Title: DS-TG: Dynamical Systems as Accurate and Efficient Solvers for Time-Dependent Differential Equations
Primary area: infrastructure, software libraries, hardware, systems, etc.
Keywords: Dynamical Systems, Time-Dependent Differential Equations

Abstract: Time-Dependent Differential Equations (TDDEs) are central to modeling dynamic processes in various scientific and engineering systems. Numerical solvers typically provide reliable solutions but are burdened by prohibitive costs due to fine-grained discretization and iterative procedures. Recent machine learning (ML)-based approaches attempt to accelerate computation through fast ML inference, yet they are often trained on trajectories produced by numerical solvers, resulting in reduced accuracy and limited generalization. Designing a TDDE solver that achieves good accuracy and high efficiency remains a fundamental challenge. In this paper, we introduce DS-TG, a novel TDDE solver that employs Dynamical Systems (DS) as Trajectory Generators (TG), exploiting the intrinsic connection between DS and TDDEs. DS-TG leverages a DS-based processor whose physical states evolve continuously in real time according to carefully designed dynamics that directly emulate the target TDDE. This approach represents a novel paradigm fundamentally distinct from traditional discrete-time methods, offering inherent advantages in both accuracy and efficiency. Specifically, the continuous evolution of DS-TG can be seen as partitioning the target trajectory into a continuum of infinitesimal time steps, thereby reducing the problem to learning the trajectory gradient at each intermediate state of evolution. Building on this foundation, we further introduce two hardware-friendly techniques to enhance the dynamics design: (1) Laplacian-style interactions for effectively capturing spatial derivatives and (2) higher-order interactions for better representing higher-order temporal derivatives. Extensive experiments across representative TDDEs demonstrate that DS-TG achieves superior accuracy while delivering up to $10^4\times$ efficiency improvement compared to baseline methods.