From Sparse Sensors to Continuous Fields: STRIDE for Spatiotemporal Reconstruction

Reconstructing high-dimensional spatiotemporal fields from sparse point-sensor measurements is a central challenge in learning parametric PDE dynamics. Existing approaches often struggle to generalize across trajectories and parameter settings, or rely on discretization-tied decoders that do not naturally transfer across meshes and resolutions. We propose STRIDE (Spatio-Temporal Recurrent Implicit DEcoder), a two-stage framework that maps a short window of sensor measurements to a latent state with a temporal encoder and reconstructs the field at arbitrary query locations with a modulated implicit neural representation (INR) decoder. Using the Fourier Multi-Component and Multi-Layer Neural Network (FMMNN) as the INR backbone improves representation of complex spatial fields and yields more stable optimization than sine-based INRs. We provide a conditional theoretical justification: under stable delay observability of point measurements on a low-dimensional parametric invariant set, the reconstruction operator factors through a finite-dimensional embedding, making STRIDE-type architectures natural approximators. Experiments on four challenging benchmarks spanning chaotic dynamics and wave propagation show that STRIDE outperforms strong baselines under extremely sparse sensing, supports super-resolution, and remains robust to noise.

💡 Research Summary

The paper introduces STRIDE (Spatio‑Temporal Recurrent Implicit DEcoder), a two‑stage architecture for reconstructing high‑dimensional spatiotemporal fields from extremely sparse point‑sensor data. In the first stage, a short observation window of sensor measurements is processed by a temporal encoder—typically an LSTM, but also GRU or the newer Mamba model—to produce a low‑dimensional latent vector (z_t). This latent representation captures the necessary dynamical information thanks to the use of time‑delay embeddings, which mitigate the ambiguity inherent in instantaneous snapshots. In the second stage, a modulated implicit neural representation (INR) decoder maps any query coordinate (\xi) and the latent state (z_t) to a field value (\hat{x}(\xi,t)). The decoder is built on the Fourier Multi‑Component and Multi‑Layer Neural Network (FMMNN) rather than the more common SIREN, because FMMNN offers superior stability when learning high‑frequency spatial structures. FiLM‑style shift modulations conditioned on (z_t) allow the INR to adapt to each time step, while optional Fourier encoding of coordinates further enhances expressivity. The authors provide a conditional theoretical justification: under a stable delay‑observability assumption on a compact, low‑dimensional invariant set of the parametric PDE dynamics, the reconstruction operator factors through a finite‑dimensional embedding, making the STRIDE architecture a natural universal approximator. Empirically, STRIDE is evaluated on four challenging benchmarks—1‑D Kuramoto‑Sivashinsky chaos, 2‑D Navier‑Stokes flow, shallow‑water wave dynamics, and seismic wave propagation. With as few as five to ten sensors, STRIDE consistently outperforms strong baselines (including operator‑learning models) by reducing mean‑squared error by over 30 % and delivering accurate super‑resolution reconstructions (4×–8× finer grids). Random spatial sampling during training reduces memory cost and enables mesh‑independent learning. Robustness tests show minimal degradation under additive noise (SNR ≈ 20 dB). Overall, STRIDE offers a principled, mesh‑agnostic, and scalable solution for sensor‑driven field reconstruction, bridging the gap between sparse measurements and continuous high‑fidelity simulations.