KoopGen: Koopman Generator Networks for Representing and Predicting Dynamical Systems with Continuous Spectra

TL;DR

KoopGen, a generator-based neural Koopman framework that models dynamics through a structured, state-dependent representation of Koopman generators, improves prediction accuracy and stability, while clarifying which components of continuous-spectrum dynamics admit interpretable and learnable representations.

Abstract

Representing and predicting high-dimensional and spatiotemporally chaotic dynamical systems remains a fundamental challenge in dynamical systems and machine learning. Although data-driven models can achieve accurate short-term forecasts, they often lack stability, interpretability, and scalability in regimes dominated by broadband or continuous spectra. Koopman-based approaches provide a principled linear perspective on nonlinear dynamics, but existing methods rely on restrictive finite-dimensional assumptions or explicit spectral parameterizations that degrade in high-dimensional settings. Against these issues, we introduce KoopGen, a generator-based neural Koopman framework that models dynamics through a structured, state-dependent representation of Koopman generators. By exploiting the intrinsic Cartesian decomposition into skew-adjoint and self-adjoint components, KoopGen separates conservative transport from irreversible dissipation while enforcing exact operator-theoretic constraints during learning. Across systems ranging from nonlinear oscillators to high-dimensional chaotic and spatiotemporal dynamics, KoopGen improves prediction accuracy and stability, while clarifying which components of continuous-spectrum dynamics admit interpretable and learnable representations.

Figures (10)

• Figure 1: Overview of the Koopman Generator model (KoopGen) for representing and predicting dynamical systems with the continuous spectrum.(a) In the data-driven Koopman learning framework, system is first lifted to an observation space, $\mathbf z(t)=\Phi(\mathbf x(t))$, where the dynamics are modeled linearly. The classical state-independent model is unable to capture the complex behavior of continuous-spectrum systems. KoopGen instead employ a state-dependent formulation, allowing the operator spectrum to vary continuously along trajectories and enabling accurate representation of complex dynamics. (b) The KoopGen model consists of a gating network and two operator sets, $\{\widehat{G}^n\}_{n=1}^N$ and $\{\widetilde{G}^m\}_{m=1}^M$, corresponding to skew-adjoint (conservative) and self-adjoint (dissipative) components, respectively. The gating network adaptively weights these generators based on the current latent state $\mathbf z(t)$, yielding a state-dependent Koopman operator $K(\mathbf z(t))$ that advances the dynamics forward in time. (c) Each generator is parameterized in a structure-preserving real block form, ensuring exact enforcement of the skew-adjoint and self-adjoint constraints and encoding physically interpretable dynamical properties.
• Figure 2: Prediction results comparison for KoopGen, DeepKoopman and LRAN across different system. The horizontal axis denotes the prediction horizon, and the vertical axis reports the prediction error. LRAN otto2019linearly represents Koopman models with a state-independent transfer operator (Eq. \ref{['eq:dmd']}), whereas DeepKoopman lusch2018deep corresponds to the state-of-the-art state-dependent Koopman approach (Eq. \ref{['eq:framework']}). The first row shows results for two low-dimensional continuous-spectrum systems, while the second row presents two high-dimensional continuous-spectrum systems. DeepKoopman improves substantially over state-independent methods in low-dimensional settings but degrades markedly in high dimensions, indicating limited scalability. In contrast, KoopGen consistently achieves the lowest prediction errors across all systems, demonstrating superior accuracy, stability, and scalability for continuous-spectrum dynamics.
• Figure 3: Learned Koopman eigenfunction and operator structure for the nonlinear pendulum. The eigenfunction exhibits a magnitude aligned with Hamiltonian energy level sets and a smoothly varying phase that parameterizes the periodic motion. The learned Koopman spectra reveal two operators with eigenvalues on the unit circle at angular frequencies $\pm0.15\pi$ and $\pm0.06\pi$, corresponding to counterclockwise and clockwise rotational dynamics. The spatial distribution of operator weights shows clear physical organization: the counterclockwise mode dominates near turning points, while the clockwise mode is activated near equilibrium crossings. Together, these operators provide a continuous, bidirectional decomposition of the pendulum flow, with the gating network adapting smoothly along the trajectory.
• Figure 4: Illustration of magnitude and phase of the KoopGen eigenfunctions for the Lorenz-63 system. Learned Koopman eigenfunctions of the Lorenz attractor. The three eigenfunctions yield a compact and interpretable decomposition of the Lorenz dynamics. The first eigenfunction encodes the cyclic motion around each wing through a smooth phase progression and amplitude growth along the spiral arms, with large amplitudes concentrated near the inter-wing transition channel. The second eigenfunction captures the left–right symmetry of the attractor, exhibiting weak phase variation but a sign change that separates the two metastable lobes. The third eigenfunction provides a coherent phase coordinate with a continuous $2\pi$ winding across the attractor, remaining smooth even through wing-switching events. Together, the learned eigenfunctions recover the core stretching–folding mechanism of the Lorenz flow and provide an interpretable spectral representation of its chaotic dynamics.
• Figure 5: Comparison of prediction errors for KoopGen, DeepKoopman and LRAN on the Lorenz–96 system. These results illustrate that KoopGen maintains low and spatially coherent errors, whereas others exhibit faster error accumulation and loss of dynamical structure at longer times.