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Spatiotemporal Observer Design for Predictive Learning of High-Dimensional Data

Tongyi Liang, Han-Xiong Li

TL;DR

This work tackles high-dimensional spatiotemporal forecasting by introducing a Spatiotemporal Observer that merges Kazantzis-Kravaris-Luenberger observer theory with deep learning. The architecture uses a spatial encoder to form latent states, a spatiotemporal observer to predict future latent representations via a linear transformation in transformed space, and a spatial decoder to reconstruct observations, all trained with a dynamical-regularized loss. The authors provide generalization bounds and a convergence theorem, and they instantiate the framework with CNN-based components, including Inception modules, to demonstrate strong one-step and multi-step forecasting across traffic, synthetic, and radar datasets. The combination of theory-guided design and neural approximations yields improved predictive accuracy and reliable convergence, offering a principled approach to spatiotemporal learning in high dimensions.

Abstract

Although deep learning-based methods have shown great success in spatiotemporal predictive learning, the framework of those models is designed mainly by intuition. How to make spatiotemporal forecasting with theoretical guarantees is still a challenging issue. In this work, we tackle this problem by applying domain knowledge from the dynamical system to the framework design of deep learning models. An observer theory-guided deep learning architecture, called Spatiotemporal Observer, is designed for predictive learning of high dimensional data. The characteristics of the proposed framework are twofold: firstly, it provides the generalization error bound and convergence guarantee for spatiotemporal prediction; secondly, dynamical regularization is introduced to enable the model to learn system dynamics better during training. Further experimental results show that this framework could capture the spatiotemporal dynamics and make accurate predictions in both one-step-ahead and multi-step-ahead forecasting scenarios.

Spatiotemporal Observer Design for Predictive Learning of High-Dimensional Data

TL;DR

This work tackles high-dimensional spatiotemporal forecasting by introducing a Spatiotemporal Observer that merges Kazantzis-Kravaris-Luenberger observer theory with deep learning. The architecture uses a spatial encoder to form latent states, a spatiotemporal observer to predict future latent representations via a linear transformation in transformed space, and a spatial decoder to reconstruct observations, all trained with a dynamical-regularized loss. The authors provide generalization bounds and a convergence theorem, and they instantiate the framework with CNN-based components, including Inception modules, to demonstrate strong one-step and multi-step forecasting across traffic, synthetic, and radar datasets. The combination of theory-guided design and neural approximations yields improved predictive accuracy and reliable convergence, offering a principled approach to spatiotemporal learning in high dimensions.

Abstract

Although deep learning-based methods have shown great success in spatiotemporal predictive learning, the framework of those models is designed mainly by intuition. How to make spatiotemporal forecasting with theoretical guarantees is still a challenging issue. In this work, we tackle this problem by applying domain knowledge from the dynamical system to the framework design of deep learning models. An observer theory-guided deep learning architecture, called Spatiotemporal Observer, is designed for predictive learning of high dimensional data. The characteristics of the proposed framework are twofold: firstly, it provides the generalization error bound and convergence guarantee for spatiotemporal prediction; secondly, dynamical regularization is introduced to enable the model to learn system dynamics better during training. Further experimental results show that this framework could capture the spatiotemporal dynamics and make accurate predictions in both one-step-ahead and multi-step-ahead forecasting scenarios.
Paper Structure (26 sections, 2 theorems, 52 equations, 9 figures, 7 tables, 2 algorithms)

This paper contains 26 sections, 2 theorems, 52 equations, 9 figures, 7 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries and problem statement
  4. Spatiotemporal Predictive Learning
  5. Kazantzis-Kravaris-Luenberger Observers
  6. Problem Statement
  7. Spatiotemporal Observer Design for Learning
  8. Theoretical Design
  9. Definition of Spatiotemporal Observer
  10. One-step-ahead Forecasting
  11. Recursive Multi-step-ahead Forecasting
  12. Neural Configurations for Learning
  13. Function Approximation via CNNs
  14. Learning with Dynamical Regularization
  15. Generalization and Convergence Analysis
  16. ...and 11 more sections

Key Result

Theorem 1

Given $n$ training samples $S=((x_1, y_1), ...,(x_n, y_n))$ drawn i.i.d. according to distribution $\mathcal{D}$. A hypothesis $h\in\mathcal{H}$ is defined by the spatiotemporal observer. Let $X=(x_1,...,x_n)$ be the input and loss function $\mathscr{L}_\eta$ be $\eta$-Lipschitz and upper bounded by where $\mathscr{R}$ is defined in Eq eq_rademancher.

Figures (9)

  • Figure 1: Spatiotemporal nature of forecasting high-dimensional data.
  • Figure 2: Conceptual framework of spatiotemporal observer for forecasting.
  • Figure 3: Details of Spatial Encoder, Spatial Decoder and Inception.
  • Figure 4: Visualization of one-step-ahead traffic flow forecasting on TaxiBJ.
  • Figure 5: Visualization of prediction examples on Moving MNIST.
  • ...and 4 more figures

Theorems & Definitions (5)

  • Definition 1: Spatiotemporal Observer
  • Theorem 1
  • proof
  • Theorem 2
  • proof