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

arXiv: https://arxiv.org/abs/2408.16650 · HTML: https://mlsanity.com/paper/2408.16650/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 29 references that are in the map.
- First posted on arXiv 29 Aug 2024.

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

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Efficiently Modeling Long Sequences with Structured State Spaces](https://mlsanity.com/paper/2111.00396/) (2021)
- [Simplified State Space Layers for Sequence Modeling](https://mlsanity.com/paper/2208.04933/) (2022)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Resurrecting Recurrent Neural Networks for Long Sequences](https://mlsanity.com/paper/2303.06349/) (2023)
- [Diagonal State Spaces are as Effective as Structured State Spaces](https://mlsanity.com/paper/2203.14343/) (2022)
- [How to Train Your HiPPO: State Space Models with Generalized Orthogonal Basis Projections](https://mlsanity.com/paper/2206.12037/) (2022)
- [Neural Operator Learning for Long-Time Integration in Dynamical Systems with Recurrent Neural Networks](https://mlsanity.com/paper/2303.02243/) (2023)
- [Physical Modeling using Recurrent Neural Networks with Fast Convolutional Layers](https://mlsanity.com/paper/2204.10125/) (2022)
- [Direct design of biquad filter cascades with deep learning by sampling random polynomials](https://mlsanity.com/paper/2110.03691/) (2021)
- [Rigid-Body Sound Synthesis with Differentiable Modal Resonators](https://mlsanity.com/paper/2210.15306/) (2022)

## Built on by (most cited first, 3 of 3)

- [Fast Differentiable Modal Simulation of Non-linear Strings, Membranes, and Plates](https://mlsanity.com/paper/2505.05940/) (2025)
- [Learning Nonlinear Dynamics in Physical Modelling Synthesis using Neural Ordinary Differential Equations](https://mlsanity.com/paper/2505.10511/) (2025)
- [Identifying the nonlinear string dynamics with port-Hamiltonian neural networks](https://mlsanity.com/paper/2605.12785/) (2026)

## Abstract (arXiv)

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.