Continuous data assimilation for hydrodynamics: consistent discretization and application to moment recovery

TL;DR

This paper introduces a novel data assimilation method for hydrodynamic models that combines relaxation nudging with a new discretization technique, enabling accurate moment recovery from sparse data, crucial for machine learning in kinetic theory.

Contribution

It presents a new continuous data assimilation approach with a specialized discretization for hydrodynamics, improving moment recovery and data preparation for machine learning models.

Findings

Effective recovery of force terms and high-resolution solutions from sparse data.

Convergence analysis under full and partial data scenarios.

Numerical experiments demonstrating the method's robustness and potential for high-dimensional systems.

Abstract

Motivated by the challenge of moment recovery in hydrodynamic approximation in kinetic theory, we propose a data-driven approach for the hydrodynamic models. Inspired by continuous data assimilation, our method introduces a relaxation-based nudging system coupled with a novel discretization technique. This approach facilitates the simultaneous recovery of both the force term and a high-resolution solution from sparsely observed data. To address potential numerical artifacts, we use kernel regression to fit the observed data. We also analyze the convergence of the proposed nudging system under both full and partial data scenarios. When applied to moment systems, the source term involves the derivative of higher-order moments, our approach serves as a crucial step for data preparation in machine-learning based moment closure models. Multiple numerical experiments demonstrate the effectiveness of our algorithm, and we discuss its potential extension to high-dimensional systems.