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

TL;DR

This paper compares State Space and Koopman-based deep learning models for string dynamics, demonstrating that Koopman models perform well in non-linear, long-sequence scenarios and discussing their generalization capabilities.

Contribution

It provides a comparative analysis of SSM and Koopman-based models for string dynamics and introduces strategies to improve their performance and generalization.

Findings

Koopman-based models match or outperform existing methods in non-linear cases.

Models show strong ability to generalize across initial conditions within training intervals.

Challenges remain in extending predictions beyond training horizons.

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.