Neural models speed up optimizing shapes for vibration frequencies
Neural networks for spectral optimization
Machine LearningArtificial Intelligence
Summary
Some problems involve finding the best shape so that the vibrations or other wave behaviors in that shape have specific properties. Traditionally, finding these shapes requires slow calculations each time. The authors created two neural networks that learn how these vibrations relate to shapes, so they can quickly predict the important numbers (eigenvalues) from a shape. This lets them find shapes that optimize these vibrations much faster, and the models even work well on new shapes not seen before.
What this means in practice
- •For architectural engineers: Design building components by quickly finding shapes with desired vibration or resonance properties without repeated heavy simulations.
- •For aerospace designers: Optimize aircraft part shapes for vibration minimization using fast spectral predictions from neural models instead of costly PDE solves.
Authors
Alexis de Villeroché, Beniamin Bogosel, Stéphane Breuils, Dorin Bucur, Jacques-Olivier Lachaud
Abstract
Given a functional dependent on the spectrum of a differential operator, we address the problem of finding a domain which optimizes this functional. PDE solvers might be used to tackle this optimization. It is however computationally expensive. We propose two neural network models which learn the spectrum directly from the geometry of the domain and can be used to optimize the domain from one or more eigenvalues. We investigate two representations. The first encodes the domain through Fourier coefficients and a light MLP, which is efficient on star-shaped geometries, achieving a precision of 0.2\%. Through a rescaling of the coefficients the designed models satisfy the scaling law of the eigenvalues. Additionally, averaging the outputs of the trained surrogates over rotations and reflections induces invariance for these transformations. The second is a model that takes the landscape function, the indicator function and the gradient of the landscape function. A Gram-Schmidt process produces orthogonal eigenfunctions as output of the model along with the associated eigenvalues. The landscape model reaches 1\% mean relative error on the first ten eigenvalues, compared with 4\% for an FNO model. Replacing the landscape by an SDF worsened both prediction and optimization errors. The trained model also generalizes from synthetic shapes to domains given as classical image dataset. The resulting surrogates of both approaches recover classical spectral optima such as the disk for the first eigenvalue or the conjectured minima of higher eigenvalues. This confirms that our models produce accurate differentiable estimates of eigenvalues, which can be used in shape optimization problems involving spectral quantities.