Distance-preserving manifold denoising for data-driven mechanics

This article introduces an isometric manifold embedding data-driven paradigm designed to enable model-free simulations with noisy data sampled from a constitutive manifold. The proposed data-driven approach iterates between a global optimization problem that seeks admissible solutions for the balance principle and a local optimization problem that finds the closest point projection of the Euclidean space that isometrically embeds a nonlinear constitutive manifold. To de-noise the database, a geometric autoencoder is introduced such that the encoder first learns to create an approximated embedding that maps the underlying low-dimensional structure of the high-dimensional constitutive manifold onto a flattened manifold with less curvature. We then obtain the noise-free constitutive responses by projecting data onto a denoised latent space that is completely flat by assuming that the noise and the underlying constitutive signal are orthogonal to each other. Consequently, a projection from the conservative manifold onto this de-noised constitutive latent space enables us to complete the local optimization step of the data-driven paradigm. Finally, to decode the data expressed in the latent space without reintroducing noise, we impose a set of isometry constraints while training the autoencoder such that the nonlinear mapping from the latent space to the reconstructed constituent manifold is distance-preserving. Numerical examples are used to both validate the implementation and demonstrate the accuracy, robustness, and limitations of the proposed paradigm.(c) 2022 Elsevier B.V. All rights reserved.

Automated summary

This article presents an isometric manifold embedding data-driven paradigm for model-free simulations with noisy data. The proposed approach solves a global optimization problem to find admissible solutions for the balance principle and a local optimization problem to project the Euclidean space onto a nonlinear constitutive manifold. A geometric autoencoder is introduced to de-noise the database by mapping the high-dimensional constitutive manifold onto a flattened manifold. Numerical examples validate the implementation and show the accuracy, robustness, and limitations of the proposed paradigm.