Neural Network Approximations of Compositional Functions With Applications to Dynamical Systems
This work provides theoretical foundations and error analysis for neural network approximations of compositional functions, which is significant for researchers and practitioners in optimal control and dynamical systems, especially for problems lacking analytic solutions.
This paper develops an algebraic framework and approximation theory for compositional functions and their neural network approximations, aiming to explain why neural networks handle high-dimensional problems. It provides error upper bounds for neural network approximations of solutions to differential equations, optimization, and optimal control problems.
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.