AI Insight
This paper introduces ZENO (Zakai Equation Neural Operator), a machine learning method that uses neural operators to solve the Zakai equation for nonlinear filtering problems in stochastic dynamical systems. The approach combines physics-informed loss functions with operator learning to infer hidden states and learn system parameters through an expectation-maximization framework. The authors validate ZENO on three simulated systems (Double-Well, Van Der Pol oscillator, and Lorenz system) and demonstrate its application to real whole-brain imaging data from C. elegans nematodes, successfully recovering locomotion-related neural activity patterns.
Why it matters
This method could improve state estimation and parameter inference for complex dynamical systems across engineering and neuroscience applications, particularly where traditional numerical methods for solving high-dimensional stochastic partial differential equations are intractable. The successful application to neural data suggests potential for advancing our understanding of brain dynamics and behavior.
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⚠️ Preprint – Noch nicht peer-reviewed
Dieser Artikel wurde noch nicht von unabhängigen Experten begutachtet. Die Ergebnisse sind vorläufig und sollten mit Vorsicht interpretiert werden.
Abstract: State inference and parameter learning for nonlinear filters is a core computation for scientific machine learning with varied applications in engineering and neuroscience. The unnormalized filtering distribution is governed by the Zakai equation, a stochastic partial differential equation (SPDE) that combines the forward Kolmogorov operator (known as the Fokker-Planck equation) with observed likelihood updates. Although solving these high-dimensional SPDEs is often impossible analytically and intractable numerically, advances in operator learning and physics-informed machine learning have made it feasible. Harnessing these advances, we introduce the Zakai Equation Neural Operator (ZENO), a model that optimizes a neural operator against a Zakai Equation based physics-informed loss to approximate the exact unnormalized filtering distribution for latent stochastic dynamical systems. We use this ZENO-inferred posterior distribution in an expectation-maximization (EM) framework to perform parameter learning and state inference. We demonstrate ZENO’s ability to accurately infer the posterior distribution and learn its corresponding SDE parameters in three simulated stochastic dynamical systems (Double-Well bistable system, Van Der Pol (VDP) oscillator, and Lorenz system) and compare its performance against established methods for latent SDE inference and learning. We show that ZENO recovers previously observed limit cycles representing forward and backward locomotion from a whole-brain imaging dataset of immobilized textit{C. elegans} nematodes. ZENO is a novel approach to the problem of nonlinear filtering and dynamical inference through the lens of partial differential equations and operator theory.
Source: ZENO: Nonlinear filtering with Zakai Equation Neural Operators