AI 中文总结
本研究针对大型固体动力学系统模态基求解成本高的问题，提出基于RRAE的深度学习方法，通过耦合模态的非线性参数框架实现模态基参数化，在1维和2维问题中验证了该方法的有效性。
AI 中文摘要
在固体动力学领域，了解系统对应的模态基至关重要，这有助于针对期望的动力学行为改进设计，例如避免特定值处的固有频率，或设计能满足期望频谱的机械系统。然而，模态基的确定涉及特征值问题的求解，对于大型系统而言，该过程计算成本高昂，尤其是在处理参数化系统设计优化时。本研究提出基于秩缩减自动编码器（Rank Reduction AutoEncoder，RRAE）的先进深度学习技术来确定模态基的参数化方法。RRAE是一种自动编码器，其潜在空间通过截断奇异值分解（Singular Value Decomposition，SVD）近似进行约束，该公式能让潜在空间高效捕获数据的主导特征，从而引导自动编码器学习数据集中表征的潜在物理行为，缓解过拟合与虚假预测。核心思路是利用RRAE针对第一特征向量识别缩减参数空间，其余模态随后通过以同一缩减参数空间为输入的神经网络进行重构，进而在非线性参数框架中耦合所有模态。所提架构通过1维和2维问题中的模态基参数化得到验证。
英文摘要
In the field of solid dynamics, knowing the corresponding modal basis of the system is capital, in order to improve design with respect to a desired dynamical behavior, such as avoiding natural frequencies at specific values or designing mechanical systems that can account for desired frequency spectrum. However, the determination of the modal basis involve the resolution of an eigenvalue problem, which can be expensive to perform for large systems, especially when dealing with a optimization of a parametric system design. In the present work, we propose to determine the parametrization of modal basis by considering an advanced Deep Learning technique based on the Rank Reduction AutoEncoder (RRAE). The RRAE is based on an autoencoder whose latent space is constrained through a truncated Singular Value Decomposition (SVD) approximation. This formulation enables the latent space to capture the dominant features of the data efficiently. As a result, the autoencoder is guided toward learning the underlying physical behavior represented across the dataset, mitigating overfitting and spurious predictions. The main idea consists of identifying a reduced parameter space using the RRAE for the first eigenvector, while the remaining modes are subsequently reconstructed through neural networks that take the same reduced parameter space as input, thereby coupling all modes in a nonlinear parametric framework. The proposed architecture is validated through the parametrization of the modal basis in 1D and 2D problems.
