Flow Matching Transport for Quasi-Monte Carlo Integration

TL;DR

This paper introduces FM-ISQMC, a novel framework combining flow matching and importance sampling with Quasi-Monte Carlo methods, providing unbiased, high-precision high-dimensional integration with rigorous convergence guarantees.

Contribution

It develops a convergence theory for QMC importance sampling with transport maps and proves that flow matching architectures meet these conditions, enabling unbiased high-order integration.

Findings

FM-ISQMC achieves an $ ext{O}(N^{-1+ ext{ε}})$ root-mean-square error rate.

Numerical experiments show FM-ISQMC surpasses error floors of direct transport methods.

The framework bridges deep generative modeling and numerical integration.

Abstract

High-dimensional integration with respect to complex target measures remains a fundamental challenge in computational science. While Flow Matching (FM) offers a powerful paradigm for constructing continuous-time transport maps, its deployment in high-precision integration is severely limited by the discretization bias inherent to numerical ODE solvers and the lack of rigorous convergence guarantees when coupled with Quasi-Monte Carlo (QMC) methods. This paper addresses these critical gaps by proposing Flow Matching Importance Sampling Quasi-Monte Carlo (FM-ISQMC), a framework designed to transform biased generative flows into unbiased, high-order integration schemes. Methodologically, we construct a transport map by composing a logistic base transformation with an Euler-discretized neural ODE field and employ importance sampling to correct for residual transport errors. Our central contribution is twofold. First, we establish a general convergence analysis for QMC importance sampling with arbitrary transport maps, identifying sufficient growth conditions for the $\mathcal{O}(N^{-1+\varepsilon})$ root-mean-square error rate. Second, we rigorously prove that the specific transport architecture of Flow Matching satisfies these conditions. Consequently, we establish a $\mathcal{O}(N^{-1+\varepsilon})$ root-mean-square error for the unbiased FM-ISQMC estimator, extending classical QMC theory to the realm of generative models. Numerical experiments validate that FM-ISQMC consistently breaks through the error floor observed in direct transport methods, delivering superior precision. This work thus bridges the divide between deep generative modeling and numerical integration.