Deep operator learning for efficient sampling from invariant measures of stochastic differential equations

2026-09-10

AI Summary

本文通过结合算子学习和流方法，提出了一种神经采样器，用于从随机微分方程的不变测度中高效采样，显著提高了高维问题的处理速度。

Abstract

We introduce an amortized neural sampler that combines operator learning with flow methods for sampling. It maps SDE coefficient functions to pushforwards from a reference measure to the invariant measures, enabling efficient sampling across families of stochastic differential equations. Our framework shifts traditional sampling cost to an initial training phase, after which new SDE instances require only one encoder pass and a few ODE solver steps, independent of mixing time. To handle problems in high dimensions, we use Lagrangian trajectory sensors for the coefficient functions and cross attention in the architecture. We also theoretically establish the expressivity and resolution invariance of our framework. Experiments on 1D and 2D SDE families show competitive accuracy with substantial speedups over MCMC in regimes with slow mixing, transfer across sensor counts, and demonstration results on a 64D interacting particle SDE where traditional grid approaches are infeasible.

Problem

• Stochastic Differential Equations
• Invariant Measures
• Efficient Sampling
• High Dimensions
• Slow Mixing

Innovation

• operator learning
• flow methods
• invariant measures
• Lagrangian trajectory sensors
• cross attention