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As demonstrated in many areas of real-life applications, neural networks havethe capability of dealing with high dimensional data. In the fields of optimalcontrol and dynamical systems, the same capability was studied and verified inmany published results in recent years. Towards the goal of revealing theunderlying reason why neural networks are capable of solving some highdimensional problems, we develop an algebraic framework and an approximationtheory for compositional functions and their neural network approximations. Thetheoretical foundation is developed in a way so that it supports the erroranalysis for not only functions as input-output relations, but also numericalalgorithms. This capability is critical because it enables the analysis ofapproximation errors for problems for which analytic solutions are notavailable, such as differential equations and optimal control. We identify aset of key features of compositional functions and the relationship between thefeatures and the complexity of neural networks. In addition to functionapproximations, we prove several formulae of error upper bounds for neuralnetworks that approximate the solutions to differential equations,optimization, and optimal control.

Neural Network Approximations of Compositional Functions With Applications to Dynamical Systems