On Fast Simulation of Dynamical System with Neural Vector Enhanced Numerical Solver

TL;DR

This paper introduces Neural Vector (NeurVec), a deep learning corrector that enhances traditional numerical solvers for dynamical systems, enabling larger time steps and significantly faster simulations without sacrificing accuracy.

Contribution

NeurVec is a novel deep learning-based corrector that improves simulation speed and accuracy for dynamical systems by compensating integration errors, even with limited training data.

Findings

NeurVec achieves speeds 10-100 times faster than traditional solvers.

It generalizes well across continuous phase spaces.

It maintains high accuracy and stability in complex systems.

Abstract

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.