Neural Galerkin Normalizing Flows for Bayesian Inference of Diffusions with Inaccessible Boundaries

2026-06-02

AI Summary

This work addresses the challenge of Bayesian inference in diffusion models under discrete observations, where analytical transition densities are generally unavailable—particularly when the diffusion matrix degenerates at inaccessible boundaries, as in stochastic volatility models satisfying the Feller condition. To overcome this, the authors propose a novel normalizing flow architecture that, for the first time, integrates a neural Galerkin framework with normalizing flows. By solving the Fokker–Planck equation offline with a Dirac mass as the initial condition, the method learns approximate transition densities between observation times. The resulting surrogate likelihood model substantially accelerates posterior inference, circumventing the need to solve partial differential equations in real time for each MCMC proposal or resorting to computationally expensive likelihood-free approaches.

Abstract

One of the primary challenges in Bayesian inference on the parameters of a diffusion model from discrete observations is the unavailability of an analytical expression for the transition density function between consecutive observation times, which is needed to derive the likelihood function. Extending previous studies that solve Fokker-Planck (FP) type partial differential equations with Normalizing Flows, we propose a new Normalizing Flow architecture to learn the transition density function of the diffusion process between two observation times. We do so by solving in a Neural Galerkin framework the associated FP equation with a Dirac mass as initial condition, over a specified training distribution of the initial datum and the coefficients of the diffusion. We specifically focus on processes whose diffusion matrix vanishes in certain inaccessible boundary regions, such as Stochastic Volatility models that satisfy a Feller condition. The product of the obtained transition densities evaluated along the observed trajectory approximates the likelihood function, thereby enabling cheap posterior sampling via Markov chain Monte Carlo (MCMC). After the offline training phase, inference becomes significantly more efficient, as it avoids the need to solve the FP equation in real time for each parameter proposed by the MCMC sampler or to rely on other likelihood-free methods for Bayesian inference that involve repeated simulation of diffusion bridges.

Problem

• Bayesian inference
• diffusion processes
• transition density
• inaccessible boundaries
• Fokker-Planck equation

Innovation

• Neural Galerkin
• Normalizing Flows
• Fokker-Planck equation
• Bayesian inference
• diffusion processes