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

NeurIPS 2026 · Image generation › Fast diffusion sampling · approach: Diffusion theory and learned samplers · 0 citations (OpenAlex)

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Learns the transition density of stochastic volatility diffusions with inaccessible boundaries by solving the Fokker-Planck equation with a bounded truncated-Gaussian-mixture normalizing flow trained via Neural Galerkin, then uses the flow-based likelihood for MCMC posterior sampling.

Problem

Bayesian inference for discretely observed diffusions requires the transition density between observations, which is generally unavailable in closed form; existing approaches are costly, biased, or limited, and PINNs struggle with advection-dominated, Dirac-initialized Fokker-Planck equations on dom

Approach

A bounded normalizing flow built from conditional mixtures of truncated Gaussians (with GRU conditioning networks) is parameterized by the SDE parameters and the Dirac initial location, and its parameters are evolved in time with a Neural Galerkin least-squares scheme so the flow approximates the transition density; the flow's density is then used directly as a likelihood surrogate for MCMC.

Key results

• On Heston data (Delta=0.5), the NF log-likelihood relative error at the true parameter is 0.0003, versus 0.0064 for Aït-Sahalia and 0.0093 for Itô-Taylor.
• For 10^4 MCMC samples on Heston, NF takes 2243 s versus 14560 s for Fourier inversion, with joint W2 distance 0.066 to the Fourier posterior.
• On SVCEV, NF's relative log-likelihood error at the true parameter is 0.0004 versus 0.0063 (A.-S.) and 0.0091 (Itô-Taylor), and MCMC takes 1389 s versus 63542 s for data augmentation.

Surprising findings

• Closed-form transition-density expansions (Aït-Sahalia) can become unstable and diverge when evaluated far from the true parameter.

Limitations

• Accuracy is not guaranteed in distribution tails, making the surrogate unsuitable for extreme-event simulation.
• Compact-support assumption on the target density; reference distribution may not capture tail decay at inaccessible boundaries.
• Zero-flux boundary conditions are verified a posteriori rather than imposed; focus is on fully observed processes.

Experimental setup

Datasets: simulated Heston trajectories (Euler-Maruyama) simulated SVCEV trajectories

Benchmarks: Heston model likelihood vs Fourier inversion SVCEV model likelihood vs data augmentation reference

Metrics: relative log-likelihood error relative L2 error of TPDF Wasserstein-2 distance between posteriors runtime

Baselines: Aït-Sahalia closed-form likelihood expansion Itô-Taylor expansion Fourier inversion of the characteristic function data augmentation with importance sampling

Builds on: Neural Galerkin Normalizing Flows Neural Galerkin scheme normalizing flows

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