A neural operator for predicting vibration frequency response curves from limited data
Authors: D. Bluedorn, A. Badawy, B. E. Saunders, D. Roettgen, A. Abdelkefi
Organizations: Department of Mechanical and Aerospace Engineering, New Mexico State University, Las Cruces, NM 88003, USA. · Knight Foundation School of Computing and Information Science, Florida International University, Miami, FL 33199, USA. · Sandia National Laboratories, Albuquerque, NM 87123, USA.
In the design of engineered components, rigorous vibration testing is essential for performance validation and identification of resonant frequencies and amplitudes encountered during operation. Performing this evaluation numerically via machine learning has great potential to accelerate design iteration and make testing workflows more efficient. However, dynamical systems are conventionally difficult to solve via machine learning methods without using physics-based regularizing loss functions. To properly perform this forecasting task, a structure that has an inspectable physical obedience can be devised without the use of regularizing terms from first principles. The method employed in this work is a neural operator integrated with an implicit numerical scheme. This architecture enables operators to learn of the underlying state-space dynamics from limited data, allowing generalization to untested driving frequencies and initial conditions. This network can infer the system's global frequency response by training on a small set of input conditions. As a foundational proof of concept, this investigation verifies the machine learning algorithm with a linear, single-degree-of-freedom system, demonstrating implicit obedience of dynamics. This approach demonstrates 99.87% accuracy in predicting the Frequency Response Curve (FRC), forecasting the frequency and amplitude of linear resonance training on 7% of the bandwidth of the solution. By training machine learning models to internalize physics information rather than trajectory, better generalization accuracy can be realized, vastly improving the timeframe for vibration studies on engineered components.
Reservoir computing has emerged as an efficient machine learning framework for predicting time series generated by dynamical systems. In contrast to other machine and deep learning approaches, a reservoir computing trains only the output layer via linear regression, leaving the reservoir (recurrent layer) untrained. This simplification makes reservoir computers easier to train and more amenable to experimentation. However, because current reservoirs consist of networks of randomly connected nodes and require the optimization of numerous hyperparameters, a framework that precisely explains how reservoir computing operates and how it can be optimized remains missing. Here, we propose a frequency-based reservoir inspired by the brain's oscillatory dynamics and its hierarchy of timescales. The frequency-based reservoir can be interpreted as an ensemble of independent oscillatory units, each processing a portion of the input's frequency content. This allows us to understand the reservoir's internal behavior by modeling it as a single unit driven by an external input. Borrowing from the theory of a nonlinear oscillator forced by complex periodic inputs, we found that units of the frequency-based reservoir selectively amplify and store specific input frequencies, which are then used for prediction. The frequency-based reservoir performs as well as or better than equivalent random reservoirs. Furthermore, the frequency-based approach can be optimized to improve short-term prediction, a property that random reservoirs lack. Finally, we show that the frequency-based reservoir can also predict complex spatiotemporal dynamics. Our results show that reservoir computing can be designed using brain properties and theoretical insights borrowed from the physics of forced nonlinear oscillators.
Arthur S Powanwe
Department of Mathematics, Western University, London, ON, Canada
Extrapolative prediction of complex nonlinear dynamics remains a central challenge in engineering. This study proposes a one-shot learning method to identify global frequency-response curves from a single excitation time history by learning governing equations. We introduce MEv-SINDy (Multi-frequency Evolutionary Sparse Identification of Nonlinear Dynamics) to infer the governing equations of non-autonomous and multi-frequency systems. The methodology leverages the Generalized Harmonic Balance (GHB) method to decompose complex forced responses into a set of slow-varying evolution equations. We validated the capabilities of MEv-SINDy on two critical Micro-Electro-Mechanical Systems (MEMS). These applications include a nonlinear beam resonator and a MEMS micromirror. Our results show that the model trained on a single point accurately predicts softening/hardening effects and jump phenomena across a wide range of excitation levels. This approach significantly reduces the data acquisition burden for the characterization and design of nonlinear microsystems.
Machine learning has opened new frontiers in purely data-driven algorithms for data assimilation in, and for forecasting of, dynamical systems; the resulting methods are showing some promise. However, in contrast to model-driven algorithms, analysis of these data-driven methods is poorly developed. In this paper we address this issue, developing a theory to underpin data-driven methods to solve smoothing problems arising in data assimilation and forecasting problems. The theoretical framework relies on two key components: (i) establishing the existence of the mapping to be learned; (ii) the properties of the operator learning architecture used to approximate this mapping. By studying these two components in conjunction, we establish novel universal approximation theorems for purely data driven algorithms for both smoothing and forecasting of dynamical systems. We work in the continuous time setting, hence deploying neural operator architectures. The theoretical results are illustrated with experiments studying the Lorenz 63, Lorenz 96 and Kuramoto-Sivashinsky dynamical systems.
Edoardo Calvello, Elizabeth Carlson, Nikola Kovachki +2
California Institute of Technology, Pasadena, CA · Oregon State University, Corvallis, OR · NVIDIA Corporation, Santa Clara, CA +1