A pseudo-reversible normalizing flow for stochastic dynamical systems with various initial distributions

TL;DR

This paper introduces a pseudo-reversible normalizing flow that efficiently models the conditional distribution of stochastic differential equations' final states, enabling flexible sampling across various initial conditions without retraining.

Contribution

The novel normalizing flow learns the conditional distribution of SDE states, allowing for a single training that generalizes to multiple initial distributions, reducing computational costs.

Findings

Model accurately captures the conditional distribution of SDEs.

The approach significantly reduces retraining efforts for different initial states.

Numerical experiments confirm the model's effectiveness and convergence.

Abstract

We present a pseudo-reversible normalizing flow method for efficiently generating samples of the state of a stochastic differential equation (SDE) with different initial distributions. The primary objective is to construct an accurate and efficient sampler that can be used as a surrogate model for computationally expensive numerical integration of SDE, such as those employed in particle simulation. After training, the normalizing flow model can directly generate samples of the SDE's final state without simulating trajectories. Existing normalizing flows for SDEs depend on the initial distribution, meaning the model needs to be re-trained when the initial distribution changes. The main novelty of our normalizing flow model is that it can learn the conditional distribution of the state, i.e., the distribution of the final state conditional on any initial state, such that the model only needs to be trained once and the trained model can be used to handle various initial distributions. This feature can provide a significant computational saving in studies of how the final state varies with the initial distribution. We provide a rigorous convergence analysis of the pseudo-reversible normalizing flow model to the target probability density function in the Kullback-Leibler divergence metric. Numerical experiments are provided to demonstrate the effectiveness of the proposed normalizing flow model.