High-order differentiable autoencoder for nonlinear model reduction

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