Bounding discretization errors of physics-informed neural network solutions in elasticity

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.