The large-scale simulation of dynamical systems is critical in numerous scientific and engineering disciplines. However, traditional numerical solvers are limited by the choice of step sizes when estimating integration, resulting in a trade-off between accuracy and computational efficiency. To address this challenge, we introduce a deep learning-based corrector called Neural Vector (NeurVec), which can compensate for integration errors and enable larger time step sizes in simulations. Our extensive experiments on a variety of complex dynamical system benchmarks demonstrate that NeurVec exhibits remarkable generalization capability on a continuous phase space, even when trained using limited and discrete data. NeurVec significantly accelerates traditional solvers, achieving speeds tens to hundreds of times faster while maintaining high levels of accuracy and stability. Moreover, NeurVec's simple-yet-effective design, combined with its ease of implementation, has the potential to establish a new paradigm for fast-solving differential equations based on deep learning.

翻译：动态系统的大规模模拟在众多科学与工程领域中至关重要。然而，传统数值求解器在估计积分时受步长选择限制，导致精度与计算效率之间存在权衡。为解决这一挑战，我们提出了一种基于深度学习的校正器——神经向量（NeurVec），它能够补偿积分误差，从而在模拟中实现更大的时间步长。我们在多种复杂动态系统基准上的大量实验表明，即使使用有限的离散数据训练，NeurVec在连续相空间上也展现出卓越的泛化能力。NeurVec显著加速了传统求解器，在保持高精度和高稳定性的同时，实现了数十至数百倍的速度提升。此外，NeurVec简单而有效的设计，结合其易于实现的特性，有望为基于深度学习的快速求解微分方程建立新范式。