Complexity Theory meets Ordinary Differential Equations
For researchers in scientific computing and analog simulation, it provides a theoretical limit on the efficiency of digital simulation of linear ODEs, showing that complexity blowup is generic.
The paper characterizes the computational complexity of simulating linear ODEs, showing that for most ODEs (except degenerate cases) there is a complexity blowup: a low-complexity input leads to an output whose approximation requires super-polynomial time. This extends previous results for first-order ODEs and is applied to a neuronal model.
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.