Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Neural Koopman prior for data assimilation

Anthony Frion, Lucas Drumetz, Mauro Dalla Mura, Guillaume Tochon, Abdeldjalil Aïssa El Bey

TL;DR

This work tackles data-driven modeling of sequential dynamics under irregular sampling by learning a neural Koopman prior that encodes dynamics as a fixed linear operator in a learned latent subspace. The approach combines an encoder/decoder with a latent matrix $\mathbf{K}$ (optionally coupled with a continuous operator $\mathbf{L}$) and a suite of losses that enforce long-horizon predictive fidelity, autoencoding accuracy, latent linearity, and optional orthogonality to promote stable, periodic-like behavior. It demonstrates that the resulting differentiable dynamical prior can be effectively used for self-supervised learning, long-term forecasting, and variational data assimilation for interpolation and denoising, with experiments on synthetic 3D fluid dynamics data and real Sentinel-2 time series. The findings indicate improved continuous representations from irregular data and strong utility as a data-driven prior in assimilation pipelines, with potential for transfer learning and richer spatial priors in remote sensing applications.

Abstract

With the increasing availability of large scale datasets, computational power and tools like automatic differentiation and expressive neural network architectures, sequential data are now often treated in a data-driven way, with a dynamical model trained from the observation data. While neural networks are often seen as uninterpretable black-box architectures, they can still benefit from physical priors on the data and from mathematical knowledge. In this paper, we use a neural network architecture which leverages the long-known Koopman operator theory to embed dynamical systems in latent spaces where their dynamics can be described linearly, enabling a number of appealing features. We introduce methods that enable to train such a model for long-term continuous reconstruction, even in difficult contexts where the data comes in irregularly-sampled time series. The potential for self-supervised learning is also demonstrated, as we show the promising use of trained dynamical models as priors for variational data assimilation techniques, with applications to e.g. time series interpolation and forecasting.

Neural Koopman prior for data assimilation

TL;DR

This work tackles data-driven modeling of sequential dynamics under irregular sampling by learning a neural Koopman prior that encodes dynamics as a fixed linear operator in a learned latent subspace. The approach combines an encoder/decoder with a latent matrix (optionally coupled with a continuous operator ) and a suite of losses that enforce long-horizon predictive fidelity, autoencoding accuracy, latent linearity, and optional orthogonality to promote stable, periodic-like behavior. It demonstrates that the resulting differentiable dynamical prior can be effectively used for self-supervised learning, long-term forecasting, and variational data assimilation for interpolation and denoising, with experiments on synthetic 3D fluid dynamics data and real Sentinel-2 time series. The findings indicate improved continuous representations from irregular data and strong utility as a data-driven prior in assimilation pipelines, with potential for transfer learning and richer spatial priors in remote sensing applications.

Abstract

With the increasing availability of large scale datasets, computational power and tools like automatic differentiation and expressive neural network architectures, sequential data are now often treated in a data-driven way, with a dynamical model trained from the observation data. While neural networks are often seen as uninterpretable black-box architectures, they can still benefit from physical priors on the data and from mathematical knowledge. In this paper, we use a neural network architecture which leverages the long-known Koopman operator theory to embed dynamical systems in latent spaces where their dynamics can be described linearly, enabling a number of appealing features. We introduce methods that enable to train such a model for long-term continuous reconstruction, even in difficult contexts where the data comes in irregularly-sampled time series. The potential for self-supervised learning is also demonstrated, as we show the promising use of trained dynamical models as priors for variational data assimilation techniques, with applications to e.g. time series interpolation and forecasting.
Paper Structure (20 sections, 1 theorem, 38 equations, 9 figures, 6 tables)

This paper contains 20 sections, 1 theorem, 38 equations, 9 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Background and related works
  3. Koopman operator theory
  4. Orthogonality regularisation
  5. Variational data assimilation
  6. Proposed methods
  7. Neural network design and training
  8. Handling irregular time series
  9. Variational data assimilation using our trained model
  10. Experiments on simulated data
  11. Interpolation from low-frequency regular data
  12. Learning to predict forward and backward
  13. Experiments on real Sentinel-2 time series
  14. Forecasting
  15. Forecasting with data assimilation
  16. ...and 5 more sections

Key Result

Theorem 1

Let $\mathbf{K} \in \mathcal{SO}(d)$, the special orthogonal group of real invertible matrices satisfying $\mathbf{KK}^T = \mathbf{K}^T\mathbf{K} = \mathbf{I}$ and with determinant equal to $+1$, and define a discrete-time dynamical system by with any initial condition $\mathbf{z}_0 \in \mathbb{R}^d$. Then there exists a continuous-time dynamical system with $\mathbf{z}(0) = \mathbf{z}_0$, and

Figures (9)

  • Figure 1: Visual representation of constrained variational data assimilation. It consists in choosing the initial condition from which the model's trajectory minimises the distance to the sampled data. One could also include a prior in the variational cost on the initial condition, such as the trajectory smoothness.
  • Figure 2: Schematic view of our architecture.
  • Figure 3: Upsampling experiment on fluid flow data. We learn a model on low-frequency data and then use the continuous representation to make a high-frequency prediction which we compare to the groundtruth on test trajectories. Top: the three tested models compared to a groundtruth trajectory. Bottom: corresponding mean squared errors over time, in a logarithmic scale.
  • Figure 4: Backward reconstructions of a test time series from a model trained with an orthogonality loss term (orthogonal) and a model trained without it (unregularised). Both models were trained on high-frequency forecasting.
  • Figure 5: Left: a temporally interpolated Fontainebleau image. Right: a non- interpolated Orléans image. The date for both images is 20/06/2018. Those are RGB compositions with saturated colors. The red square is the $150 \times 150$ pixel training area and the blue squares are test areas.
  • ...and 4 more figures

Theorems & Definitions (1)

  • Theorem 1: Discrete linear systems with special orthogonal matrices lead to periodic dynamics