Complexity Theory meets Ordinary Differential Equations

2026-04-10

AI Summary

This study investigates the phenomenon of complexity explosion in digital computer simulations of linear ordinary differential equations (ODEs), wherein low-complexity inputs yield high-complexity outputs. By integrating computational complexity theory, algebraic analysis, and numerical approximation techniques, the work provides the first precise characterization of complexity explosion for linear ODEs of arbitrary order and extends this framework to first-order linear systems, establishing a rigorous connection to Turing-machine computability. The results demonstrate that, except for a few degenerate cases, complexity explosion is inherent in the digital simulation of almost all linear ODEs. This theoretical framework is successfully applied to the leaky integrate-and-fire neuron model, revealing fundamental limitations in the digital simulation of neural dynamical systems.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Web Mining and Content Analysis: Web data generation and simulation
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications

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.

Problem

• computational complexity
• ordinary differential equations
• complexity blowup
• digital simulation
• Turing machine

Innovation

• computational complexity
• ordinary differential equations
• complexity blowup
• Turing machine
• linear ODEs