## Abstract

In this talk, an energy-based a posteriori error bound is presented for the physics-informed neural network solutions

of elasticity problems. The proposed error bound is formulated as the constitutive relation error defined by a solution

pair, consisting of a kinematically admissible displacement solution and a statically admissible stress solution. In this

work, such an admissible displacement-stress pair is obtained from a mixed form of physics-informed neural

networks. The proposed error estimator can provide an upper bound of the global error of neural network

discretization. From the perspective of neural network training, the generalization errors of physics-informed neural

networks can be quantified through the proposed energy-based error bounds in elasticity problems. The asymptotic

behavior of the physics-informed neural network solutions are discussed with the constitutive relation error bounds

as well. [1] E. Haghighat, M. Raissi, A. Moure, H. Gomez, and R. Juanes. A deep learning framework for solution

and discovery in solid mechanics. arXiv: 2003.02751, 2020. [2] M. Guo, and E. Haghighat. An energy-based error

bound of physics-informed neural network solutions in elasticity. arXiv: 2010.09088, 2020.

of elasticity problems. The proposed error bound is formulated as the constitutive relation error defined by a solution

pair, consisting of a kinematically admissible displacement solution and a statically admissible stress solution. In this

work, such an admissible displacement-stress pair is obtained from a mixed form of physics-informed neural

networks. The proposed error estimator can provide an upper bound of the global error of neural network

discretization. From the perspective of neural network training, the generalization errors of physics-informed neural

networks can be quantified through the proposed energy-based error bounds in elasticity problems. The asymptotic

behavior of the physics-informed neural network solutions are discussed with the constitutive relation error bounds

as well. [1] E. Haghighat, M. Raissi, A. Moure, H. Gomez, and R. Juanes. A deep learning framework for solution

and discovery in solid mechanics. arXiv: 2003.02751, 2020. [2] M. Guo, and E. Haghighat. An energy-based error

bound of physics-informed neural network solutions in elasticity. arXiv: 2010.09088, 2020.

Original language | English |
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Publication status | Published - 2021 |

Event | 16th U.S. National Congress on Computational Mechanics 2021 - Virtual Event, United States Duration: 25 Jul 2021 → 29 Jul 2021 Conference number: 16 |

### Conference

Conference | 16th U.S. National Congress on Computational Mechanics 2021 |
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Country/Territory | United States |

City | Virtual Event |

Period | 25/07/21 → 29/07/21 |