Distance-preserving manifold denoising for data-driven mechanics
Distance-preserving manifold denoising for data-driven mechanics
复制标题
DOI：
10.1016/j.cma.2022.115857
复制
发表时间：
2023-01-09
影响因子：
7.2
通讯作者：
Sun，WaiChing
中科院分区：
文献类型：
--
作者：
Bahmani，Bahador;Sun，WaiChing
本文介绍了一种等距流形嵌入数据驱动的范例，旨在使无模型模拟与噪声数据采样从本构流形。所提出的数据驱动的方法迭代之间的全局优化问题，寻求容许的解决方案的平衡原则和局部优化问题，找到最近的点投影的欧氏空间等距嵌入非线性本构流形。为了对数据库去噪，引入几何自动编码器，使得编码器首先学习创建近似嵌入，该近似嵌入将高维本构流形的底层低维结构映射到具有较小曲率的平坦流形上。然后，我们通过将数据投影到完全平坦的去噪潜在空间上来获得无噪声的本构响应，该去噪潜在空间通过假设噪声和基础本构信号彼此正交来实现。因此，从保守流形到这个去噪的本构潜在空间的投影使我们能够完成数据驱动范式的局部优化步骤。最后，为了在不重新引入噪声的情况下解码在潜在空间中表达的数据，我们在训练自动编码器的同时施加一组等距约束，使得从潜在空间到重构的组成流形的非线性映射是距离保持的。数值例子被用来验证的实施，并证明所提出的范例的准确性，鲁棒性和局限性。
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.