  This paper provides a new avenue for exploiting deep neural networks to
improve physics-based simulation. Specifically, we integrate the classic
Lagrangian mechanics with a deep autoencoder to accelerate elastic simulation
of deformable solids. Due to the inertia effect, the dynamic equilibrium cannot
be established without evaluating the second-order derivatives of the deep
autoencoder network. This is beyond the capability of off-the-shelf automatic
differentiation packages and algorithms, which mainly focus on the gradient
evaluation. Solving the nonlinear force equilibrium is even more challenging if
the standard Newton's method is to be used. This is because we need to compute
a third-order derivative of the network to obtain the variational Hessian. We
attack those difficulties by exploiting complex-step finite difference, coupled
with reverse automatic differentiation. This strategy allows us to enjoy the
convenience and accuracy of complex-step finite difference and in the meantime,
to deploy complex-value perturbations as collectively as possible to save
excessive network passes. With a GPU-based implementation, we are able to wield
deep autoencoders (e.g., $10+$ layers) with a relatively high-dimension latent
space in real-time. Along this pipeline, we also design a sampling network and
a weighting network to enable \emph{weight-varying} Cubature integration in
order to incorporate nonlinearity in the model reduction. We believe this work
will inspire and benefit future research efforts in nonlinearly reduced
physical simulation problems.
