Approximating Parametric PDEs: Defeating the Curse of Dimensionality Using Multilevel CNNs

Cosmas Heiss (TU Berlin)

Apr 05. 2022, 11:10 — 12:00

In this talk, we present a multilevel neural network approach for approximating the solution mapping of the parametric Darcy problem. Our architecture is constructed using a multilevel cascade of CNNs configured similarly to the multigrid F-Cycle. We show that common multilevel CNN architectures are able to approximate multigrid cycles arbitrarily well. More specifically, U-Net architectures alone are able to approximate multigrid solver steps for the parametric Darcy problem without considerable overhead, thus beating the curse of dimensionality. Furthermore, we solidify our theoretical findings by presenting state-of-the-art numerical results on a wide variety of test cases.

Further Information
Venue:
ESI Boltzmann Lecture Hall
Associated Event:
Adaptivity, High Dimensionality and Randomness (Workshop)
Organizer(s):
Carsten Carstensen (HU Berlin)
Albert Cohen (Sorbonne U, Paris)
Michael Feischl (TU Vienna)
Christoph Schwab (ETH Z├╝rich)