Functional automatic differentiation with dirac impulses

Abstract

Functional Reactive Programming (FRP) is a framework for reactive programming in a functional setting. FRP has been applied to a number of domains, such as graphical animation, graphical user interfaces, robotics, and computer vision. Recently, we have been interested in applying FRPlike principles to hybrid modeling and simulation of physical systems. As a step in that direction, we have extended an existing FRP implementation, Yampa, in two new ways that make it possible to express certain models in a very natural way, and reduces the amount of work needed to put modeling equations into a suitable form for simulation. First, we have added Dirac impulses that allow certain types of discontinuities to be handled in an easy yet rigorous manner. Second, we have adapted automatic differentiation to the setting of Yampa, and generalized it to work correctly with Dirac impulses. This allows derivatives of piecewise continuous signals to be well-defined at all points. This paper reviews the basic ideas behind automatic differentiation, in particular Jerzy Karczmarczuk's elegant version for a lazy functional language with overloading, and then considers the integration with Yampa and the addition of Dirac impulses.

Cofree coalgebras and differential linear logic

“…It is interesting to note that functional programs extended with Dirac distributions have already been considered in the literature on automatic differentiation; see Nilsson (2003). For an abstract categorical theory of distributions via monads, see Kock (2012).…”

Beautiful differentiation

“…Henrik Nilsson (2003) extended higher-order AD to work on a generalized notion of functions that includes Dirac impulses, allowing for more elegant functional specification of behaviors involving instantaneous velocity changes. These derivatives were for functions over a scalar domain (time).…”

Hybrid System Modelling and Simulation with Dirac Deltas

“…Simulators with support for Dirac impulses have been proposed in [11] and more recently in [8], but both approaches deal with the symbolic computation of impulses, and not their numerical approximation, nor evaluation or comparison is advanced for the approach.…”

“…Our research hypothesis is: (RH) a simulator which manipulates Dirac deltas symbolically (such as the one proposed in [11]) is more accurate than one that just operates with approximated impulses.…”

“…C1 builds on the work of [11], where many of the symbolic operations on Dirac deltas are derived. Here, we derive the remaining operations -decision and inversion -and provide justification to the use of distributions as a formalization of a correct impulse symbolic simulator.…”