  The proliferation of 3D representations, from explicit meshes to implicit
neural fields and more, motivates the need for simulators agnostic to
representation. We present a data-, mesh-, and grid-free solution for elastic
simulation for any object in any geometric representation undergoing large,
nonlinear deformations. We note that every standard geometric representation
can be reduced to an occupancy function queried at any point in space, and we
define a simulator atop this common interface. For each object, we fit a small
implicit neural network encoding spatially varying weights that act as a
reduced deformation basis. These weights are trained to learn physically
significant motions in the object via random perturbations. Our loss ensures we
find a weight-space basis that best minimizes deformation energy by
stochastically evaluating elastic energies through Monte Carlo sampling of the
deformation volume. At runtime, we simulate in the reduced basis and sample the
deformations back to the original domain. Our experiments demonstrate the
versatility, accuracy, and speed of this approach on data including signed
distance functions, point clouds, neural primitives, tomography scans, radiance
fields, Gaussian splats, surface meshes, and volume meshes, as well as showing
a variety of material energies, contact models, and time integration schemes.
