Learning Spatiotemporal Dynamical Systems from Point Process Observations

TL;DR

This work tackles learning spatiotemporal dynamics governed by PDEs from randomly timed and located observations by formulating a generative model that couples a low-dimensional latent ODE/PDE dynamical system with a neural intensity for point-process observations and an implicit neural representation for the spatiotemporal field. It uses amortized variational inference with a transformer-based encoder to infer the latent initial state, while a sparse evaluation grid and interpolation dramatically speed up latent trajectory simulation. The approach unifies latent dynamics, observation times/locations, and observation values into a single trainable framework and demonstrates strong predictive performance and computational efficiency on Burgers', Shallow Water, Navier–Stokes, and Scalar Flow datasets, outperforming both dense-grid spatiotemporal models and neural point-process baselines. These results highlight the practical potential of learning from unconstrained measurements in real-world sensor networks, with immediate implications for fields ranging from environmental monitoring to crowdsourced sensing.

Abstract

Spatiotemporal dynamics models are fundamental for various domains, from heat propagation in materials to oceanic and atmospheric flows. However, currently available neural network-based spatiotemporal modeling approaches fall short when faced with data that is collected randomly over time and space, as is often the case with sensor networks in real-world applications like crowdsourced earthquake detection or pollution monitoring. In response, we developed a new method that can effectively learn spatiotemporal dynamics from such point process observations. Our model integrates techniques from neural differential equations, neural point processes, implicit neural representations and amortized variational inference to model both the dynamics of the system and the probabilistic locations and timings of observations. It outperforms existing methods on challenging spatiotemporal datasets by offering substantial improvements in predictive accuracy and computational efficiency, making it a useful tool for modeling and understanding complex dynamical systems observed under realistic, unconstrained conditions.

Figures (10)

• Figure 1: Top: Previous models assume dense observations at every time point and fixed spatiotemporal grids. Bottom: Our model works with extremely sparse observations and predicts where the next observations will happen.
• Figure 2: Model diagram. Black crosses on the time axis are observation times, while white crosses on field images are the corresponding observation locations. The initial observations (context) are mapped by the encoder to the latent initial state distribution $q({\bm{z}}_1)$. The initial state ${\bm{z}}_1$ is then sampled from that distribution and evolved through time using the dynamics model. The latent trajectory is evaluated only at the sparse time grid $\tau_1,\dots,\tau_n$ and the latent state $\Tilde{{\bm{z}}}(t)$ at other time points is evaluated via interpolation. The latent state $\Tilde{{\bm{z}}}(t)$ is then mapped by $\phi$ to the spatiotemporal state ${\bm{u}}(t, {\bm{x}})$, which is used to parameterize the point process and observation distribution (via mappings $\lambda$ and $g$, respectively) for predicting subsequent observation times and locations.
• Figure 3: Examples of trajectories from our datasets. White crosses indicate the observation locations ${\bm{x}}$, marks on the horizontal time grid indicate the corresponding observation times $t$, and color-coded fields denote the system state with observations ${\bm{y}}$ at white crosses $(t,{\bm{x}})$. Row 1: Burger's (1D) dataset. The full trajectory is shown, with 1-D measurement location ${\bm{x}}$ along the vertical and time $t$ along the horizontal directions. Rows 2, 3 and 4: Shallow Water, Navier-Stokes, and Scalar Flow datasets, respectively. Due to the 2D nature of the systems, only snapshots of the system state are plotted.
• Figure 4: Test MAE ($\downarrow$) and log-likelihood ($\uparrow$) vs. context size. For Scalar Flow we use the right axis for better visibility.
• Figure 5: Test MAE ($\downarrow$) and process log-likelihood ($\uparrow$) for different temporal grid resolutions ($n$) and interpolation methods (nn = nearest neighbor, lin = linear). For Scalar Flow we use the right axis for better visibility.