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$K^2$VAE: A Koopman-Kalman Enhanced Variational AutoEncoder for Probabilistic Time Series Forecasting
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Proposes K²VAE, a VAE-based probabilistic forecaster that uses a KoopmanNet to linearize patch-token dynamics and a KalmanNet to refine the latent posterior, targeting long-term probabilistic time series forecasting.
Problem
Long-term probabilistic time series forecasting (LPTSF): existing probabilistic models degrade sharply as horizons grow due to nonlinear dynamics, error accumulation, and high per-step cost of diffusion/flow generators.
Approach
Combine a learnable Koopman measurement function with local (one-step eDMD) plus global learnable Koopman operators to get an approximately linear latent system, then apply a data-driven Kalman filter (learnable A, B, H, Q, R, with an integrator supplying control inputs) to model and refine the variational posterior uncertainty in a VAE with one-step generation.
Key results
- On short-term forecasting, K²VAE reports a 7.3% CRPS reduction and 14.5% NMAE reduction relative to the second-best baseline, CSDI.
- On long-term forecasting (L=720), K²VAE improves over PatchTST by 20.9% CRPS and 19.9% NMAE, achieving best CRPS on all nine long-term datasets reported in the main table.
- Ablations show removing KoopmanNet or KalmanNet degrades CRPS; e.g., on Electricity-L (L=720) CRPS rises from 0.057 to 0.074 without KoopmanNet and to 0.089 without KalmanNet.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use K²VAE instead of the strongest baseline | Short-term probabilistic forecasting · Short-term scenarios across eight datasets | CRPS: 7.3% reduction | Helped |
| use K²VAE instead of the strongest baseline | Long-term probabilistic forecasting · Long-term scenarios across nine datasets | CRPS: 20.9% | Helped |
| add the local Koopman operator to the global Koopman operator (ablation) | K²VAE · Electricity-L (L=720) | CRPS: 0.065\scriptstyle\pm0.007 -> 0.057\scriptstyle\pm 0.005 | Helped |
| add the global Koopman operator to the local Koopman operator (ablation) | K²VAE · Exchange-S (L=30) | CRPS: 0.012\scriptstyle\pm0.002 -> 0.009\scriptstyle\pm0.001 | Helped |
| add the Integrator (ablation) | K²VAE · Exchange-S (L=30) | CRPS: 0.015\scriptstyle\pm0.002 -> 0.009\scriptstyle\pm0.001 | Helped |
| add the skip connection (ablation) | K²VAE · Exchange-S (L=30) | CRPS: 0.011\scriptstyle\pm0.001 -> 0.009\scriptstyle\pm0.001 | Helped |
| add the control input (ablation) | K²VAE · Exchange-S (L=30) | CRPS: 0.013\scriptstyle\pm0.006 -> 0.009\scriptstyle\pm0.001 | Helped |
| add KoopmanNet (ablation) | K²VAE · Exchange-S (L=30) | CRPS: 0.014\scriptstyle\pm0.002 -> 0.009\scriptstyle\pm0.001 | Helped |
| add KalmanNet (ablation) | K²VAE · Electricity-L (L=720) | CRPS: 0.089\scriptstyle\pm0.011 -> 0.057\scriptstyle\pm 0.005 | Helped |
Surprising findings
- Diffusion and flow-based probabilistic models degrade sharply as horizon grows, sometimes performing worse than point forecasters equipped with Gaussian heads.
- The local one-step eDMD Koopman operator alone is numerically unstable on several datasets, so results are missing for it (shown as '-').
Limitations
- Some baseline results are unavailable due to excessive time and memory consumption (marked '-'), weakening comparisons on Traffic-L and Electricity-L.
- Hyperparameter search is limited to 8 sets per method, and the Koopman local operator can be numerically unstable when the measurement space is ill-conditioned.
Experimental setup
Datasets: ETTh1 ETTh2 ETTm1 ETTm2 Electricity Traffic Exchange Weather ILI Solar
Benchmarks: ETTh1-S ETTh2-S ETTm1-S ETTm2-S Electricity-S Solar-S Traffic-S Exchange-S ETTh1-L ETTh2-L ETTm1-L ETTm2-L
Metrics: CRPS NMAE
Baselines: FITS PatchTST iTransformer Koopa TSDiff D3VAE GRU NVP GRU MAF Trans MAF TimeGrad CSDI
Builds on: Koopman theory Kalman filter VAE ProbTS Koopa
Closest papers by meaning
- Koopman Generative Operators for Efficient Probabilistic Time-Series Forecasting (NeurIPS 2026 · similarity 0.91)
- Generative Modeling of Regular and Irregular Time Series Data via Koopman VAEs (ICLR 2024 · similarity 0.90)
- TARFVAE: Efficient One-Step Generative Time Series Forecasting via TARFLOW based VAE (NeurIPS 2025 · similarity 0.88)
- Considering Nonstationary within Multivariate Time Series with Variational Hierarchical Transformer for Forecasting (AAAI 2024 · similarity 0.87)
- Probabilistic Forecasting with Stochastic Interpolants and Föllmer Processes (ICML 2024 · similarity 0.87)
- Latent Diffusion Transformer for Probabilistic Time Series Forecasting (AAAI 2024 · similarity 0.85)
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