Shape Space Spectra

2024-08-19

AI Summary

Conventional feature analysis is limited to individual shapes and discrete sampling, hindering generalization and differentiable optimization across shape families. Method: We propose a differentiable spectral analysis framework for continuously parameterized shape families. Our approach employs a neural field representation covering the entire shape space, jointly optimizing signed distance functions (SDFs), variational eigenfunctions, nested orthogonality constraints, and causal gradient filtering to enable end-to-end differentiability of both eigenfunctions and eigenvalues. Contribution/Results: First, we support dynamic eigenvalue reordering and differentiable backpropagation under eigenvalue crossings. Second, our method generalizes to unseen shapes with accurate modal prediction. Third, it enables unified reduced-order modeling and gradient-driven optimization across shapes. Experiments on acoustic synthesis, motion design, and elastic dynamics simulation demonstrate substantial improvements in generalization capability and optimization efficiency compared to prior methods.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Learning with Manifolds
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

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.

Problem

• Eigenanalysis for continuously parameterized shape families
• Differentiable eigenanalysis enabling shape optimization
• Single reduced-order model for all shapes' vibration modes

Innovation

• Spatial neural fields represent eigenfunctions across shape space
• Dynamic reordering of eigenfunctions based on eigenvalues
• Differentiable eigenanalysis enables eigenfunction-aware shape optimization