Showing papers on "Gâteaux derivative published in 2019"

The differential calculus of causal functions.

Abstract: Causal functions of sequences occur throughout computer science, from theory to hardware to machine learning. Mealy machines, synchronous digital circuits, signal flow graphs, and recurrent neural networks all have behaviour that can be described by causal functions. In this work, we examine a differential calculus of causal functions which includes many of the familiar properties of standard multivariable differential calculus. These causal functions operate on infinite sequences, but this work gives a different notion of an infinite-dimensional derivative than either the Fr\'echet or Gateaux derivative used in functional analysis. In addition to showing many standard properties of differentiation, we show causal differentiation obeys a unique recurrence rule. We use this recurrence rule to compute the derivative of a simple recurrent neural network called an Elman network by hand and describe how the computed derivative can be used to train the network.

Solution of Systems of Partial Differential Equations by Using Properties of Monogenic Functions on Commutative Algebras

Abstract: Some systems of differential equations with partial derivatives are studied by using the properties of Gâteaux differentiable functions on commutative algebras. The connection between solutions of systems of partial differential equations and components of monogenic functions on the corresponding commutative algebras is shown. We also give some examples of systems of partial differential equations and find their solutions.