Deep Convolutional Neural Networks for Eigenvalue Problems in Mechanics

TL;DR

This paper demonstrates that deep convolutional neural networks significantly outperform traditional neural networks in predicting eigenvalues in mechanics, offering high accuracy with less data and the ability to generalize to higher dimensions and complex mechanical tensors.

Contribution

The study introduces a novel application of CNNs for eigenvalue problems in mechanics, showing their superior performance and data efficiency over traditional neural networks.

Findings

CNNs achieve 98% accuracy on unseen data for 1D phononic crystals.

CNNs outperform traditional networks even with fewer training samples.

CNNs can naturally represent mechanical material tensors.

Abstract

We show that deep convolutional neural networks (CNN) can massively outperform traditional densely-connected neural networks (both deep or shallow) in predicting eigenvalue problems in mechanics. In this sense, we strike out in a new direction in mechanics computations with strongly predictive NNs whose success depends not only on architectures being deep, but also being fundamentally different from the widely-used to date. We consider a model problem: predicting the eigenvalues of 1-D and 2-D phononic crystals. For the 1-D case, the optimal CNN architecture reaches $98\%$ accuracy level on unseen data when trained with just 20,000 samples, compared to $85\%$ accuracy even with $100,000$ samples for the typical network of choice in mechanics research. We show that, with relatively high data-efficiency, CNNs have the capability to generalize well and automatically learn deep symmetry operations, easily extending to higher dimensions and our 2D case. Most importantly, we show how CNNs can naturally represent mechanical material tensors, with its convolution kernels serving as local receptive fields, which is a natural representation of mechanical response. Strategies proposed are applicable to other mechanics' problems and may, in the future, be used to sidestep cumbersome algorithms with purely data-driven approaches based upon modern deep architectures.