Data assimilation as simulation-based inference

Abstract

Complex dynamical systems are found across various scientific disciplines, representing phenomena like atmospheric and oceanic behavior, brain activity, robot state in its environment, among many others. Due to the challenges that those systems may address, it is often impractical to observe their complete state, leading to the collection of partial observations. For instance, weather stations can only measure a limited number of variables like temperature and pressure, but not the entire state of the atmosphere. However, despite the limited nature of those observations, we can still use them to infer and deduce states that are consistent with the gathered data. By leveraging advanced inference methods, we can make predictions about the complete state of complex dynamical systems based on these information. In this thesis, we delve into the realm of simulation-based inference methods applied to inverse problems in high-dimensional dynamical systems. We discuss how classical methods can be adapted to our problem and investigate how existing evaluation techniques can be used to assess our estimator's performances. Unlike classical simulation-based inference problems, our focus extends to incorporating the temporal dimension of such systems and scaling consistently existing inference methods with the size of the problem. Our goal is to infer the posterior density of states in dynamic systems, using observations to condition the inference process. By accounting for the temporal aspect, we can extend our understanding of the system's behavior and make informed predictions about its future states. Eventually, we show that existing estimation methods can adapt to our problem by incorporating consistently available information related to both system dynamics and observation process. We argue that convolutional estimators are needed to allow good scaling without increasing excessively computational costs. By leveraging system's structure, we found diffusion-based estimators being promising to solve our problem. We also highlight the need of new evaluation techniques that scales correctly and propose a classifier-based posterior check that fill the lacks of other classical evaluations at the cost of harder interpretation.