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

arXiv: https://arxiv.org/abs/2208.03680 · HTML: https://mlsanity.com/paper/2208.03680/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 83 references that are in the map.
- First posted on arXiv 7 Aug 2022.

## Builds on (most cited first, 30 of 32)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Squeeze-and-Excitation Networks](https://mlsanity.com/paper/1709.01507/) (2017)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [DARTS: Differentiable Architecture Search](https://mlsanity.com/paper/1806.09055/) (2018)
- [Large Scale Incremental Learning](https://mlsanity.com/paper/1905.13260/) (2019)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [On Lazy Training in Differentiable Programming](https://mlsanity.com/paper/1812.07956/) (2018)
- [Gradient Descent Provably Optimizes Over-parameterized Neural Networks](https://mlsanity.com/paper/1810.02054/) (2018)
- [In-Datacenter Performance Analysis of a Tensor Processing Unit](https://mlsanity.com/paper/1704.04760/) (2017)
- [The Expressive Power of Neural Networks: A View from the Width](https://mlsanity.com/paper/1709.02540/) (2017)
- [Symplectic Recurrent Neural Networks](https://mlsanity.com/paper/1909.13334/) (2019)
- [Rational neural networks](https://mlsanity.com/paper/2004.01902/) (2020)
- [Learning to correct spectral methods for simulating turbulent flows](https://mlsanity.com/paper/2207.00556/) (2022)
- [DIANet: Dense-and-Implicit Attention Network](https://mlsanity.com/paper/1905.10671/) (2019)
- [Convolution-Weight-Distribution Assumption: Rethinking the Criteria of Channel Pruning](https://mlsanity.com/paper/2004.11627/) (2020)
- [Implicit energy regularization of neural ordinary-differential-equation control](https://mlsanity.com/paper/2103.06525/) (2021)
- [Instance Enhancement Batch Normalization: an Adaptive Regulator of Batch Noise](https://mlsanity.com/paper/1908.04008/) (2019)
- [Machine Learning for Prediction with Missing Dynamics](https://mlsanity.com/paper/1910.05861/) (2019)
- [ChemNODE: A Neural Ordinary Differential Equations Approach for Chemical Kinetics Solvers](https://mlsanity.com/paper/2101.04749/) (2020)
- [Adaptable Hamiltonian neural networks](https://mlsanity.com/paper/2102.13235/) (2021)
- [Sparse Symplectically Integrated Neural Networks](https://mlsanity.com/paper/2006.12972/) (2020)
- [On generalized residue network for deep learning of unknown dynamical systems](https://mlsanity.com/paper/2002.02528/) (2020)
- [Long-time predictive modeling of nonlinear dynamical systems using neural networks](https://mlsanity.com/paper/1805.12547/) (2018)
- [Blending Pruning Criteria for Convolutional Neural Networks](https://mlsanity.com/paper/2107.05033/) (2021)
- [A Multifidelity Ensemble Kalman Filter with Reduced Order Control Variates](https://mlsanity.com/paper/2007.00793/) (2020)
- [A simple contagion process describes spreading of traffic jams in urban networks](https://mlsanity.com/paper/1906.00585/) (2019)
- [FastVA: Deep Learning Video Analytics Through Edge Processing and NPU in Mobile](https://mlsanity.com/paper/2001.04049/) (2020)

## Built on by (most cited first, 9 of 9)

- [Let's Think Outside the Box: Exploring Leap-of-Thought in Large Language Models with Creative Humor Generation](https://mlsanity.com/paper/2312.02439/) (2023)
- [FMint: Bridging Human Designed and Data Pretrained Models for Differential Equation Foundation Model](https://mlsanity.com/paper/2404.14688/) (2024)
- [Innovative Thinking, Infinite Humor: Humor Research of Large Language Models through Structured Thought Leaps](https://mlsanity.com/paper/2410.10370/) (2024)
- [A Causality-aware Paradigm for Evaluating Creativity of Multimodal Large Language Models](https://mlsanity.com/paper/2501.15147/) (2025)
- [Covering Number of Real Algebraic Varieties and Beyond: Improved Bounds and Applications](https://mlsanity.com/paper/2311.05116/) (2023)
- [AttNS: Attention-Inspired Numerical Solving For Limited Data Scenarios](https://mlsanity.com/paper/2302.10184/) (2023)
- [Algorithmically Designed Artificial Neural Networks (ADANNs): Higher order deep operator learning for parametric partial differential equations](https://mlsanity.com/paper/2302.03286/) (2023)
- [Neural network-enhanced integrators for simulating ordinary differential equations](https://mlsanity.com/paper/2504.05493/) (2025)
- [FMint-SDE: A Multimodal Foundation Model for Accelerating Numerical Simulation of SDEs via Error Correction](https://mlsanity.com/paper/2510.27173/) (2025)

## Abstract (arXiv)

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.