# ISALT: Inference-based schemes adaptive to large time-stepping for locally Lipschitz ergodic systems

arXiv: https://arxiv.org/abs/2102.12669 · HTML: https://mlsanity.com/paper/2102.12669/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 47 references that are in the map.
- First posted on arXiv 25 Feb 2021.

## Builds on (most cited first, 10 of 10)

- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Physics-Informed Generative Adversarial Networks for Stochastic Differential Equations](https://mlsanity.com/paper/1811.02033/) (2018)
- [Nonparametric inference of interaction laws in systems of agents from trajectory data](https://mlsanity.com/paper/1812.06003/) (2018)
- [Data Assimilation: A Mathematical Introduction](https://mlsanity.com/paper/1506.07825/) (2015)
- [Learning interaction kernels in stochastic systems of interacting particles from multiple trajectories](https://mlsanity.com/paper/2007.15174/) (2020)
- [A Data-Driven Approach for Discovering Stochastic Dynamical Systems with Non-Gaussian Levy Noise](https://mlsanity.com/paper/2005.03769/) (2020)
- [Data-driven parameterization of the generalized Langevin equation](https://mlsanity.com/paper/1606.02596/) (2016)
- [Modeling of Missing Dynamical Systems: Deriving Parametric Models using a Nonparametric Framework](https://mlsanity.com/paper/1905.08082/) (2019)
- [The Seven-League Scheme: Deep learning for large time step Monte Carlo simulations of stochastic differential equations](https://mlsanity.com/paper/2009.03202/) (2020)

## Abstract (arXiv)

Efficient simulation of SDEs is essential in many applications, particularly for ergodic systems that demand efficient simulation of both short-time dynamics and large-time statistics. However, locally Lipschitz SDEs often require special treatments such as implicit schemes with small time-steps to accurately simulate the ergodic measure. We introduce a framework to construct inference-based schemes adaptive to large time-steps (ISALT) from data, achieving a reduction in time by several orders of magnitudes. The key is the statistical learning of an approximation to the infinite-dimensional discrete-time flow map. We explore the use of numerical schemes (such as the Euler-Maruyama, a hybrid RK4, and an implicit scheme) to derive informed basis functions, leading to a parameter inference problem. We introduce a scalable algorithm to estimate the parameters by least squares, and we prove the convergence of the estimators as data size increases. We test the ISALT on three non-globally Lipschitz SDEs: the 1D double-well potential, a 2D multi-scale gradient system, and the 3D stochastic Lorenz equation with degenerate noise. Numerical results show that ISALT can tolerate time-step magnitudes larger than plain numerical schemes. It reaches optimal accuracy in reproducing the invariant measure when the time-step is medium-large.