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Efficient Parametric SVD of Koopman Operator for Stochastic Dynamical Systems
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Proposes learning top-k Koopman singular functions for stochastic dynamics by minimizing a low-rank approximation (LoRA) Hilbert-Schmidt loss, a polynomial in second-moment matrices that avoids backpropagating through SVD/inversion, with nesting and CCA-based inference.
Problem
Learning dominant singular subspaces of the Koopman operator from trajectory data with deep networks; existing VAMPnet and DPNet objectives require backpropagation through unstable matrix inversions, SVD and eigendecompositions, giving biased gradients and poor scalability.
Approach
Directly minimize the Hilbert-Schmidt low-rank approximation error of the Koopman operator, which reduces to a loss polynomial in second-moment matrices (trace of joint moment and product of marginal second moments), so mini-batch gradients are unbiased and no matrix inverse or SVD is needed; add nesting for ordered singular functions and CCA or EDMD for inference.
Key results
- On chignolin (k=16, H=64, B=384), LoRA-seq achieves VAMP-E 12.29 versus 9.54 for VAMPnet-1 and 7.36 for DPNet-relaxed.
- At high rank k=64 with small batch B=96, VAMPnet-1 drops to 19.71 VAMP-E while LoRA variants reach about 37-38.5.
- On 1D Langevin dynamics, sequential-nested LoRA recovers the top eigenfunctions and eigenvalues reliably, while the DPNet objective implementation failed to produce successful results.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| replace DPNet-relaxed with LoRA | Koopman singular-subspace learning · Chignolin 300 K, low-rank (k=16), H=64, B=384 | VAMP-E: 7.36 -> 10.27 | Helped |
| replace VAMPnet-1 with LoRA | Koopman singular-subspace learning · Chignolin 300 K, low-rank (k=16), H=64, B=384 | VAMP-E: 9.54 -> 10.27 | Helped |
| replace DPNet-relaxed with LoRA | Koopman singular-subspace learning · Chignolin 300 K, high-rank (k=64), H=128, B=384 | VAMP-E: 7.85 -> 38.89 | Helped |
| replace VAMPnet-1 with LoRA | Koopman singular-subspace learning · Chignolin 300 K, high-rank (k=64), H=128, B=384 | VAMP-E: 34.76 -> 38.89 | Helped |
| replace DPNet-relaxed with LoRA | Koopman singular-subspace learning · Chignolin 300 K, high-rank (k=64), H=64, B=96 | VAMP-E: 6.97 -> 37.74 | Helped |
| replace VAMPnet-1 with LoRA | Koopman singular-subspace learning · Chignolin 300 K, high-rank (k=64), H=64, B=96 | VAMP-E: 19.71 -> 37.74 | Helped |
| add sequential nesting (ablation) | LoRA · Chignolin 300 K, low-rank (k=16), H=64, B=384 | VAMP-E: 10.27 -> 12.29 | Helped |
| add sequential nesting (ablation) | LoRA · Chignolin 300 K, high-rank (k=64), H=128, B=384 | VAMP-E: 38.89 -> 42.44 | Helped |
| add sequential nesting (ablation) | LoRA · Chignolin 300 K, high-rank (k=64), H=64, B=96 | VAMP-E: 37.74 -> 37.33 | No clear effect |
| add joint nesting (ablation) | LoRA · Chignolin 300 K, low-rank (k=16), H=64, B=384 | VAMP-E: 10.27 -> 10.74 | Helped |
| add joint nesting (ablation) | LoRA · Chignolin 300 K, high-rank (k=64), H=128, B=384 | VAMP-E: 38.89 -> 39.37 | Helped |
| add joint nesting (ablation) | LoRA · Chignolin 300 K, high-rank (k=64), H=64, B=96 | VAMP-E: 37.74 -> 38.50 | Helped |
Surprising findings
- VAMPnet-2 and DPNet diverged on chignolin, and VAMPnet-1 diverged under a specific PyTorch version and for some train-test splits.
- Using the right singular basis for EDMD gave results comparable to the left basis originally proposed.
Limitations
- Quality of learned dynamics is limited by data temporal resolution, preventing recovery of the slowest processes (chignolin).
- Robustly identifying coherent structures in highly non-normal or chaotic systems remains open.
- Landscape of the LoRA objective is not yet well understood theoretically.
Experimental setup
Datasets: ordered MNIST 1D Langevin dynamics simulated trajectory chignolin 300K molecular dynamics dataset
Benchmarks: ordered MNIST 1D Langevin dynamics chignolin molecular dynamics
Metrics: RMSE classification accuracy VAMP-E score test-time orthogonality eigenvalue convergence
Baselines: VAMPnet-1 VAMPnet-2 DPNet DPNet-relaxed EDMD
Builds on: VAMPnet DPNet NeuralSVD extended dynamic mode decomposition
Closest papers by meaning
- Deep-Koopman-KANDy: Dictionary Discovery for Deep-Koopman Operators with Kolmogorov-Arnold Networks for Dynamics (NeurIPS 2026 · similarity 0.90)
- ResKoopNet: Learning Koopman Representations for Complex Dynamics with Spectral Residuals (ICML 2025 · similarity 0.90)
- Finding Koopman Invariant Subspaces via Personalized PageRank (NeurIPS 2026 · similarity 0.86)
- KoNODE: Koopman-Driven Neural Ordinary Differential Equations with Evolving Parameters for Time Series Analysis (ICML 2025 · similarity 0.85)
- Balanced Neural ODEs: nonlinear model order reduction and Koopman operator approximations (ICLR 2025 · similarity 0.85)
- Operator SVD with Neural Networks via Nested Low-Rank Approximation (ICML 2024 · similarity 0.85)
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