TY - JOUR

T1 - Asymptotic Self-Similar Blow-Up Profile for Three-Dimensional Axisymmetric Euler Equations Using Neural Networks

AU - Wang, Y.

AU - Lai, C. Y.

AU - Gómez-Serrano, J.

AU - Buckmaster, T.

N1 - Funding Information:
The authors were supported by the NSF Grants No. DMS-1900149 and No. DMS-1929284, ERC Grant No. 852741, the Spanish State Research Agency, through the Severo Ochoa and María de Maeztu Program for Centers and Units of Excellence in R&D (CEX2020-001084-M), Dean for Research Fund at Princeton University, the Schmidt DataX Fund at Princeton University, a Simons Foundation Mathematical and Physical Sciences Collaborative Grant, and grant from the Institute for Advanced Study. The simulations presented in this Letter were performed using the Princeton Research Computing resources at Princeton University and School of Natural Sciences Computing resources at the Institute for Advanced Study.
Publisher Copyright:
© 2023 American Physical Society.

PY - 2023/6/16

Y1 - 2023/6/16

N2 - Whether there exist finite-time blow-up solutions for the 2D Boussinesq and the 3D Euler equations are of fundamental importance to the field of fluid mechanics. We develop a new numerical framework, employing physics-informed neural networks, that discover, for the first time, a smooth self-similar blow-up profile for both equations. The solution itself could form the basis of a future computer-assisted proof of blow-up for both equations. In addition, we demonstrate physics-informed neural networks could be successfully applied to find unstable self-similar solutions to fluid equations by constructing the first example of an unstable self-similar solution to the Córdoba-Córdoba-Fontelos equation. We show that our numerical framework is both robust and adaptable to various other equations.

AB - Whether there exist finite-time blow-up solutions for the 2D Boussinesq and the 3D Euler equations are of fundamental importance to the field of fluid mechanics. We develop a new numerical framework, employing physics-informed neural networks, that discover, for the first time, a smooth self-similar blow-up profile for both equations. The solution itself could form the basis of a future computer-assisted proof of blow-up for both equations. In addition, we demonstrate physics-informed neural networks could be successfully applied to find unstable self-similar solutions to fluid equations by constructing the first example of an unstable self-similar solution to the Córdoba-Córdoba-Fontelos equation. We show that our numerical framework is both robust and adaptable to various other equations.

UR - http://www.scopus.com/inward/record.url?scp=85163754521&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85163754521&partnerID=8YFLogxK

U2 - 10.1103/PhysRevLett.130.244002

DO - 10.1103/PhysRevLett.130.244002

M3 - Article

C2 - 37390436

AN - SCOPUS:85163754521

SN - 0031-9007

VL - 130

JO - Physical Review Letters

JF - Physical Review Letters

IS - 24

M1 - 244002

ER -