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Pathwise Learning of Stochastic Dynamical Systems with Partial Observations

Nicole Tianjiao Yang

TL;DR

This work tackles learning and inference for stochastic dynamical systems from noisy, partial observations by learning a posterior path measure on trajectory space. It derives a pathwise Zakai equation and casts posterior sampling as a stochastic control problem, then trains an observation-conditioned neural SDE to amortize the posterior path measure via a pathwise ELBO. The approach enables fast generation of data-assimilated trajectories, handles missing observations, and supports uncertainty quantification for path functionals, even when the true dynamics are unknown. Practically, it offers a scalable, parameter-free-friendly alternative to particle-based filtering for high-dimensional, multimodal, and chaotic systems, with demonstrated performance on double-well, Lorenz-63, and Lorenz-96 dynamics.

Abstract

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, the data are indirectly observed through noisy and nonlinear measurement. 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 a neural path estimation approach to solve stochastic dynamical systems based on variational inference. We first derive a stochastic control problem that solve filtering posterior path measure corresponding to a pathwise Zakai equation. We then construct a generative model that maps the prior path measure to posterior measure through the controlled diffusion and the associated Randon-Nykodym derivative. Through an amortization of sample paths of the observation process, the control is learned by an embedding of the noisy observation paths. Thus, we learn the unknown prior SDE and the control can recover the conditional path measure given the observation sample paths and we learn an associated SDE which induces the same path measure. In the end, we perform experiments on nonlinear dynamical systems, demonstrating the model's ability to learn multimodal, chaotic, or high dimensional systems.

Pathwise Learning of Stochastic Dynamical Systems with Partial Observations

TL;DR

This work tackles learning and inference for stochastic dynamical systems from noisy, partial observations by learning a posterior path measure on trajectory space. It derives a pathwise Zakai equation and casts posterior sampling as a stochastic control problem, then trains an observation-conditioned neural SDE to amortize the posterior path measure via a pathwise ELBO. The approach enables fast generation of data-assimilated trajectories, handles missing observations, and supports uncertainty quantification for path functionals, even when the true dynamics are unknown. Practically, it offers a scalable, parameter-free-friendly alternative to particle-based filtering for high-dimensional, multimodal, and chaotic systems, with demonstrated performance on double-well, Lorenz-63, and Lorenz-96 dynamics.

Abstract

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, the data are indirectly observed through noisy and nonlinear measurement. 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 a neural path estimation approach to solve stochastic dynamical systems based on variational inference. We first derive a stochastic control problem that solve filtering posterior path measure corresponding to a pathwise Zakai equation. We then construct a generative model that maps the prior path measure to posterior measure through the controlled diffusion and the associated Randon-Nykodym derivative. Through an amortization of sample paths of the observation process, the control is learned by an embedding of the noisy observation paths. Thus, we learn the unknown prior SDE and the control can recover the conditional path measure given the observation sample paths and we learn an associated SDE which induces the same path measure. In the end, we perform experiments on nonlinear dynamical systems, demonstrating the model's ability to learn multimodal, chaotic, or high dimensional systems.
Paper Structure (29 sections, 4 theorems, 62 equations, 5 figures, 2 tables)

This paper contains 29 sections, 4 theorems, 62 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Contribution and Main results
  3. Related work
  4. Bayesian filtering methods
  5. Joint parameter and state estimation
  6. Density propagation in continuous time
  7. Data assimilation via conditional generative models
  8. Organization of the paper
  9. Problem set-up
  10. Optimal filter
  11. Motivation for path-based approach
  12. Path dependence
  13. Robustness towards the perturbation in noisy observations
  14. Amortized variational inference for posterior measure
  15. Pathwise Zakai equation
  16. ...and 14 more sections

Key Result

Proposition 3.1

Under Assumption asmp:sde-asmp: novikov, for $y \in C([0,T];\mathbb{R}^{m})$ a.e., the unnormalized posterior density in eq: q-orig follows where $\beta^y (t,x):=\beta(t,x)-a(t,x)(\nabla_x h(t,x))^\top e^{-1}(t) y_t$, and $a(t,x) = \sigma(t,x) \sigma(t,x)^\top$, and

Figures (5)

  • Figure 1: Stochastic double well equation. The model is trained on synthetic data from \ref{['eq:dw-1']} on time horizon $[0,4]$ and perform inference on test noisy observation paths on time horizon $[0,8]$. Left: True test trajectories and noisy observations; Right: Estimated trajectories and the 90% confidence interval (CI) with only the noisy observations available during inference.
  • Figure 2: Estimated trajectories from stochastic double well equation. Left to right: Loss every 100 epochs, Dwell time RMSE and 90% coverage every 200 epochs. The inferred trajectories from our method can capture long-time, metastable behavior of the underlying dynamics, verifying the efficient learning of the posterior path measure.
  • Figure 3: Stochastic Lorenz-63. We compare the ground-truth trajectory against three reconstructions: (i) sample paths mean from our conditional latent SDE, (ii) a backward-sampled trajectory from a bootstrap particle filter (BPF), and (iii) a particle Gibbs (PG) conditional SMC smoother trajectory. Observations follow the nonlinear model $y_t=\arctan(x_t)+\varepsilon_t$, with PF/PG applying likelihood updates only at observed indices while propagating true dynamics \ref{['eq: lorenz63']}.
  • Figure 4: Stochastic Lorenz-96 equation. Estimated trajectories from $15$-dimensional stochastic Lorenz-96 equation. The model is trained for time $[0,2]$ with only observation from test dataset available. The observation model is $y_t = Tanh(x_t)+ N(0, \sigma^2), \sigma=0.15$. Top: Comparison of mariginal distributions at time 0.5, 1, and 1.5 for the first dimension. Bottom: True (left) and Inferenced (right) trajectories of the first 3 dimensions. 90% confidence intervals are also presented for the inferenced trajectories.
  • Figure 5: Estimated trajectories of the first 3 dimensions from $15$-dimensional stochastic Lorenz-96 equation. The model is trained for time $[0,3]$ with 20% observation randomly masked. During inference, only the noisy observation from test dataset available, (random) 20% of the observation time is missing. The observation model is $y_t = arctan(x_t)+ N(0, \sigma^2), \sigma=0.15$. 90% confidence intervals are also presented for the inferenced trajectories.

Theorems & Definitions (11)

  • Proposition 3.1
  • Proof 1
  • Definition 3.1: Admissible controls
  • Theorem 3.1
  • Proof 2
  • Corollary 3.2
  • Proof 3
  • Theorem 4.1: Pathwise ELBO
  • Proof 4
  • Remark 4.2
  • ...and 1 more