Shape Space Spectra

TL;DR

This paper introduces a novel eigenanalysis method for continuously parameterized shape families, enabling shape space modeling, optimization, and reduced-order modeling across diverse shapes using neural fields.

Contribution

It presents the first eigenanalysis approach for shape families that is agnostic to shape representation and supports differentiable optimization over shape space.

Findings

Enables vibration mode analysis across shape space including unseen shapes

Facilitates shape optimization using eigenfunction gradients

Improves reduced-order modeling for elastodynamics

Abstract

Eigenanalysis of differential operators, such as the Laplace operator or elastic energy Hessian, is typically restricted to a single shape and its discretization, limiting reduced order modeling (ROM). We introduce the first eigenanalysis method for continuously parameterized shape families. Given a parametric shape, our method constructs spatial neural fields that represent eigenfunctions across the entire shape space. It is agnostic to the specific shape representation, requiring only an inside/outside indicator function that depends on shape parameters. Eigenfunctions are computed by minimizing a variational principle over nested spaces with orthogonality constraints. Since eigenvalues may swap dominance at points of multiplicity, we jointly train multiple eigenfunctions while dynamically reordering them based on their eigenvalues at each step. Through causal gradient filtering, this reordering is reflected in backpropagation. Our method enables applications to operate over shape space, providing a single ROM that encapsulates vibration modes for all shapes, including previously unseen ones. Since our eigenanalysis is differentiable with respect to shape parameters, it facilitates eigenfunction-aware shape optimization. We evaluate our approach on shape optimization for sound synthesis and locomotion, as well as reduced-order modeling for elastodynamic simulation.