An Asynchronous multi-rate Taylor method for Delay Differential Equations

2026-06-18

AI Summary

This work addresses the computational challenges in numerically solving high-dimensional, multi-rate delay differential equations (DDEs), which are typically constrained by synchronous stepping and dynamic memory allocation. We propose the Asynchronous Adaptive Taylor Solver (AATS), which equips each coordinate with an independent local clock and leverages compile-time automatic differentiation to generate high-order Taylor polynomials. AATS evaluates only active subgraphs and stores polynomial segments in static circular buffers, thereby enabling continuous output without interpolation or runtime memory allocation. We establish, for the first time, a theory of continuous convergence for asynchronous Taylor expansions and prove that the algorithm achieves strict linear complexity, O(N). In benchmarks at scale N = 10⁴, AATS demonstrates near-theoretically linear strong scaling and substantial wall-clock speedups compared to state-of-the-art synchronous solvers such as those in Julia’s SciML ecosystem.

Abstract

The numerical simulation of high-dimensional, multi-rate Delay Differential Equations (DDEs) is fundamentally bottlenecked by synchronous time-stepping and the dynamic memory allocation required for continuous history tracking. In this paper, we introduce the Asynchronous Adaptive Taylor Solver (AATS), an event-driven integration framework designed to overcome these high-performance computing limitations. By assigning independent local clocks to individual coordinates and advancing them using high-order Taylor polynomials generated via compile-time Automatic Differentiation, AATS restricts computational work to actively evolving sub-graphs. To eliminate the severe memory overhead endemic to traditional DDE solvers, AATS utilizes statically allocated circular buffers to store polynomial segments, achieving interpolation-free continuous dense-output evaluation with a verified zero-allocation runtime memory footprint. Alongside this software architecture, we establish a novel continuous proof of convergence for asynchronous Taylor expansions and formally prove that the framework's algorithmic complexity scales linearly (O(N)). Extensive benchmarks against state-of-the-art synchronous solvers (Julia SciML) validate these theoretical bounds. On large-scale benchmarks (upto $N = 10000$ coordinates) AATS fundamentally minimizes the constant factor of algorithmic work by avoiding redundant evaluations, delivering empirically consistent with O(N) execution scaling and significant wall-clock speedups.

Problem

• Delay Differential Equations
• multi-rate
• synchronous time-stepping
• dynamic memory allocation
• high-dimensional

Innovation

• asynchronous integration
• Taylor method
• delay differential equations
• automatic differentiation
• zero-allocation memory