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

2026-05-06

AI Summary

This work addresses the challenging problem of state smoothing and parameter estimation for stochastic differential equations (SDEs) under sparse and noisy observations. The authors propose a novel approach based on conditional backward-time score functions, leveraging neural networks to model the posterior score that satisfies both the Kolmogorov backward equation and observation-induced jump conditions. This enables the construction of a posterior SDE with a corrected drift term, which is embedded within a variational EM framework to jointly optimize latent state trajectories and model parameters. By innovatively integrating dynamic neural flows with conditional score modeling, the method unifies continuous-time dynamics and discrete Bayesian updates, and introduces a likelihood-based evidence lower bound (ELBO) objective. The approach achieves highly accurate and stable inference even with extremely limited observations, significantly outperforming conventional MCMC methods while offering substantial improvements in scalability and computational efficiency.

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.

Problem

• stochastic differential equations
• latent trajectory inference
• sparse observations
• parameter estimation
• Bayesian smoothing

Innovation

• neural score learning
• stochastic differential equations
• variational smoothing
• Kolmogorov backward equation
• dynamic neural flows