# Sparse identification of nonlinear dynamics and Koopman operators with Shallow Recurrent Decoder Networks

**Authors:** Mars Liyao Gao, Jan P. Williams, J. Nathan Kutz  
**arXiv:** [2501.13329](https://arxiv.org/abs/2501.13329)  
**Date:** 2025-01-23  
**Summary page:** https://www.summarizepaper.com/en/paper/2501.13329v2/

## Key points

- Authors address challenges in modeling real-world spatio-temporal data:
- High dimensionality
- Measurement noise
- Partial observations
- Costly data collection procedures
- Leveraging Gated Recurrent Units for capturing temporal sequences of sparse sensor measurements
- Using a shallow decoder network to reconstruct the full spatio-temporal field from a latent state space
- Incorporating SINDy-based regularization to guide the latent space towards a SINDy-class functional
- Learning an interpretable generative model in a low-dimensional latent space for complex dynamics
- Discovering new physics models even for well-known systems
- Achieving robust convergence with a globally convex loss landscape
- Demonstrating superior accuracy, data efficiency, and training time compared to existing deep learning models
- Validation through experimental studies on various datasets:
- Turbulent flows, sea surface temperature sensor measurements, and direct video data
- Interpretable SINDy and Koopman models enable stable and accurate long-term video predictions, surpassing current baseline deep learning models.

## Summary

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In their paper titled "Sparse identification of nonlinear dynamics and Koopman operators with Shallow Recurrent Decoder Networks," authors Mars Liyao Gao, Jan P. Williams, and J. Nathan Kutz address the challenges associated with modeling real-world spatio-temporal data. They highlight the difficulties arising from high dimensionality, measurement noise, partial observations, and costly data collection procedures. To tackle these issues, the authors introduce a novel method called Sparse Identification of Nonlinear Dynamics with SHallow REcurrent Decoder networks (SINDy-SHRED). SINDy-SHRED leverages Gated Recurrent Units to capture the temporal sequence of sparse sensor measurements and employs a shallow decoder network to reconstruct the full spatio-temporal field from a latent state space. The algorithm incorporates a SINDy-based regularization that guides the latent space towards a SINDy-class functional, leading to the generation of a Koopman-SHRED model when restricting SINDy to a linear framework. The key contributions of SINDy-SHRED include: 
(i) learning an interpretable generative model in a parsimonious and low-dimensional latent space for complex spatio-temporal dynamics,
(ii) discovering new physics models even for well-known systems,
(iii) achieving robust convergence with a globally convex loss landscape,
(iv) demonstrating superior accuracy, data efficiency, and training time compared to existing deep learning models. The authors validate their approach through systematic experimental studies on various datasets including turbulent flows, sea surface temperature sensor measurements, and direct video data. The interpretable SINDy and Koopman models enable stable and accurate long-term video predictions, surpassing current baseline deep learning models such as Convolutional LSTM, PredRNN, ResNet, and SimVP in terms of accuracy, training time, and data requirements. Overall, the proposed method offers a promising solution for effectively modeling complex spatio-temporal dynamics by combining sparse sensing techniques with deep learning architectures.