# Pathwise Learning of Stochastic Dynamical Systems with Partial Observations

arXiv: https://arxiv.org/abs/2601.21860 · HTML: https://mlsanity.com/paper/2601.21860/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 35 of its 75 references that are in the map.
- First posted on arXiv 29 Jan 2026.

## Builds on (most cited first, 30 of 35)

- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [DeepMind Control Suite](https://mlsanity.com/paper/1801.00690/) (2018)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Neural Stochastic Differential Equations: Deep Latent Gaussian Models in the Diffusion Limit](https://mlsanity.com/paper/1905.09883/) (2019)
- [Neural Rough Differential Equations for Long Time Series](https://mlsanity.com/paper/2009.08295/) (2020)
- [On the relation between optimal transport and Schr"odinger bridges: A stochastic control viewpoint](https://mlsanity.com/paper/1412.4430/) (2014)
- [Score-based Data Assimilation](https://mlsanity.com/paper/2306.10574/) (2023)
- [Combining data assimilation and machine learning to emulate a dynamical model from sparse and noisy observations: a case study with the Lorenz 96 model](https://mlsanity.com/paper/2001.01520/) (2020)
- [Probabilistic Forecasting with Stochastic Interpolants and F"ollmer Processes](https://mlsanity.com/paper/2403.13724/) (2024)
- [Conditional Simulation Using Diffusion Schr"odinger Bridges](https://mlsanity.com/paper/2202.13460/) (2022)
- [Bayesian Learning via Neural Schr"odinger-F"ollmer Flows](https://mlsanity.com/paper/2111.10510/) (2021)
- [Coupling techniques for nonlinear ensemble filtering](https://mlsanity.com/paper/1907.00389/) (2019)
- [On Particle Methods for Parameter Estimation in State-Space Models](https://mlsanity.com/paper/1412.8695/) (2014)
- [Denoising Likelihood Score Matching for Conditional Score-based Data Generation](https://mlsanity.com/paper/2203.14206/) (2022)
- [Bayesian inference of chaotic dynamics by merging data assimilation, machine learning and expectation-maximization](https://mlsanity.com/paper/2001.06270/) (2020)
- [An Ensemble Score Filter for Tracking High-Dimensional Nonlinear Dynamical Systems](https://mlsanity.com/paper/2309.00983/) (2023)
- [Provably Convergent Schr"odinger Bridge with Applications to Probabilistic Time Series Imputation](https://mlsanity.com/paper/2305.07247/) (2023)
- [When are dynamical systems learned from time series data statistically accurate?](https://mlsanity.com/paper/2411.06311/) (2024)
- [Reflected Schr"odinger Bridge for Constrained Generative Modeling](https://mlsanity.com/paper/2401.03228/) (2024)
- [A variational approach to path estimation and parameter inference of hidden diffusion processes](https://mlsanity.com/paper/1508.00506/) (2015)
- [Neural Koopman prior for data assimilation](https://mlsanity.com/paper/2309.05317/) (2023)
- [A Score-based Nonlinear Filter for Data Assimilation](https://mlsanity.com/paper/2306.09282/) (2023)
- [McKean-Vlasov SDEs in nonlinear filtering](https://mlsanity.com/paper/2007.12658/) (2020)
- [An energy-based deep splitting method for the nonlinear filtering problem](https://mlsanity.com/paper/2203.17153/) (2022)
- [CGKN: A Deep Learning Framework for Modeling Complex Dynamical Systems and Efficient Data Assimilation](https://mlsanity.com/paper/2410.20072/) (2024)
- [Modeling Partially Observed Nonlinear Dynamical Systems and Efficient Data Assimilation via Discrete-Time Conditional Gaussian Koopman Network](https://mlsanity.com/paper/2507.08749/) (2025)
- [An optimal control approach to particle filtering](https://mlsanity.com/paper/2110.03199/) (2021)
- [Deep Learning for the Benes Filter](https://mlsanity.com/paper/2203.05561/) (2022)
- [A variational approach to sampling in diffusion processes](https://mlsanity.com/paper/2405.00126/) (2024)
- [Relative Arbitrage Opportunities in an Extended Mean Field System](https://mlsanity.com/paper/2311.02690/) (2023)

## Built on by (most cited first, 1 of 1)

- [On the Convergence of Jacobian-Free Backpropagation for Optimal Control Problems with Implicit Hamiltonians](https://mlsanity.com/paper/2602.00921/) (2026)

## Abstract (arXiv)

The reconstruction and inference of stochastic dynamical systems from data is a fundamental task in inverse problems and statistical learning. While surrogate modeling advances computational methods to approximate these dynamics, standard approaches typically require high-fidelity training data. In many practical settings, however, the data are indirectly observed through noisy and nonlinear measurements. The challenge lies not only in approximating the coefficients of the SDEs, but in simultaneously inferring the posterior updates given the observations. In this work, we present an amortized path generation method to address these challenges and solve nonlinear stochastic filtering from noisy observations. We first derive a variational inference formulation that solves filtering distribution for a given noisy observation path. This leads to a controlled SDE representation in which the feedback control is identified through the score structure of a pathwise Zakai equation. Motivated by this representation, we construct a conditional generative model that learns, in an amortized manner over observation paths, to transport a prior latent path measure toward the corresponding posterior path measure. We demonstrate the method on nonlinear stochastic systems with multimodal posterior structure, chaotic dynamics, and sparse observations, showing that the learned conditional path generator enables uncertainty quantification for both filtering marginals and trajectory-dependent functionals.