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From Sparse Sensors to Continuous Fields: STRIDE for Spatiotemporal Reconstruction

Yanjie Tong, Peng Chen

TL;DR

This work tackles reconstructing high-dimensional spatiotemporal fields from sparse sensors in parametric PDE regimes. It introduces STRIDE, a two-stage approach that first encodes a short sensor history into a latent state ${oldsymbol{z}}_t$ with a temporal encoder and then decodes a continuous field using a modulated implicit neural representation (INR) backbone (FMMNN) conditioned on ${oldsymbol{z}}_t$, enabling queries at arbitrary locations with resolution-invariant outputs. A conditional theory based on stable delay observability and Mañé projection argues that the reconstruction operator factors through a finite-dimensional embedding, justifying STRIDE-type architectures. Empirically, STRIDE-FMMNN achieves lower reconstruction errors than baselines across four challenging benchmarks (Kuramoto–Sivashinsky, FlowAO, SWE, Seismic), remains robust to noise, and supports super-resolution, albeit with higher computational cost due to INR querying.

Abstract

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.

From Sparse Sensors to Continuous Fields: STRIDE for Spatiotemporal Reconstruction

TL;DR

This work tackles reconstructing high-dimensional spatiotemporal fields from sparse sensors in parametric PDE regimes. It introduces STRIDE, a two-stage approach that first encodes a short sensor history into a latent state with a temporal encoder and then decodes a continuous field using a modulated implicit neural representation (INR) backbone (FMMNN) conditioned on , enabling queries at arbitrary locations with resolution-invariant outputs. A conditional theory based on stable delay observability and Mañé projection argues that the reconstruction operator factors through a finite-dimensional embedding, justifying STRIDE-type architectures. Empirically, STRIDE-FMMNN achieves lower reconstruction errors than baselines across four challenging benchmarks (Kuramoto–Sivashinsky, FlowAO, SWE, Seismic), remains robust to noise, and supports super-resolution, albeit with higher computational cost due to INR querying.

Abstract

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.
Paper Structure (25 sections, 2 theorems, 29 equations, 17 figures, 9 tables)

This paper contains 25 sections, 2 theorems, 29 equations, 17 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Method: STRIDE
  3. Setting and Observation Model
  4. Two-Stage Architecture
  5. Architectural Choices
  6. Training and Normalization
  7. Theoretical Justification
  8. Setting
  9. Point-sensor observation and delay observability
  10. A continuous reconstruction operator
  11. Approximation by a STRIDE-type architecture
  12. Experiments
  13. Experimental Setup
  14. Kuramoto--Sivashinsky Equation.
  15. Flow Around an Obstacle.
  16. ...and 10 more sections

Key Result

Lemma 3.3

If the map $({\boldsymbol{\mu}},{\boldsymbol{x}})\mapsto F_{\boldsymbol{\mu}}^k{\boldsymbol{x}}$ is continuous on $\mathcal{A}$ and point evaluation ${\boldsymbol{x}}\mapsto {\boldsymbol{x}}({\boldsymbol{\xi}})$ is continuous on $X$, then $T$ is continuous on the compact set $Y$. Moreover, if $({\bo

Figures (17)

  • Figure 1: Overview of STRIDE. A temporal encoder maps a window of point-sensor observations ${\boldsymbol{y}}_{t-k:t}$ to a latent state ${\boldsymbol{z}}_t$. A conditional spatial decoder, e.g., modulated INR decoder (FMMNN with shift modulation), then evaluates $\hat{{\boldsymbol{x}}}({\boldsymbol{\xi}},t)$ at arbitrary query locations ${\boldsymbol{\xi}}$ (optionally Fourier-encoded), enabling discretization- and resolution-invariant reconstruction across parameterized trajectories.
  • Figure 2: Divergent trajectories in the chaotic KS system. The spatiotemporal fields are initialized with nearly identical frequency parameters ($2.99, 3.00, \text{ and } 3.01$), illustrating the system's sensitive dependence on initial conditions.
  • Figure 3: Example trajectory of the velocity field in the FlowAO case, showing unsteady flow patterns influenced by time-varying angles of attack and inflow intensities.
  • Figure 4: Example trajectory of the surface elevation in the SWE case, demonstrating the propagation of a tsunami-like wave generated from a Gaussian source.
  • Figure 5: Sample of velocity map.
  • ...and 12 more figures

Theorems & Definitions (5)

  • Lemma 3.3: Continuity and stability of $T$
  • Theorem 3.4: Operator factorization and approximation
  • proof
  • proof
  • Remark A.1: Approximate embedding and an error floor