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SOLVING PDEs ON COMPLEX GEOMETRIES USING PHYSICS-INFORMED NEURAL NETWORKS (PINNs)

DNN DNN DNN

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

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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