Shape Space Spectra

TL;DR

Shape Space Spectra introduces a differentiable, discretization-agnostic framework to compute eigenfunctions across continuously parameterized shape spaces by learning neural-field representations conditioned on a shape code. It develops a novel trio of techniques—joint training, gradient causal filtering, and shape-dependent causal sorting—to maintain consistent eigenmodes across space where eigenvalues cross, enabling a single reduced-order model for an entire shape family. The approach extends from single-shape eigenanalysis to shape-space eigenanalysis, supports elasticity, and yields practical benefits in reduced-space simulations, differentiable shape optimization, and differentiable modal sound synthesis. By enabling eigenanalysis over arbitrary shape representations (including neural implicit fields) and ensuring differentiability with respect to shape parameters, the method facilitates fast queries, warm-starts for PDE solvers, and robust design across diverse geometries. Empirical results show competitive agreement with discrete operators, rotation invariance, and substantial speedups for large shape-space queries, highlighting its potential for broad applications in physics-informed design and spectral geometry.

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.

Figures (26)

• Figure 1: Elastic eigenfunctions over Shape Space. With a single neural model,we can compute eigenfunctions for a bridge over the entire shape-space shown in Fig. \ref{['fig:teaser']}.
• Figure 2: Eigenfunctions Across Different Representations Our method is discretization-agnostic; we have demonstrated the calculation of eigenfunctions for point clouds, signed distance fields, tetrahedral meshes, and neural implicit representations.
• Figure 3: Laplace Eigenfunctions for a Teapot Shape Space. The eigenfunctions of the Laplace operator describe the low-frequency heat distributions. We demonstrate these eigenfunctions across different teapot shapes in the shape space.
• Figure 4: Boundary Condition. Minimizing the Dirichlet energy on a shape naturally enforces a vanishing Neumann boundary condition. To illustrate this, we visualize the isolines of the eigenfunctions. Notably, the gradients along the boundary are nearly zero, highlighting the fulfillment of the Neumann condition.
• Figure 5: Comparison with eigenfunctions from cotangent Laplacians. When training on a single shape, our method converges to results consistent with traditional eigenanalysis. We compare the eigenfunctions obtained using our approach with those derived from a cotan Laplace (linear finite element on triangle mesh) matrix. The accuracy of our method matches that of existing techniques that rely on matrix construction from point cloud sampling.