Tutorial

Abstract: In these lectures, I will cover some topics in the rapidly growing field of the integration of machine learning and data assimilation. I will cover:

• Learning a forecast model from partial, noisy observations with the aid of data assimilation.

• Applying data assimilation to a learned forecast model.

• Learning new probabilistic filtering algorithms and state estimators.

The lectures will contain material from lecture notes from a course taught at Caltech in 2024.

Abstract: High-dimensional nonlinear filtering plays a crucial role in a wide range of applications, including atmospheric and oceanic data assimilation, robotics and autonomous systems, neuroscience, and personalized medicine. This lecture introduces modern strategies to address this challenge and we will be highlighting methods such as ensemble-based techniques and sequential Monte Carlo approaches. We will discuss both theoretical underpinnings and practical implementations, illustrating key ideas through real-world examples. Participants will also gain insight into current research trends and emerging directions in this rapidly evolving area.

Abstract: This compact three-part tutorial offers an end-to-end introduction to data assimilation (DA) for geoscientists and modelers and is designed to equip participants with both the conceptual framework and practical approaches necessary to integrate observational data into their oceanographic and atmospheric models effectively. Session 1 establishes the Bayesian foundations of DA and contrasts variational (3D-Var/4D-Var) with sequential (Kalman filter) approaches. Session 2 examines modern ensemble and hybrid methods, including Ensemble Kalman Filters, Particle Filters, and Ensemble-Variational (EnVar) frameworks. Session 3 

applies these techniques to a Red Sea case study, demonstrating a complete DA workflow, from inputs processing to result interpretation, while highlighting common pitfalls and best 

practices for operational forecasting.

Abstract: In these series of tutorials, we will introduce downscaling data assimilation algorithms for weather and climate prediction based on discrete coarse spatial scale measurements of the state variables (or only part of them, depending on the underlying model). The algorithm is based on linear nudging of the coarse spatial scales in the algorithm’s solution toward the coarse spatial scales corresponding to the observed measurements of the unknown reference solution. The algorithm’s solution can be initialized arbitrary and is shown to converge at an exponential rate toward the exact unknown reference solution. This shows that the dynamics of the nudged algorithm is globally stable (not chaotic) unlike the dynamics of the model that governs the unknown reference solution. Capitalizing on this fact, we will also prove uniform in time error estimates of the numerical discretization of these algorithms, which makes them reliable upon implementation computationally. Furthermore, we will also present a recent improvement of this algorithm by employing nonlinear nudging, which yields super exponential convergence rate toward the unknown exact reference solution. Notably, this algorithm applies to all dissipative systems, however, we will show by examples that it does not work for non-disspative systems.

As a product of this theoretical development, we will show that the trajectories in the global attractor of the Navier-Stokes equations are determined uniquely by a scalar parameter obeying a 1D ODE equation.