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

• URL: http://arxiv.org/abs/2501.13329v1
• Date: Thu, 23 Jan 2025 02:18:13 GMT
• Title: Sparse identification of nonlinear dynamics and Koopman operators with Shallow Recurrent Decoder Networks
• Authors: Mars Liyao Gao, Jan P. Williams, J. Nathan Kutz,
• Abstract summary: We present S Identification of Dynamics with SHallow REcurrent Decoder networks (SINDy-SHRED), a method to jointly solve the sensing and model identification problems. SINDy-SHRED uses Gated Recurrent Units (GRUs) to model the temporal sequence of sensor measurements along with shallow decoder network to reconstruct the full field from the latent state space. We conduct a systematic experimental study including synthetic PDE data, real-world sensor measurements for sea surface temperature, and direct video data.
• Score: 3.1484174280822845
• License:
• Abstract: Spatiotemporal modeling of real-world data poses a challenging problem due to inherent high dimensionality, measurement noise, and expensive data collection procedures. In this paper, we present Sparse Identification of Nonlinear Dynamics with SHallow REcurrent Decoder networks (SINDy-SHRED), a method to jointly solve the sensing and model identification problems with simple implementation, efficient computation, and robust performance. SINDy-SHRED uses Gated Recurrent Units (GRUs) to model the temporal sequence of sensor measurements along with a shallow decoder network to reconstruct the full spatiotemporal field from the latent state space using only a few available sensors. Our proposed algorithm introduces a SINDy-based regularization; beginning with an arbitrary latent state space, the dynamics of the latent space progressively converges to a SINDy-class functional, provided the projection remains within the set. In restricting SINDy to a linear model, the architecture produces a Koopman-SHRED model which enforces a linear latent space dynamics. We conduct a systematic experimental study including synthetic PDE data, real-world sensor measurements for sea surface temperature, and direct video data. With no explicit encoder, SINDy-SHRED and Koopman-SHRED enable efficient training with minimal hyperparameter tuning and laptop-level computing; further, it demonstrates robust generalization in a variety of applications with minimal to no hyperparameter adjustments. Finally, the interpretable SINDy and Koopman models of latent state dynamics enables accurate long-term video predictions, achieving state-of-the-art performance and outperforming all baseline methods considered, including Convolutional LSTM, PredRNN, ResNet, and SimVP.

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