  Real-time simulation of elastic structures is essential in many applications,
from computer-guided surgical interventions to interactive design in mechanical
engineering. The Finite Element Method is often used as the numerical method of
reference for solving the partial differential equations associated with these
problems. Yet, deep learning methods have recently shown that they could
represent an alternative strategy to solve physics-based problems 1,2,3. In
this paper, we propose a solution to simulate hyper-elastic materials using a
data-driven approach, where a neural network is trained to learn the non-linear
relationship between boundary conditions and the resulting displacement field.
We also introduce a method to guarantee the validity of the solution. In total,
we present three contributions: an optimized data set generation algorithm
based on modal analysis, a physics-informed loss function, and a Hybrid
Newton-Raphson algorithm. The method is applied to two benchmarks: a cantilever
beam and a propeller. The results show that our network architecture trained
with a limited amount of data can predict the displacement field in less than a
millisecond. The predictions on various geometries, topologies, mesh
resolutions, and boundary conditions are accurate to a few micrometers for
non-linear deformations of several centimeters of amplitude.
