Abstract: Partial Differential Equations (PDEs) model a large variety of phenomena of interest in Phyiscs and Engineering. Despite their remarkable success over many decades, numerical methods for approximating PDEs can incur a very high computational cost. This limitation has provided the impetus for the design of fast and accurate Machine Learning/AI based neural PDE surrogates which can learn the PDE solution operator from data. In this talk, we review some latest developments in the field of Neural Operators, which are widely used as an ML paradigm for PDEs and discuss state of the art neural operators based on convolutions or attention. We will discuss graph and transformer based architectures for PDEs on arbitrary domains and conditional Diffusion models for PDEs with chaotic multiscale solutions. Finally, the issue of sample complexity is addressed by the design of general purpose Foundation models for PDEs.

Bio-Sketch: Siddhartha Mishra is a Chair Professor for Applied Mathematics at ETH Zurich, where he heads the Computational and Applied Mathematics Laboratoray (CAMLab). He is also the Director of Computational Science Zurich and a core faculty member of the ETH AI center. Mishra's research is focussed on the design and analysis of numerical and AI/ML algorithms for simulating physical systems and their applications to astrophysics, geophysics, climate science, engineering and biology. Mishra is an elected Fellow of the European Academy of Sciences and the recipient of awards such as the Collatz Prize of ICIAM, Dahlquist Prize of SIAM, von Mises Prize of GAMM and Rossler Prize of ETH. He has been an invited speaker at leading international conferences including the International Congress of Mathematicians (ICM) and International Congress of Basic Sciences.