Past Event: CSEM Student Forum

Recent developments on Neural Galerkin schemes for solving PDEs and doing model order reduction.

Marshall Zhang,

Abstract

Some recent developments on Neural Galerkin schemes for solving PDEs and doing model order reduction. These methods project the solution to a non-linear manifold (typically parametrized by neural network), to overcome the Kolmogorov barrier which means achieving a fast error decay even if solution manifolds are not smooth and the Kolmogorov n-width decays slowly as in transport-dominated problems. More specifically, I'm going to cover the following 6 papers by Dr. Benjamin Peherstofer:
   1. Breaking the Kolmogorov Barrier with Nonlinear Model Reduction (B Peherstorfer, May 2022)
   2. Randomized Sparse Neural Galerkin Schemes for Solving Evolution Equations with Deep Networks. (Jules Berman, Benjamin Peherstorfer, Nov 2023)
   3. Neural Galerkin schemes with active learning for high-dimensional evolution equations (Joan Bruna, Benjamin Peherstorfer, Eric Vanden-Eijnden, January 2024)
   4. CoLoRA: Continuous Low-Rank Adaptation for Reduced Implicit Neural Modeling of Parameterized Partial Differential Equations (Jules Berman, Benjamin Peherstorfer, February 2024)
   5. Sequential-in-time training of nonlinear parametrizations for solving time-dependent partial differential equations.(H. Zhang, Yifan Chen, E. Vanden-Eijnden, B. Peherstorfer, April 2024)
   6. Nonlinear model reduction with Neural Galerkin schemes on quadratic manifold(Philipp Weder, Paul Schwerdtner, Benjamin Peherstorfer, December 2024)