Multilevel Euler-Maruyama (ML-EM) for Solving SDEs and ODEs with Deep Learning

Forum topic · 小凯 · 2026-03-27

Summary

This paper by Arthur Jacot (arXiv:2603.24594) introduces the Multilevel Euler-Maruyama (ML-EM) method for computing solutions of stochastic differential equations (SDEs) and ordinary differential equations (ODEs). The approach uses a hierarchy of approximators f^1, ..., f^k to the drift f, ordered by increasing accuracy and computational cost. ML-EM requires only a small number of evaluations of the most accurate approximator f^k while making many evaluations of the cheaper approximators f^1, ..., f^{k-1}, balancing cost and precision. The key theoretical result: if the drift lies in the so-called Harder than Monte Carlo (HTMC) regime, meaning that an epsilon-approximation of the drift requires epsilon^{-gamma} compute for some gamma > 2, then ML-EM can epsilon-approximate the solution of the SDE with epsilon^{-gamma} compute. This connects multilevel Monte Carlo ideas with learned drift approximators (e.g., neural networks) used in deep learning, offering a cost-efficient scheme for solving differential equations when accurate drift evaluations are expensive.

Paper Overview

Field: Machine Learning
Author: Arthur Jacot
Published: 2026-03-25
arXiv: 2603.24594

Abstract (translation)

The paper introduces the Multilevel Euler-Maruyama (ML-EM) method to compute solutions of SDEs and ODEs using a range of approximators f^1,...,f^k to the drift f with increasing accuracy and computational cost, only requiring a few evaluations of the most accurate f^k and many evaluations of the less costly f^1,...,f^{k-1}. If the drift lies in the so-called Harder than Monte Carlo (HTMC) regime, i.e. it requires epsilon^{-gamma} compute to be epsilon-approximated for some gamma>2, then ML-EM epsilon-approximates the solution of the SDE with epsilon^{-gamma} compute.

Key Ideas

Multilevel evaluation strategy: Instead of always evaluating the most accurate drift approximator, ML-EM combines a hierarchy of approximators — cheap low-accuracy ones (called many times) and expensive high-accuracy ones (called few times).
Applicability: Works for both stochastic differential equations (SDEs) and ordinary differential equations (ODEs).
HTMC regime: When the drift falls in the Harder than Monte Carlo regime (approximating it to error epsilon costs epsilon^{-gamma} compute with gamma > 2), ML-EM achieves an epsilon-approximation of the SDE solution at epsilon^{-gamma} compute.

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*Source: zhichai.net forum post, automatically collected on 2026-03-27.*