A plausible Parametrization of Modal Basis for Dynamical Systems Analysis

2026-08-01

AI Summary

This work addresses the computational bottleneck in traditional approaches for large-scale parametric dynamical systems, which require repeated and expensive eigenvalue solves to construct modal bases, thereby hindering efficient design optimization. To overcome this limitation, the authors propose a coupled architecture that integrates a rank-reduced autoencoder (RRAE) with a deep neural network. The RRAE leverages truncated singular value decomposition to define a unified low-dimensional parameter space, which serves as input to the neural network for jointly reconstructing all modal bases. This framework enables a nonlinear, physics-consistent parameterization of the modal basis while entirely eliminating the need for repeated eigenvalue computations. Validated on both 1D and 2D dynamical systems, the method demonstrates high accuracy and computational efficiency, effectively capturing dominant physical features and mitigating overfitting.

Abstract

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.

Problem

• modal basis
• dynamical systems
• eigenvalue problem
• parametric design
• computational cost

Innovation

• Rank Reduction AutoEncoder
• modal basis parametrization
• Singular Value Decomposition
• deep learning for dynamics
• reduced-order modeling