Curriculum learning for data-driven modeling of dynamical systems.

The reliable prediction of the temporal behavior of complex systems is key in numerous scientific fields. This strong interest is however hindered by modeling issues: Often, the governing equations describing the physics of the system under consideration are not accessible or, when known, their solution might require a computational time incompatible with the prediction time constraints. Not surprisingly, approximating complex systems in a generic functional format and informing it ex-nihilo from available observations has become common practice in the age of machine learning, as illustrated by the numerous successful examples based on deep neural networks. However, generalizability of the models, margins of guarantee and the impact of data are often overlooked or examined mainly by relying on prior knowledge of the physics. We tackle these issues from a different viewpoint, by adopting a curriculum learning strategy. In curriculum learning, the dataset is structured such that the training process starts from simple samples toward more complex ones in order to favor convergence and generalization. The concept has been developed and successfully applied in robotics and control of systems. Here, we apply this concept for the learning of complex dynamical systems in a systematic way. First, leveraging insights from the ergodic theory, we assess the amount of data sufficient for a-priori guaranteeing a faithful model of the physical system and thoroughly investigate the impact of the training set and its structure on the quality of long-term predictions. Based on that, we consider entropy as a metric of complexity of the dataset; we show how an informed design of the training set based on the analysis of the entropy significantly improves the resulting models in terms of generalizability and provide insights on the amount and the choice of data required for an effective data-driven modeling.

Shallow neural networks for fluid flow reconstruction with limited sensors.

In many applications, it is important to reconstruct a fluid flow field, or some other high-dimensional state, from limited measurements and limited data. In this work, we propose a shallow neural network-based learning methodology for such fluid flow reconstruction. Our approach learns an end-to-end mapping between the sensor measurements and the high-dimensional fluid flow field, without any heavy preprocessing on the raw data. No prior knowledge is assumed to be available, and the estimation method is purely data-driven. We demonstrate the performance on three examples in fluid mechanics and oceanography, showing that this modern data-driven approach outperforms traditional modal approximation techniques which are commonly used for flow reconstruction. Not only does the proposed method show superior performance characteristics, it can also produce a comparable level of performance to traditional methods in the area, using significantly fewer sensors. Thus, the mathematical architecture is ideal for emerging global monitoring technologies where measurement data are often limited.