Source-linked AI summary
Discovery of Nonlinear Multiscale Systems: Sampling Strategies and Embeddings
Kathleen Champion, Steven L. Brunton, J. Nathan Kutz
TL;DR
The paper addresses how to discover interpretable models of nonlinear dynamical systems when behavior spans multiple time scales and measurements may be complete or incomplete. It combines modified SINDy and HAVOK strategies with temporal sampling approaches, showing that multiscale governing dynamics can be discovered from full or partial observations while reducing data requirements.
Problem
Model discovery must recover interpretable governing equations from data despite nonlinear multiscale behavior and incomplete measurements.
Method
The paper uses sparse-regression SINDy for full-state measurements and time-delay-embedding HAVOK for incomplete measurements, together with multiscale sampling and modeling strategies.
Results
The methods discover distinct slow and fast dynamics, keep SINDy data requirements approximately constant as time scales separate, and produce accurate HAVOK models for multiscale systems.
Takeaways & Limitations
The combined SINDy and HAVOK modifications provide computationally scalable strategies for extracting interpretable models from multiscale systems with full or partial state measurements.
Takeaways & Limitations
SINDy requires measured states that contribute to the dynamics and a library rich enough to span the governing function.
Abstract
from arXiv · showhide
A major challenge in the study of dynamical systems is that of model discovery: turning data into models that are not just predictive, but provide insight into the nature of the underlying dynamical system that generated the data. This problem is made more difficult by the fact that many systems of interest exhibit diverse behaviors across multiple time scales. We introduce a number of data-driven strategies for discovering nonlinear multiscale dynamical systems and their embeddings from data. We consider two canonical cases: (i) systems for which we have full measurements of the governing variables, and (ii) systems for which we have incomplete measurements. For systems with full state measurements, we show that the recent sparse identification of nonlinear dynamical systems (SINDy) method can discover governing equations with relatively little data and introduce a sampling method that allows SINDy to scale efficiently to problems with multiple time scales. Specifically, we can discover distinct governing equations at slow and fast scales. For systems with incomplete observations, we show that the Hankel alternative view of Koopman (HAVOK) method, based on time-delay embedding coordinates, can be used to obtain a linear model and Koopman invariant measurement system that nearly perfectly captures the dynamics of nonlinear quasiperiodic systems. We introduce two strategies for using HAVOK on systems with multiple time scales. Together, our approaches provide a suite of mathematical strategies for reducing the data required to discover and model nonlinear multiscale systems.
1 Introduction
Model discovery seeks interpretable governing equations from data, but nonlinear systems with coupled multiple time scales make this difficult. The paper addresses this challenge with SINDy for full measurements and HAVOK for incomplete observations.
- Motivation: Model discovery aims to recover governing equations or conserved quantities from measurement data rather than only predict system behavior.Interpretable models can improve understanding, prediction, and control of dynamical systems.
- Motivation: Deep neural networks can predict dynamics effectively, but their resulting models generally lack interpretable governing equations in the original variables.Symbolic regression is interpretable but computationally expensive, whereas sparse regression offers a more tractable alternative.
- Multiscale challenge: Multiple time scales complicate modeling because microscale and macroscale dynamics can be coupled through feedback.The paper notes that time scales must be disambiguated in a principled way.
- Paper scope: The paper studies nonlinear multiscale discovery for both full-state measurements and incomplete measurements with latent variables.It focuses on SINDy for full measurements and HAVOK time-delay embeddings for incomplete observations.
- Paper scope: When full measurements are available, SINDy uses sparse regression to identify governing equations; with incomplete measurements, HAVOK produces a linear model on a delay-embedded attractor.Figure 1 contrasts the two measurement settings and their corresponding discovery strategies.
2 Systems with full state measurements
For full-state measurements, the paper evaluates SINDy’s data requirements on uniscale systems and introduces burst sampling to reduce the cost of discovering multiscale dynamics. Across four example systems, SINDy identifies governing equations from fractions of an attractor, while burst sampling keeps sample requirements approximately constant as time scales separate.
- SINDy method: SINDy identifies nonlinear governing equations by selecting sparse terms from a candidate-function library, provided the measured states and library adequately represent the dynamics.The method targets the correct structural form of the governing equations rather than only predictive accuracy.
- Uniscale systems: SINDy’s baseline data requirements are assessed by varying sampling rate, duration, and trajectory location, then checking whether the correct nonzero coefficients are recovered.The experiments use a single trajectory after convergence to each system’s attractor.
- Lorenz system: 85% of a period is the Lorenz system’s average baseline sampling duration, with successful identification requiring 70–110% depending on the attractor portion sampled.Below baseline sampling rate or duration, increasing the other quantity does not compensate.
- Periodic oscillators: 25% of a period is the Duffing oscillator’s average requirement at sufficiently high sampling rate, while the Van der Pol oscillator averages around 20%.Duffing requirements range from 15–35%; Van der Pol requirements range from 5–35% depending on where the trajectory is sampled.
- Rossler system: The Rossler system requires 65% of a period on average, with baseline durations ranging from 35–95% across attractor portions, and can be identified without sampling its departure from the x–y plane.These results show that identification depends substantially on the sampled trajectory region.
- Multiscale systems: Uniform sampling requirements increase approximately linearly with frequency ratio F, whereas burst sampling keeps SINDy’s data requirement approximately constant as time scales separate.Burst sampling uses fine-resolution short bursts spread over a long duration, preserving derivative estimates while covering slow dynamics.
3 Incomplete measurements of the state space and latent variables
For incomplete measurements, HAVOK uses time-delay embeddings and DMD to construct linear models without requiring the full governing state. The paper extends HAVOK to multiscale systems through delay spacing and an iterative fast-then-slow modeling strategy.
- HAVOK uses Hankel time-delay embeddings and SVD to obtain dominant coordinates that are approximately Koopman invariant, thereby linearizing the dynamics in a new coordinate system.
- For quasiperiodic systems, HAVOK provides a linear model sufficient for reconstruction and long-term prediction without a nonlinear forcing term or prior knowledge of the true governing variables.
- Adequate HAVOK models require enough delay coordinates and a delay duration D = (q −1)∆t large enough to capture sufficient oscillation duration, with Fourier-like modes.
- 3.2.1 Method 1: Delay spacing: Delay spacing reduces the multiscale embedding cost: unlike standard models, it can significantly reduce error without increasing snapshot-matrix size as the frequency ratio grows.
- 3.2.2 Method 2: Iterative modeling: The iterative strategy first models fast dynamics from short recordings, subtracts predicted fast behavior from longer, slower-sampled data, and then models the slow dynamics.
4 Discussion
The discussion presents SINDy and HAVOK modifications as interpretable strategies for discovering multiscale dynamics from either full-state or partial measurements. It emphasizes temporal sampling as central to computational scalability and data efficiency.
- The paper develops interpretable multiscale models for both full-state measurements through SINDy and partial measurements through time-delay embedding and HAVOK.
- The modified SINDy and HAVOK architectures discover distinct coupled dynamics at well-separated slow and fast time scales.
- Robust temporal sampling strategies make multiscale discovery computationally scalable and provide guidelines for the minimum data required for accurate models.
- The methods and data are released as open-source software to support reproducibility.