In this talk, I will discuss two topics that I have been exploring at the intersection of deep learning and dynamical systems.
    I will first present my recent research on ML-based dynamics learning and surrogate modeling. To circumvent difficulties faced by dynamics models from first principles or standard neural networks, a recent research direction has been considering a hybrid approach, where physics laws and geometric properties are encoded in the design of the deep learning architectures or in the learning process. Available physics prior knowledge can be used to construct physics-constrained neural networks with improved design and efficiency and a better generalization capacity, which can take advantage of the function approximation power of deep learning methods to deal with incomplete knowledge. Here, I will introduce two different ways to incorporate prior knowledge about the structure of a dynamical system in a strong way to obtain structure-preserving deep learning architectures for dynamics learning and surrogate modeling.
    In the second part of the talk, I will discuss a specific way to leverage deep learning techniques to accelerate the computation of high-resolution solutions of parametric partial differential equations. In numerous contexts, high-resolution solutions are required to capture faithfully essential dynamics which occur at small spatiotemporal scales, but these solutions can be very difficult and slow to obtain using traditional numerical integration methods due to limited computational resources. Here, I will introduce a new approach based on the use of neural operators to obtain high-resolution solution operators.