Solving Reflected Diffusion Models: A PINN-based Method

Liyuan Xu, Junwei Lu

Keywords: machine learning, generative models, stochastic differential equations, neural network

Abstract: Score-based reflected diffusion models generate approximations of high-dimensional data distributions while respecting the known constraints of the data distribution by learning a reversed reflected stochastic differential equation evolving within the support of the data. Similar to standard diffusion models, the theoretical convergence of reflected diffusion models is based on bounded errors of score estimations. However, the existence and attainability of low-error score estimators have not yet been studied in the reflected diffusion setting. In this paper, we construct a novel score estimator using the Physics-Informed Neural Network (PINN), solving reflected diffusion models in a deep-learning fashion. We proceed to derive a uniform theoretical error bound of $O(N^{-\frac{1}{4}})$ for the score function on a training dataset of sample size $N$ at any time $t\in[0,T]$ of the diffusion process. This result fills the gap between theory and practice in the score estimation of the reflected diffusion model. Moreover, its independence of dimension ensures the performance of our estimator in large-sample scenarios under high-dimensional settings.

Primary Area: generative models

Submission Number: 10119