Complexity Theory meets Ordinary Differential Equations

TL;DR

This paper characterizes the computational complexity involved in simulating linear ordinary differential equations (ODEs), revealing that most ODEs exhibit a complexity blowup that affects simulation efficiency.

Contribution

It extends previous complexity characterizations from first order to arbitrary order ODEs and discusses implications for simulating analog systems like neuronal models.

Findings

Most ODEs exhibit complexity blowup leading to super-polynomial simulation times.

A criterion for complexity blowup is derived for a subclass of first-order linear ODE systems.

Application to neuronal dynamics demonstrates practical relevance.

Abstract

This contribution investigates the computational complexity of simulating linear ordinary differential equations (ODEs) on digital computers. We provide an exact characterization of the complexity blowup for a class of ODEs of arbitrary order based on their algebraic properties, extending previous characterization of first order ODEs. Complexity blowup indeed arises in most ODEs (except for certain degenerate cases) and means that there exists a low complexity input signal, which can be generated on a Turing machine in polynomial time, leading to a corresponding high complexity output signal of the system in the sense that the computation time for determining an approximation up to $n$ significant digits grows faster than any polynomial in $n$. Similarly, we derive an analogous blowup criterion for a subclass of first-order systems of linear ODEs. Finally, we discuss the implications for the simulation of analog systems governed by ODEs and exemplarily apply our framework to a simple model of neuronal dynamics$-$the leaky integrate-and-fire neuron$-$heavily employed in neuroscience.