Zecheng Zhang: From Neural Single Operator to Neural Multi-Operator Foundation Models

In this talk, I will explore the machine learning approach to solving complex physical systems modeled by partial differential equations (PDEs). Since many PDE-solving problems can be framed as operator approximations, we will focus on operator learning. The discussion will begin by extending the universal approximation theorem to make it invariant to discretization, followed by an examination of distributed algorithms that can further improve the network flexibility to handle complex multiscale problems. To improve the network's ability to extrapolate, we will delve into multi-operator learning, particularly in designing foundation models that can address previously unseen problems. To mathematically quantify of these approximations, the talk will conclude with a discussion of neural scaling laws, focusing on the convergence of operator learning networks and the analysis of generalization error.