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Learning Spatiotemporal Dynamical Systems from Point Process Observations
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Proposes a variational latent-ODE model that jointly learns spatiotemporal PDE dynamics and the random point-process observation times/locations from sparse sensor data, with an interpolated sparse time grid to speed up ODE solving.
Problem
Modeling spatiotemporal dynamical systems when observations arrive at random times and locations (point process observations), where existing neural dynamics models assume dense or fixed grids and existing point-process models ignore system dynamics.
Approach
Combines a transformer amortized encoder over observations, a neural ODE latent dynamics model, an implicit neural representation decoder, and a neural intensity for a non-homogeneous Poisson process; latent states are solved on a sparse auxiliary grid and linearly/nearest interpolated to dense observation times, yielding up to 9x faster latent evaluation and 4x faster training.
Key results
- Ours achieves Burgers' MAE 0.043 and log-lik 3.77, outperforming the best dynamics baseline CNN-ODE (MAE 0.076) and the best point-process baseline AutoSTPP on the joint task.
- Sparse-grid latent evaluation gives up to 9x faster latent state evaluation and up to 4x faster training than sequential evaluation.
- On Scalar Flow, AutoSTPP (log-lik 8.49) slightly beats the proposed model (8.41) in log-likelihood, while the proposed model achieves MAE 0.092 vs CNN-ODE 0.091.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use the proposed model instead of CNN-ODE | spatiotemporal forecasting · Burgers' | MAE: 0.076 ± 0.000 -> 0.043 ± 0.002 | Helped |
| use the proposed model instead of AutoSTPP | spatiotemporal point process modeling · Shallow Water | Log-lik. (per event): 3.37 ± 0.01 -> 3.62 ± 0.05 | Helped |
| use the proposed model instead of AutoSTPP | spatiotemporal point process modeling · Navier-Stokes | Log-lik. (per event): 5.91 ± 0.01 -> 5.96 ± 0.01 | Helped |
| use the proposed model instead of CNN-ODE | spatiotemporal forecasting · Scalar Flow | MAE: 0.091 ± 0.001 -> 0.092 ± 0.002 | Mixed |
| use the proposed model instead of AutoSTPP | spatiotemporal point process modeling · Scalar Flow | Log-lik. (per event): 8.49 ± 0.05 -> 8.41 ± 0.04 | Hurt |
| remove the point process model (ablation) | Ours · Burgers', Shallow Water, Navier-Stokes | MAE: 0.042 -> 0.043; 0.012 -> 0.012; 0.071 -> 0.071 | No clear effect |
| remove the point process model (ablation) | Ours · Scalar Flow | MAE: 0.090 -> 0.089 | Hurt |
| add the observation model (ablation) | Ours · Burgers', Shallow Water, Navier-Stokes, Scalar Flow | Log-lik.: 3.70 -> 3.77; 3.56 -> 3.67; 5.92 -> 5.96; 6.87 -> 8.43 | Helped |
| increase context size (ablation) | Ours · all datasets | MAE | Helped |
| increase nearest-neighbor interpolation grid resolution (ablation) | Ours · all datasets | MAE | Helped |
| increase linear interpolation grid resolution (ablation) | Ours · all datasets | MAE | No clear effect |
| increase latent space dimension (ablation) | Ours · all datasets | MAE | Helped |
Surprising findings
- With linear interpolation the model performs best with only n=2 latent time points, suggesting that complex latent trajectories are not needed.
- Removing the point process model has practically no effect on observation MAE, while removing the observation model reduces process likelihood.
Limitations
- Assumes non-overlapping observation times and a Poisson process, which may not hold for high-frequency sensors.
- Assumes no feedback between observation events and system dynamics.
- Requires sufficiently dense sensor placement to capture intensity in high-intensity regions.
Experimental setup
Datasets: Burgers' (PDEBench) Shallow Water (PDEBench) Navier-Stokes with transport (PhiFlow-simulated) Scalar Flow
Benchmarks: Burgers' Shallow Water Navier-Stokes with transport Scalar Flow
Metrics: mean absolute error event-averaged log-likelihood
Baselines: FEN NSPDE CNN-ODE Constant Intensity DSTPP NSTPP AutoSTPP Median predictor
Builds on: neural ODE latent ODE variational autoencoder neural point processes implicit neural representations
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