It is widely known that neural networks (NNs) are universal approximators of functions. However, a less known but powerful result is that a NN can accurately approximate any nonlinear operator. This universal approximation theorem of operators is suggestive of the potential of deep neural networks (DNNs) in learning operators of complex systems.
In this talk, I will present the deep operator network (DeepONet) to learn various 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), (Fourier-)MIONet for multiple-input operators, and multifidelity DeepONet. 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, hypersonics, and geological carbon sequestration. Deep learning models are usually limited to interpolation scenarios, and I will quantify the extrapolation complexity and develop a complete workflow to address the challenge of extrapolation for deep neural operators.
Lu Lu is an Assistant Professor in the Department of Statistics and Data Science at Yale University. Prior to joining Yale, he was an Assistant Professor in the Department of Chemical and Biomolecular Engineering at University of Pennsylvania from 2021 to 2023 and 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. His current research interest lies in scientific machine learning, including theory, algorithms, 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 the 2020 Joukowsky Family Foundation Outstanding Dissertation Award of Brown University.