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SOLVING PDEs ON COMPLEX GEOMETRIES USING
PHYSICS-INFORMED NEURAL NETWORKS (PINNs)
Deep neural network, ADF to an annulus, and solution to
Laplace BVP
Use of neural networks to solve partial differential equations (PDEs) was introduced by
Lagaris et al. (1998), but only over the past few
years have we seen a surge in the applications of deep neural networks (two or more
hidden layers) for the solution of low- and high-dimension
PDEs over bounded domains in Rd. This has been driven by two
major contributions: (1)
Raissi et al. (2019),
who referred to the approach as Physics-Informed
Neural Networks (PINNs), and used a collocation approach to solve forward
and inverse problems, and (2)
E and Yu (2018)
(also see arXiv)
who proposed a deep Ritz (variational) formulation to solve
boundary-value problems. PINNs have approximation power that can
recover h-, p- and r-adaptive
finite element solutions and they are well-suited to solve forward, inverse, parametric
design and high-dimensional problems, which makes them a powerful and attractive choice.
From my prior work on meshfree methods and well-known issues pertaining to the
satisfaction of essential boundary conditions in meshfree Galerkin methods, it stands to reason that this is also pertinent in PINNs. With an eye on solving solid continua
problems over complex geometries, we introduced an
approach to exactly impose boundary conditions in PINNs that is based on
approximate distance functions (ADFs) and the theory of R-functions.
The PINN ansatz is formed so that all boundary conditions for scalar
PDEs are met—this improves network training and accuracy of the
PINN solution.
Publications
S. Berrone, C. Canuto, M. Pintore and
N. Sukumar (2023),
"Enforcing Dirichlet Boundary Conditions in Physics-Informed
Neural Networks and Variational Physics-Informed Neural Networks,"
Heliyon, Vol. 9, Article e18820.
![[HTML]](Gifs/pubs.gif)
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N. Sukumar and A. Srivastava (2022),
"Exact Imposition of Boundary Conditions with
Distance Functions in Physics-Informed Deep Neural Networks,"
Computer Methods in Applied Mechanics and Engineering,
Vol. 389, Article 114333.
This method has been implemented in NVIDIA Modulus (April 2022 Release)
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Presentations
- "Exact Imposition of Boundary Conditions
in PINNs to Solve PDEs," Invited Seminar,
Department of Civil and Environmental Engineering,
University of Pittsburgh,
Pittsburgh, PA, November 2023.
- "Exact Imposition of Boundary Conditions
in PINNs to Solve PDEs," Invited Seminar,
Department of Applied Mathematics,
University of Waterloo,
Ontario, Canada, October 2023.
- "Exact Imposition of Boundary Conditions
in PINNs to Solve PDEs," Invited Lecture,
BIRS Scientific Machine Learning Workshop, Banff, Canada, June 2023.
- "Recent Advances in Exact Imposition of Boundary Conditions
in Physics-Informed Deep Neural Networks to Solve PDEs," Invited Seminar,
Department of Mechanical Engineering,
Boston University,
Boston, MA, May 2023.
- "Recent Advances in Exact Imposition of Boundary Conditions
in Physics-Informed Deep Neural Networks to Solve PDEs," Rhodes
Information Initiative Seminar,
Duke University,
Durham, NC, May 2023.
- "Recent Advances in Exact Imposition of Boundary Conditions
in Physics-Informed Deep Neural Networks to Solve PDEs," Joint Materials
and Mechanics Seminar,
Brown University,
Providence, RI, May 2023.
- "Recent Advances in Exact Imposition of Boundary Conditions
in Physics-Informed Deep Neural Networks to Solve PDEs," Invited Seminar,
Dassault Systemes Simulia
Corporation, Johnston, RI, March 2023.
- "Exact Imposition of Boundary Condition
in Physics-Informed Deep Neural Networks to Solve PDEs,"
SEMM Seminar,
Department of Civil &
Environmental Engineering, University of California,
Berkeley, CA, October 2022.
- "Exact Imposition of Boundary Condition
in Physics-Informed Deep Neural Networks to Solve PDEs
(with A. Srivastava),"
USACM Thematic Conference on
Meshfree and Novel Finite Element Methods, Berkeley, CA, September 2022.
- "Recent Advances in Exact Imposition of Boundary Condition
in Physics-Informed Deep Neural Networks to Solve PDEs,"
Invited Seminar,
Center for Machine Intelligence and
Data Science,
IIT Bombay, Mumbai, India, September 2022.
- "Recent Advances in Exact Imposition of Boundary Condition
in Physics-Informed Deep Neural Networks to Solve PDEs," Invited Seminar,
Department of Computational and Data Sciences,
Indian Institute of Science, Bengaluru, India, July 2022.
- "Recent Advances in Polyhedral Virtual Element Methods and
Physics-Informed Deep Neural Networks to Solve PDEs," Invited Seminar,
Department of Mechanical Engineering,
Indian Institute of Science, Bengaluru, India, July 2022.
- "Recent Advances in Polyhedral Virtual Element Methods and
Physics-Informed Deep Neural Networks to Solve PDEs," Invited Seminar,
T-3 Division, Los Alamos National Laboratory,
Los Alamos, NM, May 2022.
- "Recent Advances in Polyhedral Virtual Element Methods and
Physics-Informed Deep Neural Networks to Solve PDEs," Invited Seminar,
Engineering Sciences Center,
Sandia National Laboratories,
Albuquerque, NM, May 2022.
- "Exact Imposition of Boundary Conditions with
Distance Functions in Physics-Informed Deep Neural Networks,"
Invited Seminar,
Mechanics and Computation Seminar,
Stanford University,
Stanford, CA, February 2022.
- "Meshfree Analysis on Complex Geometries Using
Physics-Informed Deep Neural Networks,"
Invited Seminar,
Instituto Superior Ténico,
University of Lisbon, Portugal, January 2022.
- "Meshfree Analysis on Complex Geometries Using
Physics-Informed Deep Neural Networks,"
Invited Seminar,
Sandia National Laboratories,
Albuquerque, NM, June 2021.
- "Exact Imposition of Boundary Conditions with
Distance Functions in Physics-Informed Deep Neural Networks
(with A. Srivastava),"
CRUNCH Group Seminar,
Brown University,
Providence, RI, May 2021.
Useful Links
© Copyright 2021-, N. Sukumar. All rights reserved.
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