Input-to-State Stable Bundle Koopman Neural ODEs for Learning Controlled Dynamics under Environmental Constraints

Technical Summary: ISS-BKNO for Learning Controlled Dynamics under Environmental Constraints

Problem Statement
The paper addresses the challenge of learning accurate, certifiably stable models for controlled nonlinear dynamical systems operating under varying environmental conditions. Existing approaches typically address stability, extrinsic inputs, or environmental constraints in isolation. Specifically:

• Koopman methods offer linear representations but often lose explicit spectral structure when using deep learning, complicating stability certificates.
• Stable Neural ODEs provide continuous-depth models but frequently lack guarantees against distributional shifts or external disturbances.
• Environment-aware learning struggles to separate geometric constraints (fibers) from dynamics while maintaining global convergence guarantees.

The authors identify a gap in methods that simultaneously provide: (i) an explicit environment-conditioned latent coordinate system separating fiber geometry from dynamics; (ii) an Input-to-State Stability (ISS) certificate checkable via a semidefinite program; and (iii) a unified training objective enforcing these properties jointly.

Methodology: The ISS-BKNO Framework
The proposed framework, ISS-BKNO (Input-to-State Stable Bundle Koopman Neural ODE), integrates Koopman operator identification, Neural ODEs, fiber bundle geometry, and ISS certification into a three-stage lifting pipeline:

1. Bundle-Aware Encoder:
   The physical state $x$ and environmental signal $\xi$ are mapped to a fiber $F_\xi$ within a lifted space $Z$. This encoder, $\Phi(x, \xi)$, ensures that latent representations for different environments are geometrically separated, adhering to the fiber bundle structure $\pi: Z \to E$.

2. Environment-Conditioned Koopman Backbone:
   The latent state is further lifted to an observable space $\psi$ via $\Psi(z)$. The dynamics are modeled as:
   $$\dot{\psi} = K(\xi)\psi + B(\xi)u + r_\theta(\psi, u, \xi)$$
   Here, $K(\xi)$ and $B(\xi)$ are matrices conditioned on the environment $\xi$ (generated by small MLP heads). Crucially, $K(\xi)$ is parameterized to be Hurwitz by construction using a Persidskii-type decomposition:
   $$K(\xi) = -\frac{1}{2}Q(\xi)^\top Q(\xi) + S(\xi)$$
   where $S(\xi)$ is skew-symmetric. This ensures the symmetric part of $K(\xi)$ is negative semi-definite, guaranteeing contraction toward the nominal solution.

3. Residual Neural ODE Correction:
   A residual term $r_\theta$ corrects the linear approximation. Its Jacobian is constrained to satisfy a quadratic sector bound, ensuring the nonlinearity does not violate stability conditions.

Stability Certification and Training
The framework employs a Lyapunov-based ISS regularization. A quadratic Lyapunov candidate $V(\psi) = \psi^\top P \psi$ is learned jointly with the model. The training objective includes a penalty term ($L_{ISS}$) that enforces a Lyapunov decrease condition:
$$\dot{V} \leq -\alpha V + \gamma |w|^2$$
This condition is formulated as a Linear Matrix Inequality (LMI) involving the learned parameters $P$, the gain $\gamma$, and the sector multiplier $\lambda$. Unlike post-hoc verification, this LMI is part of the differentiable training loss, driving the optimizer to minimize the ISS gain $\gamma$ while satisfying stability constraints. The total loss combines prediction error, ISS regularization, reconstruction loss, and Control Barrier Function (CBF) penalties.

Key Contributions

1. Bundle-Aware Koopman Lifting: A novel encoder that maps states to environment-specific fibers, separating geometric constraints from dynamic evolution.
2. Lyapunov-Based ISS Regularization: A differentiable loss function that enforces a quadratic decrease condition during training, yielding a closed-form ISS gain upon convergence.
3. Theoretical Guarantees:
   • Proof of fiber invariance for the bundle coordinate map.
   • Proof of ISS for latent dynamics with an explicit gain formula.
   • Derivation of an approximation error bound relating prediction RMSE to the Extended Dynamic Mode Decomposition (EDMD) residual.
4. Empirical Validation: Comprehensive testing on four benchmark tasks (pendulum, cart-pole, unicycle navigation, and a 7-DOF manipulator) demonstrating superior performance over existing baselines.

Experimental Results
Experiments were conducted on four tasks involving varying physical parameters (e.g., rod length, mass, friction, payload) and external disturbances. The framework was compared against five baselines: vanilla Neural ODE (NODE), Koopman-EDMD, Koopman+ISS (post-hoc), ControlSynth Neural ODE (CSODE), and ICODE.

• Prediction Accuracy: ISS-BKNO achieved the lowest Root Mean Square Error (RMSE) across all tasks. Notably, it showed a 58% reduction in prediction RMSE compared to the vanilla Neural ODE.
• Robustness: Under sinusoidal disturbances, ISS-BKNO maintained a bounded error envelope consistent with theoretical bounds, whereas models without explicit ISS certificates (like NODE) accumulated persistent offsets.
• Stability Feasibility: The method achieved a 100% LMI feasibility rate during testing. In contrast, the post-hoc Koopman+ISS baseline failed feasibility checks in 6% of test episodes due to spectral radius violations under distributional shifts.
• Generalization: Performance degradation was most graceful near the boundaries of the training distribution, outperforming baselines that lacked explicit fiber structures.

Significance and Claims
The paper claims that ISS-BKNO provides a unified solution to the disjointed problems of stability, environmental adaptation, and geometric constraint handling in data-driven control. By integrating fiber bundle geometry with a Hurwitz-parameterized Koopman backbone and a differentiable ISS regularizer, the framework eliminates the need for post-hoc stability checks. The authors assert that this approach enables the deployment of learning-based controllers in safety-critical settings by providing global convergence guarantees and computable ISS gains directly from the training process. The theoretical results tie prediction accuracy directly to the EDMD residual, offering a principled criterion for dictionary selection.