# A neural operator for predicting vibration frequency response curves from limited data

arXiv: https://arxiv.org/abs/2603.10149 · HTML: https://mlsanity.com/paper/2603.10149/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 44 references that are in the map.
- First posted on arXiv 10 Mar 2026.

## Builds on (most cited first, 11 of 11)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Stiff Neural Ordinary Differential Equations](https://mlsanity.com/paper/2103.15341/) (2021)
- [Lagrangian operator inference enhanced with structure-preserving machine learning for nonintrusive model reduction of mechanical systems](https://mlsanity.com/paper/2404.05040/) (2024)
- [Predictive Limitations of Physics-Informed Neural Networks in Vortex Shedding](https://mlsanity.com/paper/2306.00230/) (2023)
- [Training Stiff Neural Ordinary Differential Equations with Implicit Single-Step Methods](https://mlsanity.com/paper/2410.05592/) (2024)
- [Generalized Lagrangian Neural Networks](https://mlsanity.com/paper/2401.03728/) (2024)
- [Stability-Informed Initialization of Neural Ordinary Differential Equations](https://mlsanity.com/paper/2311.15890/) (2023)
- [Data-driven Nonlinear Model Reduction to Spectral Submanifolds in Mechanical Systems](https://mlsanity.com/paper/2110.01929/) (2021)
- [Deep Neural ODE Operator Networks for PDEs](https://mlsanity.com/paper/2510.15651/) (2025)
- [Non-intrusive Balancing Transformation of Highly Stiff Systems with Lightly-damped Impulse Response](https://mlsanity.com/paper/2109.10408/) (2021)
- [Lagrangian neural networks for nonholonomic mechanics](https://mlsanity.com/paper/2411.00110/) (2024)

## Abstract (arXiv)

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.