Multilevel Euler-Maruyama for Solving SDEs and ODEs with Deep Learning (arXiv 2603.24594)

Forum topic · 小凯 · 2026-03-27

Summary

This paper by Arthur Jacot, posted on 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, with increasing accuracy and computational cost. ML-EM requires only a few evaluations of the most accurate approximator f^k, while making many evaluations of the cheaper approximators f^1,...,f^{k-1}. The key theoretical result concerns the Harder than Monte Carlo (HTMC) regime: when the drift requires epsilon^{-gamma} compute to be epsilon-approximated for some gamma > 2, ML-EM can epsilon-approximate the solution of the SDE with epsilon^{-gamma} compute, matching the cost of approximating the drift itself. This makes ML-EM relevant for deep learning settings where drift evaluations via neural networks are expensive, offering a multilevel sampling strategy that balances accuracy and cost. The post was collected from zhichai.net on 2026-03-27 and includes the paper's original English abstract alongside a Chinese summary.

Paper Overview

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

Introduction

This post introduces the paper *Multilevel Euler-Maruyama for Solving SDEs and ODEs with Deep Learning* by Arthur Jacot, with a Chinese summary followed by the original abstract.

Chinese Summary (translated)

The paper introduces the Multilevel Euler-Maruyama (ML-EM) method, which computes solutions of SDEs and ODEs using a series of approximators f^1,...,f^k to the drift f with increasing accuracy and computational cost. It only requires a few evaluations of the most accurate f^k and many evaluations of the cheaper 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 where gamma > 2 — then ML-EM can epsilon-approximate the solution of the SDE with epsilon^{-gamma} compute.

Original Abstract

> We introduce the Multilevel Euler-Maruyama (ML-EM) method 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.