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.
Abstract (translated)
现实世界数据的时空建模面临挑战，原因在于其固有的高维度、测量噪声以及昂贵的数据采集过程。本文介绍了一种名为“基于稀疏非线性动力学识别与浅层递归解码器网络的方法”（Sparse Identification of Nonlinear Dynamics with SHallow REcurrent Decoder networks, SINDy-SHRED）的新方法，该方法能够同时解决传感和模型识别问题，并且具有简单实现、高效计算以及鲁棒性能的特点。SINDy-SHRED使用门控循环单元（Gated Recurrent Units, GRUs）来建模传感器测量的时间序列，并结合浅层解码器网络从仅有少量可用传感器的状态空间中重建完整的时空场。 该方法引入了基于SINDy的正则化技术：从任意初始状态空间开始，通过逐步调整动态过程，最终使得隐含空间的动力学逐渐收敛于一类SINDy功能。在将SINDy限制为线性模型的情况下，架构生成一个Koopman-SHRED模型，该模型强制执行线性的隐藏空间动力学。 我们进行了一系列系统实验研究，包括合成PDE数据、海面温度的真实传感器测量以及直接视频数据。没有显式编码器的情况下，SINDy-SHRED和Koopman-SHRED方法允许使用极小的超参数调整和笔记本级别计算实现高效的训练；此外，在各种应用中展示了优秀的泛化能力，并且在这些应用中几乎没有进行任何或很少的超参数调整。 最后，通过可解释的SINDy及Koopman模型对隐含状态动态的建模，实现了准确的长时间视频预测，在性能上超越了所有考虑中的基线方法（包括卷积LSTM、PredRNN、ResNet和SimVP），达到了最先进的水平。
URL
https://arxiv.org/abs/2501.13329
