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Course Correcting Koopman Representations
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Proposes Periodic Reencoding, an inference-time mechanism for Koopman autoencoders that decodes and re-encodes the latent state every k steps, to curb long-horizon drift and latent trajectory crossings on dynamical systems and D4RL locomotion.
Problem
Long-horizon future state prediction with Koopman autoencoders: latent-space unrolls drift and produce self-intersecting trajectories and cannot capture switching dynamics between fixed points.
Approach
Instead of unrolling the linear latent dynamics for the full horizon, periodically decode and re-encode the state (every k steps or Δt in continuous time) at inference, applying local linear Koopman dynamics with course correction; training uses multi-step alignment, reconstruction and prediction losses with a bilinear-discretized continuous Koopman operator.
Key results
- On D4RL Hopper-v2 expert, 300-step state prediction MSE drops from 0.250 without reencoding to 0.079 with periodic reencoding for a linear decoder, and to 0.012 for the nonlinear-decoder model with periodic reencoding.
- On the Pendulum 100-step benchmark, periodic reencoding reduces linear-decoder MSE from 0.0512 to 0.0042 (scaled x100).
- In semi-open-loop control on Hopper expert, total normalized reward rises from 18.40 without feedback/reencoding to 53.5 with periodic reencoding, versus 96.05 for behavior cloning with full state.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use periodic reencoding at inference instead of no reencoding (ablation) | Koopman Autoencoder · Pendulum, 100 steps | MSE: 0.0512 -> 0.0042 | Helped |
| Use periodic reencoding at inference instead of no reencoding (ablation) | Koopman Autoencoder · Duffing Oscillator, 100 steps | MSE: 0.1152 -> 0.0112 | Helped |
| Use periodic reencoding at inference instead of no reencoding (ablation) | Koopman Autoencoder · Lotka-Volterra, 100 steps | MSE: 0.0113 -> 0.0072 | Helped |
| Use periodic reencoding at inference instead of no reencoding (ablation) | Koopman Autoencoder · Parabolic Attractor, 100 steps | MSE: 0.0205 -> 0.0292 | Hurt |
| Use periodic reencoding at inference instead of no reencoding (ablation) | Koopman Autoencoder · Hopper-v2 state prediction, 300 steps | MSE: 0.250 ± 0.05 -> 0.079 ± 0.03 | Helped |
| Use a nonlinear rather than linear decoder (ablation) | Koopman Autoencoder with periodic reencoding · Hopper-v2 expert, 300 steps | MSE: 0.079 ± 0.03 -> 0.012 ± 0.00 | Helped |
| Use nonlinear rather than linear latent dynamics with periodic reencoding and a linear decoder (ablation) | Koopman Autoencoder · Pendulum, 100 steps | MSE: 0.0042 -> 0.0025 | Helped |
| Use nonlinear rather than linear latent dynamics with periodic reencoding and a linear decoder (ablation) | Koopman Autoencoder · Duffing Oscillator, 100 steps | MSE: 0.0112 -> 0.0022 | Helped |
| Use nonlinear rather than linear latent dynamics with periodic reencoding and a linear decoder (ablation) | Koopman Autoencoder · Lotka-Volterra, 100 steps | MSE: 0.0072 -> 0.0040 | Helped |
| Use nonlinear rather than linear latent dynamics with periodic reencoding and a linear decoder (ablation) | Koopman Autoencoder · Parabolic Attractor, 100 steps | MSE: 0.0292 -> 0.0496 | Hurt |
| Add a prediction loss during training (ablation) | Koopman Autoencoder for dynamical systems · Dynamical systems benchmarks | prediction performance | Hurt |
| Normalize observations and actions as preprocessing (ablation) | Koopman Autoencoder · D4RL state prediction | prediction performance | Hurt |
Surprising findings
- Diagonal and skew-symmetric Koopman matrices failed in experiments, and single-step Koopman objectives performed worse than the MLP baseline.
- Best prediction results used a linear decoder on the dynamical systems, and Periodic Reencoding did not help the Parabolic Attractor, which admits a globally linear feature map.
Limitations
- Periodic reencoding fails to stabilize models with low-dimensional latents and deep latent-to-latent transition networks.
- Reencoding at every step accumulates encoder errors faster than repeated Koopman applications, and the reencoding period is a hyperparameter selected by search.
- Evaluation is limited to a few low-dimensional systems and three D4RL locomotion environments.
Experimental setup
Datasets: D4RL Hopper-v2 D4RL HalfCheetah-v2 D4RL Walker2d-v2 simulated dynamical systems (Parabolic Attractor, Duffing, Lotka-Volterra, Pendulum, Lorenz'63)
Benchmarks: D4RL Parabolic Attractor Duffing Oscillator Lotka-Volterra Pendulum Lorenz'63
Metrics: mean squared error total reward until termination
Baselines: Koopman autoencoder without reencoding reencoding at every step nonlinear latent dynamics MLP multi-step MLP behavior cloning
Builds on: Dynamic Mode Decomposition extended DMD Koopman autoencoders S4
Closest papers by meaning
- KoNODE: Koopman-Driven Neural Ordinary Differential Equations with Evolving Parameters for Time Series Analysis (ICML 2025 · similarity 0.87)
- ResKoopNet: Learning Koopman Representations for Complex Dynamics with Spectral Residuals (ICML 2025 · similarity 0.87)
- Identifying latent state transitions in non-linear dynamical systems (ICLR 2025 · similarity 0.86)
- Deep-Koopman-KANDy: Dictionary Discovery for Deep-Koopman Operators with Kolmogorov-Arnold Networks for Dynamics (NeurIPS 2026 · similarity 0.86)
- KoopSTD: Reliable Similarity Analysis between Dynamical Systems via Approximating Koopman Spectrum with Timescale Decoupling (ICML 2025 · similarity 0.86)
- Generative Modeling of Regular and Irregular Time Series Data via Koopman VAEs (ICLR 2024 · similarity 0.85)
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