A Lagrangian Conditional Gaussian Koopman Network for Data Assimilation and Prediction

Abstract: Lagrangian data assimilation aims to recover hidden Eulerian flow fields from sparse, indirect observations of moving tracers. This problem is challenging because tracer trajectories are nonlinearly coupled with the underlying flow, making posterior inference computationally intractable in realistic, high-dimensional systems. In this work, we develop a Lagrangian conditional Gaussian Koopman network (LaCGKN), a structure-preserving, data-driven framework for joint data assimilation and prediction from Lagrangian observations. LaCGKN embeds Eulerian flow dynamics into a low-dimensional latent space governed by a nonlinear stochastic system with conditional Gaussian structure, enabling analytic posterior updates without ensemble forecasting. Unlike existing conditional Gaussian Koopman formulations that assume direct Eulerian observations, the Lagrangian setting imposes additional demands on the latent representation, which must simultaneously encode the flow dynamics and mediate nonlinear tracer-flow interactions. To address these challenges, the LaCGKN incorporates three key components: (i) tracer homogenization to enforce permutation equivariance and generalize across varying numbers of tracers; (ii) Fourier positional encoding to capture spatial dependence and reconstruct local flow features at moving tracer locations; and (iii) an SVD-inspired low-rank parameterization of the latent transition operator, which reduces model complexity while retaining expressiveness. An application to a two-layer quasi-geostrophic flow with surface tracer observations shows that LaCGKN achieves accurate and efficient Lagrangian data assimilation and prediction, without reliance on ensemble methods or the governing physical model. These results establish the LaCGKN as a unified and computationally tractable alternative to both traditional model-based approaches and purely black-box data-driven methods.