# The Seven-League Scheme: Deep learning for large time step Monte Carlo simulations of stochastic differential equations

arXiv: https://arxiv.org/abs/2009.03202 · HTML: https://mlsanity.com/paper/2009.03202/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Sep 2020.

## Built on by (most cited first, 3 of 3)

- [Monte Carlo Simulation of SDEs using GANs](https://mlsanity.com/paper/2104.01437/) (2021)
- [ISALT: Inference-based schemes adaptive to large time-stepping for locally Lipschitz ergodic systems](https://mlsanity.com/paper/2102.12669/) (2021)
- [FMint-SDE: A Multimodal Foundation Model for Accelerating Numerical Simulation of SDEs via Error Correction](https://mlsanity.com/paper/2510.27173/) (2025)

## Abstract (arXiv)

We propose an accurate data-driven numerical scheme to solve Stochastic Differential Equations (SDEs), by taking large time steps. The SDE discretization is built up by means of a polynomial chaos expansion method, on the basis of accurately determined stochastic collocation (SC) points. By employing an artificial neural network to learn these SC points, we can perform Monte Carlo simulations with large time steps. Error analysis confirms that this data-driven scheme results in accurate SDE solutions in the sense of strong convergence, provided the learning methodology is robust and accurate. With a method variant called the compression-decompression collocation and interpolation technique, we can drastically reduce the number of neural network functions that have to be learned, so that computational speed is enhanced. Numerical experiments confirm a high-quality strong convergence error when using large time steps, and the novel scheme outperforms some classical numerical SDE discretizations. Some applications, here in financial option valuation, are also presented.