate that typical test accuracy metrics do not effectively capture the ability of an NN controller to stabilize a system. In particular, some NNs with high test accuracy can fail to stabilize the dynamics. To address this we propose two NN architectures which locally approximate a linear quadratic regulator (LQR). Numerical simulations confirm our intuition that the proposed architectures reliably produce stabilizing feedback controllers without sacrificing optimality. In addition, we introduce a preliminary theoretical result describing some stability properties of such NN-controlled systems.

Wei Kang, Qi Gong

As demonstrated in many areas of real-life applications, neural networks have the capability of dealing with high dimensional data. In the fields of optimal control and dynamical systems, the same capability was studied and verified in many published results in recent years. Towards the goal of revealing the underlying reason why neural networks are capable of solving some high dimensional problems, we develop an algebraic framework and an approximation theory for compositional functions and their neural network approximations. The theoretical foundation is developed in a way so that it supports the error analysis for not only functions as input-output relations, but also numerical algorithms. This capability is critical because it enables the analysis of approximation errors for problems for which analytic solutions are not available, such as differential equations and optimal control. We identify a set of key features of compositional functions and the relationship between the features and the complexity of neural networks. In addition to function approximations, we prove several formulae of error upper bounds for neural networks that approximate the solutions to differential equations, optimization, and optimal control.

Tenavi Nakamura-Zimmerer, Qi Gong, Wei Kang

In this paper we propose a new computational method for designing optimal regulators for high-dimensional nonlinear systems. The proposed approach leverages physics-informed machine learning to solve high-dimensional Hamilton-Jacobi-Bellman equations arising in optimal feedback control. Concretely, we augment linear quadratic regulators with neural networks to handle nonlinearities. We train the augmented models on data generated without discretizing the state space, enabling application to high-dimensional problems. We use the proposed method to design a candidate optimal regulator for an unstable Burgers' equation, and through this example, demonstrate improved robustness and accuracy compared to existing neural network formulations.

Tenavi Nakamura-Zimmerer, Daniele Venturi, Qi Gong et al.

Uncertainty propagation in nonlinear dynamic systems remains an outstanding problem in scientific computing and control. Numerous approaches have been developed, but are limited in their capability to tackle problems with more than a few uncertain variables or require large amounts of simulation data. In this paper, we propose a data-driven method for approximating joint probability density functions (PDFs) of nonlinear dynamic systems with initial condition and parameter uncertainty. Our approach leverages on the power of deep learning to deal with high-dimensional inputs, but we overcome the need for huge quantities of training data by encoding PDF evolution equations directly into the optimization problem. We demonstrate the potential of the proposed method by applying it to evaluate the robustness of a feedback controller for a six-dimensional rigid body with parameter uncertainty.