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Generative Modeling of Regular and Irregular Time Series Data via Koopman VAEs
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Proposes KoVAE, a VAE for regular and irregular time series generation whose sequential prior is a learned linear (Koopman/DMD-style) latent dynamical system with eigenvalue-based spectral constraints.
Problem
Generating realistic regular and irregularly sampled time series; GAN-based models are unstable and mode-collapse prone, while existing VAE generators use non-sequential Gaussian priors and make it hard to incorporate dynamical domain knowledge.
Approach
Replace the nonlinear sequential Gaussian prior of a sequential VAE with a linear latent dynamical prior: a learnable matrix A (DMD-like, solved in closed form) propagates latent codes, a predictive loss ties the prior to the posterior, eigenvalues can be penalized toward chosen moduli for physics-constrained generation, and an NCDE embedding handles irregular sampling.
Key results
- On regular data, KoVAE achieves the best discriminative scores on all four datasets (e.g., Stocks 0.009 vs GT-GAN 0.077), with average relative improvement of 58% over the second-best method.
- On irregular data with 30/50/70% observations dropped, KoVAE gives the best discriminative scores in most settings, with mean relative improvements of about 48%, 55%, and 43%.
- In the physics-constrained pendulum experiment, eigenvalue-constrained KoVAE matches the true distribution better than unconstrained KoVAE, with two eigenvalues near modulus 1.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use KoVAE instead of the strongest reported baseline | regular time-series generation · Sines, Stocks, Energy, and MuJoCo | discriminative score: 0.011 -> 0.005; 0.036 -> 0.009; 0.221 -> 0.143; 0.245 -> 0.076 | Helped |
| Use KoVAE instead of the strongest reported baseline | regular time-series generation · Sines, Stocks, Energy, and MuJoCo | MAE: 0.093 -> 0.093; 0.037 -> 0.037; 0.254 -> 0.251; 0.039 -> 0.038 | Mixed |
| Use KoVAE instead of GT-GAN | irregular time-series generation with 30% missing observations · Sines, Stocks, Energy, and MuJoCo | discriminative score: 0.363 -> 0.035; 0.251 -> 0.162; 0.333 -> 0.280; 0.249 -> 0.123 | Helped |
| Use KoVAE instead of GT-GAN | irregular time-series generation with 30% missing observations · Sines, Stocks, Energy, and MuJoCo | MAE: 0.099 -> 0.074; 0.021 -> 0.019; 0.066 -> 0.049; 0.048 -> 0.043 | Helped |
| Use KoVAE instead of GT-GAN | irregular time-series generation with 50% missing observations · Sines, Stocks, Energy, and MuJoCo | discriminative score: 0.372 -> 0.030; 0.265 -> 0.092; 0.317 -> 0.298; 0.270 -> 0.117 | Helped |
| Use KoVAE instead of GT-GAN | irregular time-series generation with 50% missing observations · Sines, Stocks, Energy, and MuJoCo | MAE: 0.101 -> 0.072; 0.018 -> 0.019; 0.064 -> 0.049; 0.056 -> 0.042 | Mixed |
| add the linear predictive prior (nonzero alpha) (ablation) | KoVAE · regular Sines, Stocks, Energy, and MuJoCo | discriminative score: 0.006 -> 0.005; 0.023 -> 0.009; 0.155 -> 0.143; 0.087 -> 0.076 | Helped |
| add the linear predictive prior (nonzero alpha) (ablation) | KoVAE · irregular 30% missing, Sines, Stocks, Energy, and MuJoCo | discriminative score: 0.038 -> 0.035; 0.172 -> 0.162; 0.295 -> 0.280; 0.131 -> 0.123 | Helped |
| add the linear predictive prior (nonzero alpha) (ablation) | KoVAE · irregular 50% missing, Sines, Stocks, Energy, and MuJoCo | discriminative score: 0.040 -> 0.030; 0.181 -> 0.092; 0.306 -> 0.298; 0.108 -> 0.117 | Mixed |
| retain the GRU when using alpha=0 (ablation) | KoVAE without the linear prior · irregular 30% missing, Sines, Stocks, Energy, and MuJoCo | discriminative score: 0.268 -> 0.038; 0.272 -> 0.172; 0.436 -> 0.295; 0.267 -> 0.131 | Helped |
| retain the GRU when using alpha=0 (ablation) | KoVAE without the linear prior · irregular 50% missing, Sines, Stocks, Energy, and MuJoCo | discriminative score: 0.275 -> 0.040; 0.225 -> 0.181; 0.444 -> 0.306; 0.305 -> 0.108 | Helped |
Surprising findings
- Unconstrained KoVAE learns a Koopman spectrum with all eigenvalues far below 1 on the pendulum, and on Energy the learned dynamics are unstable (largest eigenvalue much greater than 1).
- TimeVAE matches or beats KoVAE on some predictive metrics (e.g., Sines 0.093 tie, MuJoCo predictive worse than GT-GAN on some irregular settings), so KoVAE does not win everywhere.
Limitations
- Reported KoVAE gains are not uniform: KoVAE loses on predictive Stocks 50% and discriminative Energy 70%.
- The Energy dataset yields unstable learned linear dynamics, and the authors leave spectral constraints for generation improvement to future work.
- Weather conditional generation lacks fine-scale spatial detail due to absence of spatial constraints.
Experimental setup
Datasets: Sines Stocks Energy MuJoCo ERA5 reanalysis temperature simulated nonlinear pendulum
Benchmarks: Sines Stocks Energy MuJoCo
Metrics: discriminative score predictive score (MAE) t-SNE probability density function relative improvement
Baselines: TimeGAN GT-GAN TimeVAE CR-VAE RCGAN C-RNN-GAN T-Forcing P-Forcing WaveNet WaveGAN
Builds on: TimeGAN GT-GAN neural controlled differential equations dynamic mode decomposition variational autoencoder
Closest papers by meaning
- $K^2$VAE: A Koopman-Kalman Enhanced Variational AutoEncoder for Probabilistic Time Series Forecasting (ICML 2025 · similarity 0.90)
- Time Series Generation Under Data Scarcity: A Unified Generative Modeling Approach (NeurIPS 2025 · similarity 0.89)
- KoNODE: Koopman-Driven Neural Ordinary Differential Equations with Evolving Parameters for Time Series Analysis (ICML 2025 · similarity 0.87)
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- Bayesian Basis Function Approximation for Scalable Gaussian Process Priors in Deep Generative Models (ICML 2025 · similarity 0.87)
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