Data Assimilation in Chaotic Systems Using Deep Reinforcement Learning

Data assimilation (DA) plays a pivotal role in diverse applications, ranging from climate predictions and weather forecasts to trajectory planning for autonomous vehicles. A prime example is the widely used ensemble Kalman filter (EnKF), which relies on linear updates to minimize variance among the ensemble of forecast states. Recent advancements have seen the emergence of deep learning approaches in this domain, primarily within a supervised learning framework. However, the adaptability of such models to untrained scenarios remains a challenge. In this study, we introduce a novel DA strategy that utilizes reinforcement learning (RL) to apply state corrections using full or partial observations of the state variables. Our investigation focuses on demonstrating this approach to the chaotic Lorenz '63 system, where the agent's objective is to minimize the root-mean-squared error between the observations and corresponding forecast states. Consequently, the agent develops a correction strategy, enhancing model forecasts based on available system state observations. Our strategy employs a stochastic action policy, enabling a Monte Carlo-based DA framework that relies on randomly sampling the policy to generate an ensemble of assimilated realizations. Results demonstrate that the developed RL algorithm performs favorably when compared to the EnKF. Additionally, we illustrate the agent's capability to assimilate non-Gaussian data, addressing a significant limitation of the EnKF.

Nonlinear dimensionality reduction then and now: AIMs for dissipative PDEs in the ML era

This study presents a collection of purely data-driven workflows for constructing reduced-order models (ROMs) for distributed dynamical systems. The ROMs we focus on, are data-assisted models inspired by, and templated upon, the theory of Approximate Inertial Manifolds (AIMs); the particular motivation is the so-called post-processing Galerkin method of Garcia-Archilla, Novo and Titi. Its applicability can be extended: the need for accurate truncated Galerkin projections and for deriving closed-formed corrections can be circumvented using machine learning tools. When the right latent variables are not a priori known, we illustrate how autoencoders as well as Diffusion Maps (a manifold learning scheme) can be used to discover good sets of latent variables and test their explainability. The proposed methodology can express the ROMs in terms of (a) theoretical (Fourier coefficients), (b) linear data-driven (POD modes) and/or (c) nonlinear data-driven (Diffusion Maps) coordinates. Both Black-Box and (theoretically-informed and data-corrected) Gray-Box models are described; the necessity for the latter arises when truncated Galerkin projections are so inaccurate as to not be amenable to post-processing. We use the Chafee-Infante reaction-diffusion and the Kuramoto-Sivashinsky dissipative partial differential equations to illustrate and successfully test the overall framework.