  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.
