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ResKoopNet: Learning Koopman Representations for Complex Dynamics with Spectral Residuals
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Proposes ResKoopNet, which trains a neural-network dictionary by minimizing the ResDMD spectral residual so the Koopman spectrum (discrete and continuous) is learned rather than only filtered.
Problem
Approximating the Koopman operator spectrum from data: EDMD suffers spectral pollution and misses continuous spectra, and ResDMD only filters precomputed eigenpairs, so it cannot discover the full spectrum (spectral inclusion problem).
Approach
Parameterize the Koopman dictionary with a feedforward neural network and minimize the total spectral residual over eigenpairs by gradient descent, alternating with a closed-form Koopman matrix update K = (G+sigma I)^{-1}A, so the dictionary itself is learned to yield accurate eigenpairs and pseudospectra.
Key results
- On the pendulum, ResKoopNet recovers the unit-circle spectrum with N_K=300 observables, whereas ResDMD needed about 964 (original) or roughly 460-500 observables to approximate it.
- On mouse visual cortex data, ResKoopNet gives a significantly lower Davies-Bouldin index than Hankel-DMD, EDMD+RBF, and Kernel-ResDMD across five mice (exact values in figures).
- On the turbulence airfoil data, ResKoopNet's lowest-residual Koopman mode recovers the global pressure field structure, which Kernel-ResDMD and Hankel-DMD did not.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Learn the dictionary by minimizing spectral residuals instead of filtering precomputed spectra | ResKoopNet vs. classical methods · Pendulum | spectral recovery | Helped |
| Use ResKoopNet instead of Kernel-ResDMD | Koopman mode recovery · Airfoil turbulence | recovery of dominant pressure-field structure | Helped |
| Use ResKoopNet instead of Hankel-DMD | Koopman mode recovery · Airfoil turbulence | recovery of dominant pressure-field structure | Helped |
| Use learned spectral-residual eigenfunctions instead of the compared methods | Koopman eigenfunction clustering · Mice visual cortex | Davies-Bouldin index | Helped |
| Use Hankel-DMD instead of ResKoopNet | Koopman eigenfunction clustering · Mice visual cortex | Davies-Bouldin index | Hurt |
| Use EDMD with RBF basis instead of ResKoopNet | Koopman eigenfunction clustering · Mice visual cortex | Davies-Bouldin index | Hurt |
| Use Kernel-ResDMD instead of ResKoopNet | Koopman eigenfunction clustering · Mice visual cortex | Davies-Bouldin index | Hurt |
| Vary hidden layers from 1 to 4 and neurons per layer from 250 to 350 (ablation) | ResKoopNet · Pendulum | spectral approximation | Mixed |
| Use about 3 hidden layers with 300 neurons each (ablation) | ResKoopNet · Pendulum | spectral approximation | Helped |
| Increase the number of neurons and hidden layers (ablation) | ResKoopNet · Pendulum | spectral approximation | Helped |
Surprising findings
- ResDMD can approximate the pendulum unit-circle spectrum with roughly 460-500 basis functions, well below the 964 used in the original ResDMD work.
Limitations
- Neural-network training makes ResKoopNet substantially more computationally expensive (tens of minutes to hours) than EDMD and ResDMD.
- Does not account for stochastic dynamics, unlike VAMP and SDMD.
- Performance depends on neural architecture and training configuration; convergence analysis assumes strong convexity that does not hold in practice.
Experimental setup
Datasets: simulated pendulum trajectories turbulent airfoil pressure field dataset Sensorium 2023 mice calcium imaging
Benchmarks: pendulum Koopman spectrum airfoil turbulence Koopman modes Sensorium 2023 mouse neural state clustering
Metrics: spectral residual Davies-Bouldin index spectrum visual agreement with unit circle
Baselines: EDMD EDMD-DL Hankel-DMD ResDMD Kernel-ResDMD EDMD with RBF basis
Builds on: ResDMD EDMD Hankel-DMD EDMD-DL Kernel-ResDMD
Closest papers by meaning
- Deep-Koopman-KANDy: Dictionary Discovery for Deep-Koopman Operators with Kolmogorov-Arnold Networks for Dynamics (NeurIPS 2026 · similarity 0.91)
- Efficient Parametric SVD of Koopman Operator for Stochastic Dynamical Systems (NeurIPS 2025 · similarity 0.90)
- KoopSTD: Reliable Similarity Analysis between Dynamical Systems via Approximating Koopman Spectrum with Timescale Decoupling (ICML 2025 · similarity 0.89)
- Course Correcting Koopman Representations (ICLR 2024 · similarity 0.87)
- KoNODE: Koopman-Driven Neural Ordinary Differential Equations with Evolving Parameters for Time Series Analysis (ICML 2025 · similarity 0.86)
- Finding separatrices of dynamical flows with Deep Koopman Eigenfunctions (NeurIPS 2025 · similarity 0.86)
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