Variational Smoothing and Inference for SDEs from Sparse Data with Dynamic Neural Flows

TL;DR

This paper introduces a neural network-based variational method for efficient smoothing and parameter inference in stochastic differential equations from sparse, noisy data, overcoming classical limitations.

Contribution

It develops a novel posterior SDE characterization using neural scores trained to satisfy PDE and jump conditions, enabling scalable inference and joint smoothing and parameter estimation.

Findings

Accurate inference with very few observations.

Significant scalability improvements over classical MCMC methods.

Stable joint smoothing and parameter estimation in nonlinear systems.

Abstract

Stochastic differential equations (SDEs) provide a flexible framework for modeling temporal dynamics in partially observed systems. A central task is to calibrate such models from data, which requires inferring latent trajectories and parameters from sparse, noisy observations. Classical smoothing methods for this problem are often limited by path degeneracy and poor scalability. In this work, we developed a novel method based on characterization of the posterior SDE in terms of conditional backward-in-time score defined as the gradient of a function solving a Kolmogorov backward equation with multiplicative updates at observation times. We learn this conditional score using neural networks trained to satisfy both the governing PDE and the observation-induced jump conditions, thereby integrating continuous-time dynamics with discrete Bayesian updates. The resulting score induces a posterior SDE with the same diffusion coefficient but a modified drift, enabling efficient posterior trajectory sampling. We further derive a likelihood-based objective for learning the SDE parameters, yielding an evidence lower bound (ELBO) for joint state smoothing and parameter estimation. This leads to a variational EM-style procedure, where the neural conditional score is optimized to approximate the smoothing distribution, followed by a maximization step over the SDE parameters using samples from the induced posterior. Experiments on nonlinear systems demonstrate accurate and stable inference with a very few observations demonstrating significant improved scalability compared to classical MCMC methods.