Solving Time-Dependent Differential Equations with Physical Dynamical Systems

Time-Dependent Differential Equations (TDDEs) model dynamical processes across science and engineering, but time-critical applications require solvers that deliver high-fidelity trajectories under stringent latency constraints. Most existing TDDE solvers are limited by time discretization, forcing a latency-accuracy trade-off where smaller step sizes capture high-fidelity trajectories but incur prohibitive runtime, while larger steps meet real-time budgets at the cost of trajectory distortion. Dynamical System Machines (DSMs) offer a promising alternative by computing through continuous physical evolution, yet existing DSMs struggle to capture the spatiotemporal complexity of TDDEs. This work introduces DS-TS, a novel TDDE solver that is both accurate and efficient by leveraging the unique computational advantages of DSMs. DS-TS integrates three key innovations: (1) Excitatory-Inhibitory Inspired Coupling to better model complex spatial interactions; (2) State-aware Dynamic Nonlinearity to enable rich inter-node interactions and state-dependent spatiotemporal correlations; and (3) Hierarchical Temporal Integration to capture high-order temporal dependencies. Experiments demonstrate that DS-TS achieves high-fidelity solutions while delivering orders-of-magnitude improvements in speed ($\sim 10^3\times$) and energy efficiency ($\sim 10^5\times$) compared to baseline solvers.

This paper introduces DS-TS, a fast and energy-efficient approach to solve time-dependent differential equations, which are used to model dynamical systems that change over time. Traditional solvers rely on time discretization: small timesteps are accurate but slow, while large timesteps are fast but less reliable. DS-TS addresses this limitation by leveraging Dynamical System Machines, which compute through continuous physical evolution rather than step-by-step digital calculation. DS-TS incorporates new mechanisms to better capture the spatiotemporal complexity of TDDEs. Experiments show that DS-TS can produce accurate trajectories while being about 1,000× faster and 100,000× more energy-efficient than baseline solvers.