It is widely known that neural networks (NNs) are universal approximators of continuous functions. However, a less known but powerful result is that a NN can accurately approximate any nonlinear continuous operator. This universal approximation theorem of operators is suggestive of the structure and potential of deep neural networks (DNNs) in learning continuous operators or complex systems from streams of scattered data. In this talk, I will present the deep operator network (DeepONet) to learn various explicit operators, such as integrals and fractional Laplacians, as well as implicit operators that represent deterministic and stochastic differential equations. I will also present several extensions of DeepONet, such as DeepM&Mnet for multiphysics problems, DeepONet with proper orthogonal decomposition (POD-DeepONet), MIONet for multiple-input operators, and multifidelity DeepONet. More generally, DeepONet can learn multiscale operators spanning across many scales and trained by diverse sources of data simultaneously. I will demonstrate the effectiveness of DeepONet and its extensions to diverse multiphysics and multiscale problems, such as nanoscale heat transport, bubble growth dynamics, high-speed boundary layers, electroconvection, and hypersonics.

Bio: Lu Lu is an Assistant Professor in the Department of Chemical and Biomolecular Engineering at University of Pennsylvania. He is also a faculty of Penn Institute for Computational Science and of Graduate Group in Applied Mathematics and Computational Science. Prior to joining Penn, he was an Applied Mathematics Instructor in the Department of Mathematics at Massachusetts Institute of Technology from 2020 to 2021. He obtained his Ph.D. degree in Applied Mathematics at Brown University in 2020, master's degrees in Engineering, Applied Mathematics, and Computer Science at Brown University, and bachelor's degrees in Mechanical Engineering, Economics, and Computer Science at Tsinghua University in 2013. Lu has a multidisciplinary research background with research experience at the interface of applied mathematics, physics, computational biology, and computer science. The goal of his research is to model and simulate physical and biological systems at different scales by integrating modeling, simulation, and machine learning, and to provide strategies for system learning, prediction, optimization, and decision making in real time. His current research interest lies in scientific machine learning, including theory, algorithms, and software, and its applications to engineering, physical, and biological problems. His broad research interests focus on multiscale modeling and high performance computing for physical and biological systems. Lu has received the 2022 U.S. Department of Energy Early Career Award, and 2020 Joukowsky Family Foundation Outstanding Dissertation Award of Brown University.