Flow Matching Transport for Quasi-Monte Carlo 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.