Department of Computational and Data Sciences
Department Seminar
Speaker : Prof. Siddhartha Mishra, Chair Professor at ETH Zurich, Switzerland
Title : “Solving PDEs with Neural Operators ”
Date & Time : January 05, 2024, 11:00 AM
Venue : # 102, CDS Seminar Hall
ABSTRACT
Partial differential equations (PDEs) are ubiquitous as mathematical models in Science and Engineering. Solutions to both forward and inverse problems for PDEs can be encapsulated in terms of the so-called solution operators, i.e, mapping between infinite-dimensional function spaces, that map inputs such as coefficients, sources, initial and boundary conditions to the PDE solution. Learning the solution operator from data falls under the rubric of “operator learning”, a rapidly evolving field within machine learning. In contrast to standard deep learning, the inputs and outputs in operator learning are infinite-dimensional. Hence, special attention needs to be paid to the correspondence between the continuous operator and its discrete realizations. By expanding on notions of continuous-discrete equivalence in signal processing and harmonic analysis, we introduce Representation equivalent Neural Operators (ReNOs) and show how they are a suitable framework of structure preserving operator learning. Moreover, a concrete instantiation of ReNOs, the convolutional neural operator (CNO) is presented and demonstrated as the state of the art machine learning surrogate for a wide variety of PDE benchmarks. If time permits, we will discuss further applications such as learning PDE inverse problems with Neural Inverse Operators.
BIOGRAPHY
Siddhartha Mishra is a Chair Professor at ETH Zurich, Switzerland, where he heads the Computational and Applied Mathematics Laboratory (CamLab), within the Seminar for Applied Mathematics at the Department of Mathematics. He is also the director of Computational Science Zurich and a core faculty member of the ETH AI center. Mishra’s research interests are in numerical analysis of PDEs, scientific computing and machine learning and in applications to fluid dynamics, geophysics, astrophysics, climate science and engineering. For his contributions to research, Mishra has received many awards and honors, including the Collatz Prize of ICIAM (2019), the Germund Dahlquist Prize of SIAM (2021), the Rossler Prize of ETH (2023) and the Infosys Prize (2019). Mishra has also been a keynote speaker at leading international conferences such as the International Conference of Mathematicians (ICM) in 2018.
Host Faculty: Prof. Sashikumaar Ganesan
ALL ARE WELCOME
