Model reduction for the material point method via an implicit neural representation of the deformation map

Understand

This work proposes a model-reduction approach for the material point method on nonlinear manifolds.

• Our technique approximates the $\textit{kinematics}$ by approximating the deformation map using an implicit neural representation that restricts deformation trajectories to reside on a low-dimensional manifold.
• By explicitly approximating the deformation map, its spatiotemporal gradients -- in particular the deformation gradient and the velocity -- can be computed via analytical differentiation.
• In contrast to typical model-reduction techniques that construct a linear or nonlinear manifold to approximate the (finite number of) degrees of freedom characterizing a given spatial discretization, the use of an implicit neural representation enables the proposed method to approximate the $\textit{continuous}$ deformation map.