ABSTRACT

In this talk, I will present our recent advancements in solving partial differential equations (PDEs) and discovering models in complex systems. The first part focuses on our approaches for approximating PDE solutions with rigorous guarantees, addressing challenges such as rough and noisy forcing terms. This includes the development of novel loss functions, such as the Negative Sobolev Norm for Gaussian Process (GP) methods and physics-informed neural networks (PINNs), along with a sparse GP approach to enhance computational efficiency. The second part highlights our work on uncovering unknown parameters and dynamics in scientific systems, utilizing hypergraph-based models and GP frameworks.