Variational Augmented Invertible Koopman Autoencoder for probabilistic time series forecasting

Anthony Frion; Lucas Drumetz; Guillaume Tochon; Mauro Dalla Mura; Ali Can Bekar; Abdeldjalil Aïssa El Bey

cs.LGSep 29, 2026

Variational Augmented Invertible Koopman Autoencoder for probabilistic time series forecasting

Authors: Anthony Frion, Lucas Drumetz, Guillaume Tochon, Mauro Dalla Mura, Ali Can Bekar, Abdeldjalil Aïssa El Bey

Organizations: Institute of coastal modelling Helmholtz-Zentrum Hereon · IMT Atlantique Lab-STICC, UMR CNRS 6285, Brest, France · LRE EPITA, Le Kremlin-Bicêtre, France · Université Grenobles Alpes Grenoble INP GIPSA-lab, Grenoble, France

Abstract

Neural Koopman autoencoder models have been shown to successfully build a latent embedding with linear dynamics for arbitrary dynamical systems, enabling strong performance in long-term time series forecasting. However, these models usually work in a deterministic setting, which does not allow the quantification of the uncertainty of their predictions. Thus, we propose the new Variational Augmented Invertible Koopman AutoEncoder (VAIKAE), in which the latent embedding follows a Gaussian distribution instead of being deterministic. A key property of the VAIKAE architecture is that it leverages normalizing flow models, enabling the use of likelihood computations in the state space of dynamical systems for training a model. We further propose new strategies for uncertainty-aware latent data assimilation with a trained VAIKAE model. The effectiveness of our methods is demonstrated in a series of experiments on long-term time series forecasting benchmarks.

Figures & tables

Figure 1: Graphical representation of the VAIKAE architecture. The two dashed lines represent sampling from a diagonal Gaussian distribution, while the full lines represent deterministic operations.

$\mathbf{Model}$		$\mathbf{VAIKAE}$	$\mathbf{D^{3}U}$	$\mathbf{TMDM}$	$\mathbf{TimeDiff}$	$\mathbf{CSDI}$	$\mathbf{TimeGrad}$
ETTm1	MSE	0.370	0.363	0.607	0.796	0.867	1.716
	MAE	0.385	0.386	0.558	0.577	0.690	1.057
	CRPS	0.299	0.285	0.429	0.454	0.773	0.665
ETTm2	MSE	0.253	0.241	0.524	0.284	1.291	1.385
	MAE	0.319	0.302	0.493	0.342	0.576	0.732
	CRPS	0.247	0.243	0.380	0.316	0.625	0.785

Table 1: Summary of probabilistic long-term time series forecasting results. For each dataset and metric, the best result is in bold and the second best result is underlined .

Figure 2: Representative examples from the test sets of ETTm1 (top left), ECL (top right), Weather (bottom left) and Solar (bottom right). For each example, we show the 96-step input context, the subsequent 192-step groundtruth and the median VAIKAE prediction with corresponding

50\%

and

90\%

confidence intervals.

	CRPS on the Fontainebleau area	CRPS on the Orléans area
VAIKAE	$\mathbf{0.0241}$	$\mathbf{0.0473}$
VIKAE	$0.0304$	$0.0567$
VKAE	$0.0276$	$0.0540$
KAE ensemble	$0.0258$	$0.0530$

Table 2: Forecasting CRPS for different methods and areas

Figure 3: Probabilistic performance of three methods, the first of which leverages only the initial observation while the two others assimilate all the blue-colored datapoints. We show only the B7 band, from the infrared domain, which is the most energetic in the dataset, yet we remind that our methods jointly manipulate the 10 spectral bands of pixelwise reflectance vectors. The dark shadings represent 50% confidence intervals while the light shadings represent 90% confidence intervals.

Figure 4: Summary of 4DVar-CRPS extrapolation results on the Orléans area. From left to right: 1) RGB composition of a true image 2) Mean squared error on the B7 spectral band, averaged over the extrapolation time range 3) Corresponding variance 4) Corresponding spread-skill ratio. Since the ideal value is 1, white marks a good calibration while blue and red respectively indicate overconfidence and underconfidence.

	CRPS on the Fontainebleau area	CRPS on the Orléans area
single-obs	$0.0241$	$0.0457$
4DVar	${0.0153}$	${0.0273}$
4DVar-CRPS	$\mathbf{0.0140}$	$\mathbf{0.0244}$

Table 3: Forecasting CRPS obtained when predicting from a single observation or assimilating on multiple observations with two different latent data assimilation methods.

Figure 5: 4DVar-CRPS results on four randomly selected pixels from the test Orléans area. We consider the B7 spectral band, which is the most energetic in the dataset, although our method jointly processes the 10 spectral bands of the pixels. For each subfigure, the dark and light shading respectively correspond to the 50% and 90% confidence intervals of the predicted probability distributions.

Figure 6: Spread-skill plots for time series extrapolation with 4DVar-CRPS on the Fontainebleau (train) and Orléans (test) areas.

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 7: Visual representation of the AIKAE architecture, copied with permission from Frion et al. (2025) .

Figure 8: Left: an image from the forest of Fontainebleau. Right: an image from the forest of Orléans. The date for both images is 20/06/2018. Those are RGB compositions with saturated colors. The red square is the $300\times 300$ pixel training area and the blue square is the $300\times 300$ pixel test area.

Figure 9: Each subfigure corresponds to a randomly selected pixel from the Orléans area, for which we assimilate the blue-colored observations and sample 100 state trajectories from the inferred distribution.

Probabilistic Time Series Forecasting

Probabilistic time series forecasting estimates distributions of future values or trajectories to quantify uncertainty and support risk-aware decisions.

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Probabilistic Time Series Forecasting Uncertainty Quantification Uncertainty Calibration Variational Autoencoders

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Real-world time series exhibit temporally structured uncertainty: volatility clusters in turbulent regimes, dissipates in stable periods, and shifts abruptly around structural breaks. Yet many probabilistic forecasting methods estimate predictive uncertainty as an independent per-step quantity, leaving the evolution and persistence of volatility regimes under-modeled. We formalize this missing dimension as temporal uncertainty dynamics and instantiate it in the Volatility Dynamics Variational Autoencoder (VolDy-VAE), a non-autoregressive generative forecaster with a location-scale decoder. VolDy-VAE combines a location path for mean prediction with a recurrent scale path that transfers and evolves a volatility hidden state from the look-back window to the forecasting horizon, enabling temporally coherent predictive variances. This design yields an adaptive attenuation mechanism: high-variance observations receive lower influence on the location estimate while their uncertainty is preserved through explicit scale predictions. We further provide a simplified regime-switching analysis showing that, when variances are known or consistently estimated, the volatility-aware objective reduces to inverse-variance weighting, whereas MSE-based estimators remain unbiased but statistically inefficient. Experiments on nine benchmarks show that VolDy-VAE improves forecasting accuracy and uncertainty calibration over competitive probabilistic and point-forecasting baselines while maintaining low inference latency; plug-in studies further indicate that the VolDy principle can benefit GAN, Koopman VAE, and Transformer backbones. The source code is publicly available at https://github.com/wangyijunlyy/VolDy-VAE.

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Probabilistic Time Series Forecasting

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