Session: 16-01-01: Poster Session: NSF-Funded Research (Grad & Undergrad)

Paper Number: 100157

100157 - Distance-Preserving Manifold Denoising for Data-Driven Mechanics 

This work introduces a new data-driven approach that leverages a manifold embedding generated by neural networks to improve constitutive-law-free simulations with limited and noisy data. Following the distance-minimization method, 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 an admissible solution onto the data manifold. We achieve this, in the local optimization step, by training deep neural networks to map data from the constitutive manifold onto a lower-dimensional Euclidean vector space where the local distance minimization becomes much simpler. As such, we establish the relation between the norm of the mapped Euclidean vector space and the metric of the manifold and lead to a more geometrically consistent notion of distance for the material data. The continuous global optimization step is formulated in the original space which leads to a cheaper optimization problem in terms of computational cost and simpler in terms of finite element implementation. An autoencoder framework is utilized for communicating between the embedding space and ambient space during the algorithm iterations. In the case of clean data, we further reduce the computational complexity by utilizing an analytically bijective neural network. 
We restrict the encoder mapping function to reconstruct the target unknown data manifold while it approximately preserves the local distance between data points on the hyperplane and the ambient data space. The distance preservation is handled by constraining the jacobian matrix of the encoder mapping function. We denoise the data in the embedding space by a linear projection onto the hyper-plan which acts similar to the classical dimension reduction but in the latent space. We empirically show that the introduced isometric condition may have a regularization effect that reduces the chance of overfitting when the data is noisy; this type of regularization effect is different from than classical L2 or L1 norm regularizer. These treatments, in return, allow us to bypass the combinatorial optimization, which may considerably speed up the model-free simulations when data are abundant and of high dimensions. Meanwhile, the learning of embedding also improves the robustness of the algorithm when the data is sparse or distributed unevenly in the parametric space. Numerical experiments are provided to demonstrate and measure the performance of the manifold embedding technique under different circumstances. Results obtained from the proposed method and those obtained via the classical energy norms are compared.

Presenting Author: Bahador Bahmani Columbia University

Presenting Author Biography: Bahador is a Ph.D. candidate in Civil Engineering and Engineering Mechanics at Columbia University. His research is at the intersection of machine learning and computational mechanics.

Authors:

Bahador Bahmani Columbia University
Waiching Sun Columbia University