Dynamical system prediction from sparse observations using deep neural networks with Voronoi tessellation and physics constraint

TL;DR

By integrating Voronoi tessellations with spatio-temporal deep learning models, DSOVT is adept at predicting dynamical systems with unstructured, sparse, and time-varying observations and incorporates physics constraints during the training process for dynamical systems with explicit formulas.

Abstract

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.

Figures (20)

• Figure 1: Schematic representation of physics-constrained CED-LSTM model employing Voronoi tessellation for enhanced state field mapping from sparse observations.
• Figure 2: Schematic representation of physics-constrained ConvLSTM model employing Voronoi tessellation to capture spatial dependencies and predict future state fields in dynamical systems. The process starts at time $t$, using $S_{in}$ and $S_{out}$ as the lengths of the input and output sequences, respectively.
• Figure 3: Illustration of the LSTM process for extracted NOAA SST latent space prediction. This figure provides a visual comprehension of the LSTM framework's operational mechanism in processing and predicting temporal sequences within the oceanographic context.
• Figure 4: Reconstructed Fields from Voronoi tessellation, including the actual fluid dynamics visualization. The figure is organized into four rows depicting: (1) NOAA state fields, (2) CED reconstructed fields from Voronoi tessellations, (3) Error maps showing discrepancies between state and predicted fields, and (4) Voronoi tessellations derived from 200 time-varying sensor data. Each column represents a different step.
• Figure 5: Comparison of NOAA SST state fields with LSTM and CED-based predictions across different steps. Each column represents a progression of steps: Step 10, Step 50, Step 90, and Step 130. The arrangement within each column follows this order: the first row displays the NOAA SST state field, the second row shows LSTM predictions, the third row depicts the error maps comparing the LSTM predictions to the state field, and the fourth row presents the CED-based reconstructions.