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Latent assimilation with implicit neural representations for unknown dynamics

Zhuoyuan Li, Bin Dong, Pingwen Zhang

TL;DR

The paper tackles data assimilation for systems with high dimensionality and partially unknown dynamics by introducing Latent Assimilation with Implicit Neural Representations (LAINR). It combines Spherical Implicit Neural Representations (SINR) for continuous, mesh-free encoding on the sphere with Neural ODE-based latent dynamics and data-driven uncertainty estimation, enabling robust, time-flexible assimilation. Empirical results on a spherical shallow-water model and ERA5 data show that LAINR outperforms AutoEncoder-based approaches in reconstruction, prediction, and assimilation, and it remains effective with unstructured or zero-shot observations. The framework thus offers a scalable, flexible tool for complex geophysical DA tasks with irregular sampling and evolving dynamics, improving practical applicability and reliability of data assimilation in real-world scenarios.

Abstract

Data assimilation is crucial in a wide range of applications, but it often faces challenges such as high computational costs due to data dimensionality and incomplete understanding of underlying mechanisms. To address these challenges, this study presents a novel assimilation framework, termed Latent Assimilation with Implicit Neural Representations (LAINR). By introducing Spherical Implicit Neural Representations (SINR) along with a data-driven uncertainty estimator of the trained neural networks, LAINR enhances efficiency in assimilation process. Experimental results indicate that LAINR holds certain advantage over existing methods based on AutoEncoders, both in terms of accuracy and efficiency.

Latent assimilation with implicit neural representations for unknown dynamics

TL;DR

The paper tackles data assimilation for systems with high dimensionality and partially unknown dynamics by introducing Latent Assimilation with Implicit Neural Representations (LAINR). It combines Spherical Implicit Neural Representations (SINR) for continuous, mesh-free encoding on the sphere with Neural ODE-based latent dynamics and data-driven uncertainty estimation, enabling robust, time-flexible assimilation. Empirical results on a spherical shallow-water model and ERA5 data show that LAINR outperforms AutoEncoder-based approaches in reconstruction, prediction, and assimilation, and it remains effective with unstructured or zero-shot observations. The framework thus offers a scalable, flexible tool for complex geophysical DA tasks with irregular sampling and evolving dynamics, improving practical applicability and reliability of data assimilation in real-world scenarios.

Abstract

Data assimilation is crucial in a wide range of applications, but it often faces challenges such as high computational costs due to data dimensionality and incomplete understanding of underlying mechanisms. To address these challenges, this study presents a novel assimilation framework, termed Latent Assimilation with Implicit Neural Representations (LAINR). By introducing Spherical Implicit Neural Representations (SINR) along with a data-driven uncertainty estimator of the trained neural networks, LAINR enhances efficiency in assimilation process. Experimental results indicate that LAINR holds certain advantage over existing methods based on AutoEncoders, both in terms of accuracy and efficiency.
Paper Structure (63 sections, 7 theorems, 86 equations, 15 figures, 6 tables, 2 algorithms)

This paper contains 63 sections, 7 theorems, 86 equations, 15 figures, 6 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related works and contributions
  3. Reduced-order models
  4. Latent assimilation
  5. Implicit neural representations
  6. Latent assimilation
  7. Embeddings from physical spaces to latent spaces
  8. Classical linear ROMs
  9. AutoEncoders
  10. Evolution of latent dynamics
  11. Assimilation in latent spaces
  12. LAINR framework
  13. Spherical implicit neural representations
  14. Implicit neural representations
  15. Spherical filters
  16. ...and 48 more sections

Key Result

Theorem 4.1

Each coordinate of the output of eq:MFN-updates is given by a finite linear combination of sinusoidal bases for some coefficients $\bar{\alpha}_t^{(j)}$, $\bar{\bm\omega}_t^{(j)}$, $\bar{\bm\phi}_t^{(j)}$ and $\bar{\beta}^{(j)}$ only dependent on the network parameters. fathony2021multiplicative

Figures (15)

  • Figure 1: The pipeline of the LAINR framework.
  • Figure 2: Illustrations of classical DA and LA frameworks with colored forward operators (red) and observation operators (blue). The temporal indices here are omitted for simplicity.
  • Figure 3: Illustration of the inclusion relationships projected into the spectral modes $(m,\ell)$. Note that the range for $\mathcal{S}_{\mathrm{SINR}}$ is only for reference.
  • Figure 4: Visualization of the SINR architecture with modulation adjustments and the spherical filters.
  • Figure 5: Grids for training and assimilation
  • ...and 10 more figures

Theorems & Definitions (17)

  • Theorem 4.1
  • Definition 4.2: spherical filters
  • Remark
  • Lemma 4.3
  • proof
  • Definition 4.4: Spherical Implicit Neural Representations (SINRs)
  • Proposition 4.5: expansion of SINRs
  • proof
  • Proposition 4.6: representability of SINRs
  • proof
  • ...and 7 more