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KoNODE: Koopman-Driven Neural Ordinary Differential Equations with Evolving Parameters for Time Series Analysis
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Proposes KoNODE, a three-level Koopman-driven neural ODE in which the time-evolving ODE parameters are modeled as nonlinear observations of an intrinsic linear Koopman system, evaluated on synthetic oscillators, robot motion, and multivariate forecasting.
Problem
Existing neural ODE time series models use static parameters or model only surface-level state dynamics, limiting long-term prediction and generalization to new systems.
Approach
Model the time-varying parameters of a NODE as a nonlinear observable of a finite-dimensional linear Koopman system dw/dt = Aw, with the parameter map h(w) and the matrix A learned end-to-end via the adjoint method, plus an RNN-based initial observable estimator for sequence inputs.
Key results
- On the Non-linear Oscillator task KoNODE reaches test MSE 0.085±0.031 versus 0.381±0.104 for the next best (Hypernet).
- On six multivariate forecasting datasets KoNODE is best on five, e.g., ETTH2 MSE 0.289 vs 0.357 for Koopa.
- On the damped oscillator family KoNODE achieves 0.810±0.009 (×10^-2) vs 0.895 for ANODEV2, and on Imitation S it gets 0.005±0.002 vs 0.010 for CoDA.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add Koopman-driven evolving ODE parameters (KoNODE) vs. static-parameter NODE | NODE · Non-linear Oscillator | test MSE: 0.497 -> 0.085 | Helped |
| Use KoNODE with Koopman-modeled parameter dynamics | time-evolving parameter ODE baselines (HyperNet, ANODEV2) · Non-linear Oscillator | test MSE: best baseline 0.381 -> 0.085 | Helped |
| Koopman-driven KoNODE vs. ODE-based NODE variants on damped oscillator family | NODE family · Damped Oscillator Family | test MSE (x10^-2): HBNODE 0.912 -> KoNODE 0.810 | Helped |
| Koopman-driven evolving parameters vs. Koopa and other forecasters on ETT/Weather/CUP | multivariate forecasting · Six long-term forecasting datasets | test MSE: KoNODE best on 5 of 6; Koopa best on ETTH1 (0.605 vs 0.678) | Mixed |
| Koopman-driven KoNODE on robot motion prediction | robot motion datasets · Imitation C / cube pick / S | test MSE: Imitation S: CODA 0.010 -> KoNODE 0.005 | Helped |
| Set latent dimension m to 10 vs larger values (10 to 50) for parameter Koopman observables (ablation) | KoNODE · Non-linear Oscillator | test MSE | Mixed |
Surprising findings
- Koopa performs best on ETTH1 (0.605 vs KoNODE 0.678), the one dataset where KoNODE is not best.
Limitations
- Koopman dimension m must be chosen by hand (10 for simple systems, 100 for complex), and performance degrades when m is large.
- Authors note the method is not yet accelerated and does not handle irregularly sampled data.
Experimental setup
Datasets: Duffing oscillator synthetic data Spiral curve synthetic data Robot motion (Khansari-Zadeh & Billard 2011: Imitation S, cube pick, Imitation C) CUP WEATHER ETTH1 ETTH2 ETTM1 ETTM2
Benchmarks: Duffing nonlinear oscillator Damped oscillator family Robot motion (Imitation S, cube pick, Imitation C) CUP WEATHER ETTH1 ETTH2 ETTM1 ETTM2
Metrics: test MSE MAE running time memory
Baselines: NODE Latent ODE Neural Process Hypernet ANODEV2 HBNODE LEADS CoDA OINNs MoNODE DeepVAR Autoformer
Builds on: Neural ODEs Koopman operator theory ANODEV2 HBNODE Koopa
Closest papers by meaning
- Deep-Koopman-KANDy: Dictionary Discovery for Deep-Koopman Operators with Kolmogorov-Arnold Networks for Dynamics (NeurIPS 2026 · similarity 0.89)
- SKOLR: Structured Koopman Operator Linear RNN for Time-Series Forecasting (ICML 2025 · similarity 0.88)
- Generative Modeling of Regular and Irregular Time Series Data via Koopman VAEs (ICLR 2024 · similarity 0.87)
- Course Correcting Koopman Representations (ICLR 2024 · similarity 0.87)
- Balanced Neural ODEs: nonlinear model order reduction and Koopman operator approximations (ICLR 2025 · similarity 0.87)
- One-Step Offline Distillation of Diffusion-based Models via Koopman Modeling (NeurIPS 2025 · similarity 0.87)
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