Abstract:
This talk addresses the challenge of solving Partial Differential Equations (PDEs) with smooth or non-smooth solutions by formulating variational PDE formulations resulting in a soft-constrained optimization problem. The flexibility of the variational formulation enables us to use hybrid non-linear surrogates to approximate discontinuous shocks while solving forward or inverse PDE problems.
We first explore general concepts and tools necessary for solving PDEs under a variational formulation with general non-linear surrogates and boundary conditions. We then compare the numerical performance of Physics Informed Neural Networks (PINNs) as surrogates against Polynomial Surrogate Models (PSMs). Our goal is to open up the discussion regarding the class of problems that genuinely require the use of Neural Networks. Our findings indicate that PSMs outperform PINNs by several orders of magnitude in both accuracy and runtime.
Furthermore, we introduce a new method for approximating discontinuous functions using modified global spectral methods. We extend this method to solve PDEs with non-smooth solutions, providing an innovative solution to a highly challenging problem.
Juan Esteban Suarez is a PHD student at the Center for Advanced Systems Understanding (CASUS) and the department of Mathematics at the Technical University of Dresden in the field of Scientific Machine Learning. His research project is focused on studying the properties of the Sobolev gradient flow in variational problems with applications in scientific computing, using ML tools such as PINNs and autoencoders.
The Machine Learning Seminar is a regular weekly seminar series aiming to harbour presentations of fundamental and methodological advances in data science and machine learning as well as to discuss application areas presented by domain specialists. The uniqueness of the seminar series lies in its attempt to extract common denominators between domain areas and to challenge existing methodologies. The focus is thus on theory and applications to a wide range of domains, including Computational Physics and Engineering, Computational Biology and Life Sciences, Computational Behavioural and Social Sciences. More information about the ML Seminar, together with video recordings from past meetings you will find here: https://legato-team.eu/seminars/