Deep convolutional neural networks for eigenvalue problems in mechanics

By D. Finol, Yan Lu, Vijay Mahadevan, Ankit Srivastava. 2018

73 citations recorded by the source corpus.

We show that deep convolutional neural networks (CNNs) can massively outperform traditional densely connected neural networks (NNs) (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 one‐dimensional (1D) and two‐dimensional (2D) phononic crystals. For the 1D 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.

Subfields: Biomedical Engineering, Civil and Structural Engineering, Statistical and Nonlinear Physics

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Citations in the corpus

How to See Hidden Patterns in Metamaterials with Interpretable Machine Learning (2021, background): “Because of the expensive computational cost of numerical simulations, these works train machine learning models (mostly deep learning models) to approximate the simulation results [26, 4, 27, 9, 35].”
Tailoring Structure‐Borne Sound through Bandgap Engineering in Phononic Crystals and Metamaterials: A Comprehensive Review (2022, methodology): “, to calculate the dispersion behavior, as shown in Refs [446,471]) as well as the optimization step.”
Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks (2021, unknown): “Independent of research in PINNs, some of these modern architectures have also been applied to mechanics problems [88].”
A high-bias, low-variance introduction to Machine Learning for physicists (2018, unknown): “In relativistic hydrodynamics, deep learning has been shown to capture features of non-linear evolution and has the potential to accelerate numerical simulations (Huang et al., 2018), while in mechanics CNNs have been used to predict eigenvalues of photonic crystals (Finol et al., 2018).”