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

arXiv: https://arxiv.org/abs/2409.00458 · HTML: https://mlsanity.com/paper/2409.00458/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 83 references that are in the map.
- First posted on arXiv 31 Aug 2024.

## Builds on (most cited first, 23 of 23)

- [SegNet: A Deep Convolutional Encoder-Decoder Architecture for Image Segmentation](https://mlsanity.com/paper/1511.00561/) (2015)
- [Convolutional LSTM Network: A Machine Learning Approach for Precipitation Nowcasting](https://mlsanity.com/paper/1506.04214/) (2015)
- [Spatio-Temporal Graph Convolutional Networks: A Deep Learning Framework for Traffic Forecasting](https://mlsanity.com/paper/1709.04875/) (2017)
- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [Machine learning with data assimilation and uncertainty quantification for dynamical systems: a review](https://mlsanity.com/paper/2303.10462/) (2023)
- [Global field reconstruction from sparse sensors with Voronoi tessellation-assisted deep learning](https://mlsanity.com/paper/2101.00554/) (2021)
- [Deep convolutional recurrent autoencoders for learning low-dimensional feature dynamics of fluid systems](https://mlsanity.com/paper/1808.01346/) (2018)
- [Fully Convolutional Mesh Autoencoder using Efficient Spatially Varying Kernels](https://mlsanity.com/paper/2006.04325/) (2020)
- [On the Convergence of the SINDy Algorithm](https://mlsanity.com/paper/1805.06445/) (2018)
- [GNN-Surrogate: A Hierarchical and Adaptive Graph Neural Network for Parameter Space Exploration of Unstructured-Mesh Ocean Simulations](https://mlsanity.com/paper/2202.08956/) (2022)
- [DeepKriging: Spatially Dependent Deep Neural Networks for Spatial Prediction](https://mlsanity.com/paper/2007.11972/) (2020)
- [Learning Sparse Dynamical Systems from a Single Sample Trajectory](https://mlsanity.com/paper/1904.09396/) (2019)
- [Grasping Extreme Aerodynamics on a Low-Dimensional Manifold](https://mlsanity.com/paper/2305.08024/) (2023)
- [Efficient deep data assimilation with sparse observations and time-varying sensors](https://mlsanity.com/paper/2310.16187/) (2023)
- [Latent-space time evolution of non-intrusive reduced-order models using Gaussian process emulation](https://mlsanity.com/paper/2007.12167/) (2020)
- [HOPE: Hierarchical Spatial-temporal Network for Occupancy Flow Prediction](https://mlsanity.com/paper/2206.10118/) (2022)
- [Separable Convolutional LSTMs for Faster Video Segmentation](https://mlsanity.com/paper/1907.06876/) (2019)
- [A non-intrusive reduced order modeling framework for quasi-geostrophic turbulence](https://mlsanity.com/paper/1906.11617/) (2019)
- [Bounded nonlinear forecasts of partially observed geophysical systems with physics-constrained deep learning](https://mlsanity.com/paper/2202.05750/) (2022)
- [Multi-fidelity physics constrained neural networks for dynamical systems](https://mlsanity.com/paper/2402.02031/) (2024)
- [Adaptive Machine Learning for Time-Varying Systems: Low Dimensional Latent Space Tuning](https://mlsanity.com/paper/2107.06207/) (2021)
- [Dependent relevance determination for smooth and structured sparse regression](https://mlsanity.com/paper/1711.10058/) (2017)
- [On the uncertainty analysis of the data-enabled physics-informed neural network for solving neutron diffusion eigenvalue problem](https://mlsanity.com/paper/2303.08455/) (2023)

## Built on by (most cited first, 5 of 5)

- [Machine learning for modelling unstructured grid data in computational physics: a review](https://mlsanity.com/paper/2502.09346/) (2025)
- [Improving Long-term Autoregressive Spatiotemporal Predictions: A Proof of Concept with Fluid Dynamics](https://mlsanity.com/paper/2508.18565/) (2025)
- [Rooftop Wind Field Reconstruction Using Sparse Sensors: From Deterministic to Generative Learning Methods](https://mlsanity.com/paper/2603.13077/) (2026)
- [Flow Reconstruction from Sparse Measurements in Urban Drainage Networks: An Application and Evaluation of Data-Driven Sparse Sensing](https://mlsanity.com/paper/2511.04556/) (2025)
- [Learning to Reconstruct Temperature Field from Sparse Observations with Implicit Physics Priors](https://mlsanity.com/paper/2512.01196/) (2025)

## Abstract (arXiv)

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.