Towards Efficient Modelling of String Dynamics: A Comparison of State Space and Koopman based Deep Learning Methods

TL;DR

The paper investigates efficient neural modelling of string dynamics by contrasting State Space Models with Koopman-based deep learning approaches on linear and nonlinear stiff-string systems. Using datasets generated at multiple sample rates and varied initial conditions, it demonstrates that a Koopman-based model can achieve competitive or superior performance in nonlinear, long-horizon scenarios, while highlighting extrapolation challenges beyond training horizons. The work details architectures, training strategies, and a comprehensive comparative evaluation against baselines including LRU, S5, FRNN, FGRU, and DMD, offering practical guidance for physics-informed sequence modelling in musical acoustics. It also discusses limitations (notably extrapolation and missing velocity inputs) and outlines future directions such as incorporating velocity data, parameter conditioning, and extending to higher dimensions for broader applicability.

Abstract

This paper presents an examination of State Space Models (SSM) and Koopman-based deep learning methods for modelling the dynamics of both linear and non-linear stiff strings. Through experiments with datasets generated under different initial conditions and sample rates, we assess the capacity of these models to accurately model the complex behaviours observed in string dynamics. Our findings indicate that our proposed Koopman-based model performs as well as or better than other existing approaches in non-linear cases for long-sequence modelling. We inform the design of these architectures with the structure of the problems at hand. Although challenges remain in extending model predictions beyond the training horizon (i.e., extrapolation), the focus of our investigation lies in the models' ability to generalise across different initial conditions within the training time interval. This research contributes insights into the physical modelling of dynamical systems (in particular those addressing musical acoustics) by offering a comparative overview of these and previous methods and introducing innovative strategies for model improvement. Our results highlight the efficacy of these models in simulating non-linear dynamics and emphasise their wide-ranging applicability in accurately modelling dynamical systems over extended sequences.

Figures (4)

• Figure 1: Koopman-based architecture, with optional varying processing for the generated states, as indicated within the dashed box. This involves an MLP, where the eigenvalue radii serve as input, and its output is applied to the state sequence through element-wise multiplication.
• Figure 2: Non-linear dynamics for an unseen uniform noise-like initial condition in the range 0 to 1. The top row for each model shows the displacement evolution along the string for 400 time steps (100). The bottom row for each model displays the spectrum of the same section at a single point ($\approx24\cm$). All models were trained on 400 time steps from the same dataset.
• Figure 3: Evolution of the predicted displacement and extrapolation beyond the training horizon for a uniform noise-like initial condition (0 to 1). The top row shows the Koopmanvar extrapolation with absolute error (in centimetres). The middle plots zoom in on the first 0.25 seconds (left) and 0.25 seconds after the training horizon (right). The bottom plots show the corresponding spectrum at a single position on the string ($\approx24$ cm) for the time spans covered in the middle plots.
• Figure 4: Mean absolute error in centimetres per timestep across 100 unseen test trajectories for each architecture for 4000 time steps. Results are shown for a random noise-like initial condition (0 to 1) (a) for a linear string and (b) for a tension-modulated string at a single position on the string ($\approx24$ cm). The regions shaded in red represent the predictions after the training time step horizon. The results for the FGRU and FRNN models are not included as these could not be trained with more than 400 steps. The error curves are smoothed for easier visualisation.