wPINNs: Weak Physics Informed Neural Networks for Approximating Entropy Solutions of Hyperbolic Conservation Laws
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Date
2024-04
Publication Type
Journal Article
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yes
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Abstract
Physics informed neural networks (PINNs) require regularity of solutions of the underlying PDE to guarantee accurate approximation. Consequently, they may fail at approximating discontinuous solutions of PDEs such as nonlinear hyperbolic equations. To ameliorate this, we propose a novel variant of PINNs, termed as weak PINNs (wPINNs) for accurate approximation of entropy solutions of scalar conservation laws. wPINNs are based on approximating the solution of a min -max optimization problem for a residual, defined in terms of Kruzkhov entropies, to determine parameters for the neural networks approximating the entropy solution as well as test functions. We prove rigorous bounds on the error incurred by wPINNs and illustrate their performance through numerical experiments to demonstrate that wPINNs can approximate entropy solutions accurately.
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published
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Journal / series
Volume
62 (2)
Pages / Article No.
811 - 841
Publisher
SIAM
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Software
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Date created
Subject
PINNs; hyperbolic conservation laws; deep learning
Organisational unit
03851 - Mishra, Siddhartha / Mishra, Siddhartha
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Funding
770880 - Computation and analysis of statistical solutions of fluid flow (EC)
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