Despite the success of various methods in addressing the issue of spatial reconstruction of dynamical systems with sparse observations, spatio-temporal prediction for sparse fields remains a challenge. Existing Kriging-based frameworks for spatio-temporal sparse field prediction fail to meet the accuracy and inference time required for nonlinear dynamic prediction problems. In this paper, we introduce the Dynamical System Prediction from Sparse Observations using Voronoi Tessellation (DSOVT) framework, an innovative methodology based on Voronoi tessellation which combines convolutional encoder–decoder (CED) and long short-term memory (LSTM) and utilizing Convolutional Long Short-Term Memory (ConvLSTM). By integrating Voronoi tessellations with spatio-temporal deep learning models, DSOVT is adept at predicting dynamical systems with unstructured, sparse, and time-varying observations. CED-LSTM maps Voronoi tessellations into a low-dimensional representation for time series prediction, while ConvLSTM directly uses these tessellations in an end-to-end predictive model. Furthermore, we incorporate physics constraints during the training process for dynamical systems with explicit formulas. Compared to purely data-driven models, our physics-based approach enables the model to learn physical laws within explicitly formulated dynamics, thereby enhancing the robustness and accuracy of rolling forecasts. Numerical experiments on real sea surface data and shallow water systems clearly demonstrate our framework’s accuracy and computational efficiency with sparse and time-varying observations.

中文翻译：

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使用具有 Voronoi 镶嵌和物理约束的深度神经网络从稀疏观测中预测动力学系统。


尽管各种方法在解决使用稀疏观测的动力系统空间重建问题方面取得了成功，但稀疏场的时空预测仍然是一个挑战。现有的基于 Kriging 的时空稀疏场预测框架无法满足非线性动态预测问题所需的精度和推理时间。在本文中，我们介绍了使用 Voronoi Tessellation （DSOVT） 框架从稀疏观测中预测动态系统，这是一种基于 Voronoi Tessellation 的创新方法，它结合了卷积编码器-解码器 （CED） 和长短期记忆 （LSTM） 并利用卷积长短期记忆 （ConvLSTM）。通过将 Voronoi 镶嵌与时空深度学习模型集成，DSOVT 擅长通过非结构化、稀疏和时变观测来预测动态系统。CED-LSTM 将 Voronoi 镶嵌映射到用于时间序列预测的低维表示，而 ConvLSTM 直接在端到端预测模型中使用这些镶嵌。此外，我们在动力学系统的训练过程中加入了具有明确公式的物理约束。与纯粹的数据驱动模型相比，我们基于物理的方法使模型能够在明确制定的动力学中学习物理定律，从而提高滚动预测的稳健性和准确性。对真实海面数据和浅水系统的数值实验清楚地证明了我们的框架在稀疏和时变观测下的准确性和计算效率。