Non-intrusive structural-preserving sequential data assimilation

Data assimilation (DA) methods combine model predictions with observational data to improve state estimation in dynamical systems, inspiring their increasingly prominent role in geophysical and climate applications. Classical DA methods assume that the governing equations modeling the dynamics are known, which is unlikely for most real world applications. Machine learning (ML) provides a flexible alternative by learning surrogate models directly from data, but standard ML methods struggle in noisy and data-scarce environments, where meaningful extrapolation requires incorporating physical constraints. Recent advances in structure-preserving ML architectures, such as the development of the entropy-stable conservative flux form network (ESCFN), highlight the critical role of explicitly encoding physical structure (e.g., conservation laws and entropy conditions) into surrogate models, thereby improving learning stability and accuracy. Structural information has also been shown to improve DA performance. Gradient-based measures of spatial variability, in particular, can help refine ensemble updates in discontinuous systems. Motivated by both of these recent innovations, this investigation proposes a new non-intrusive, structure-preserving sequential data assimilation (NSSDA) framework that leverages structure at both the forecast and analysis stages. We use the ESCFN to construct a surrogate model to preserve physical laws during forecasting, and a structurally informed ensemble transform Kalman filter (SETKF) to embed local statistical structure into the assimilation step, assuming partial structural information such as the transformation between physical and conserved variables. Our method operates in a highly constrained environment, using only a single noisy trajectory for both training and assimilation. Numerical experiments where the unknown dynamics correspond respectively to the shallow water and Euler equations demonstrate significantly improved predictive accuracy.