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GrINd: Grid Interpolation Network for Scattered Observations

Andrzej Dulny, Paul Heinisch, Andreas Hotho, Anna Krause

TL;DR

This work introduces GrINd (Grid Interpolation Network for Scattered Observations), a novel network architecture that leverages the high-performance of grid-based models by mapping scattered observations onto a high-resolution grid using a Fourier Interpolation Layer.

Abstract

Predicting the evolution of spatiotemporal physical systems from sparse and scattered observational data poses a significant challenge in various scientific domains. Traditional methods rely on dense grid-structured data, limiting their applicability in scenarios with sparse observations. To address this challenge, we introduce GrINd (Grid Interpolation Network for Scattered Observations), a novel network architecture that leverages the high-performance of grid-based models by mapping scattered observations onto a high-resolution grid using a Fourier Interpolation Layer. In the high-resolution space, a NeuralPDE-class model predicts the system's state at future timepoints using differentiable ODE solvers and fully convolutional neural networks parametrizing the system's dynamics. We empirically evaluate GrINd on the DynaBench benchmark dataset, comprising six different physical systems observed at scattered locations, demonstrating its state-of-the-art performance compared to existing models. GrINd offers a promising approach for forecasting physical systems from sparse, scattered observational data, extending the applicability of deep learning methods to real-world scenarios with limited data availability.

GrINd: Grid Interpolation Network for Scattered Observations

TL;DR

This work introduces GrINd (Grid Interpolation Network for Scattered Observations), a novel network architecture that leverages the high-performance of grid-based models by mapping scattered observations onto a high-resolution grid using a Fourier Interpolation Layer.

Abstract

Predicting the evolution of spatiotemporal physical systems from sparse and scattered observational data poses a significant challenge in various scientific domains. Traditional methods rely on dense grid-structured data, limiting their applicability in scenarios with sparse observations. To address this challenge, we introduce GrINd (Grid Interpolation Network for Scattered Observations), a novel network architecture that leverages the high-performance of grid-based models by mapping scattered observations onto a high-resolution grid using a Fourier Interpolation Layer. In the high-resolution space, a NeuralPDE-class model predicts the system's state at future timepoints using differentiable ODE solvers and fully convolutional neural networks parametrizing the system's dynamics. We empirically evaluate GrINd on the DynaBench benchmark dataset, comprising six different physical systems observed at scattered locations, demonstrating its state-of-the-art performance compared to existing models. GrINd offers a promising approach for forecasting physical systems from sparse, scattered observational data, extending the applicability of deep learning methods to real-world scenarios with limited data availability.
Paper Structure (16 sections, 1 theorem, 10 equations, 4 figures, 2 tables)

This paper contains 16 sections, 1 theorem, 10 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Method
  4. Fourier Interpolation Layer
  5. NeuralPDE
  6. GrINd
  7. Experiments
  8. Data
  9. Baseline Models
  10. Model Configuration
  11. Training
  12. Results and Discussion
  13. Interpolation accuracy
  14. DynaBench
  15. Conclusion and Future Work
  16. ...and 1 more sections

Key Result

theorem thmcountertheorem

Given a periodic complex function $u\colon \Omega \rightarrow \mathbb{C}$ on the interval box $\Omega = [0, 1]^M$, such that $\int_{\Omega}||u||^2d\mathbf{x} < \infty$, the following series: with converges pointwise to $u$ as $N_i\rightarrow\infty$, i.e.

Figures (4)

  • Figure 1: Summary of our approach. The low-resolution observations are first mapped onto a high-resolution grid using a Fourier Interpolation Layer. In this high resolution space a predictive model (NeuralPDE) forecasts the evolution of the system which is then mapped back to the original observation space.
  • Figure 2: Fourier Interpolation Layer. We use a Fourier series approximation of the original function $u$ by fitting the Fourier coefficients $c_{\mathbf{k}}$ using LLS. The Fourier coefficients can be used to evaluate the approximation at any arbitrary collection of points $\mathbf{x}$ thus making it possible to interpolate the function onto a high-resolution grid.
  • Figure 3: Results of the interpolation experiment. The plot shows the Mean Squared Error between the interpolation given by our Fourier Interpolation Layer and the full resolution dataset as a function of the number of fourier frequencies used. The lowest interpolation error for each dataset has been marked with a dot.
  • Figure 4: Rollout results on the DynaBench dataset for 16 steps. The graph and point cloud baselines are plotted with solid lines, the grid baselines with dotted lines, the persistence baseline with a dash-dotted line and our model with a dashed line. Results of our model have been averaged over 10 runs. Baseline results taken from Dulny2023. For better readability MSE scores over 2.0 are not displayed.

Theorems & Definitions (2)

  • theorem thmcountertheorem
  • proof