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Learning deep Koopman operators with convex stability constraints

Marc Mitjans, Liangting Wu, Roberto Tron

TL;DR

The paper tackles stability guarantees for data-driven Koopman models by introducing a new sufficient, piecewise-linear stability condition that decouples across matrix rows and can be enforced with a control barrier function–based projected gradient descent during learning. By combining a neural lifting of observables with a learnable basis change, the approach yields convex, row-wise constraints that are suitable for quadratic programming projections and can be integrated into an iterative optimization loop. Empirically, the KoopmanQP method achieves competitive predictive performance on LASA handwriting trajectories while offering faster training than SOC baselines and providing flexible incorporation of application-specific constraints. This work enables stable, learnable Koopman representations with practical optimization tooling for nonlinear dynamical systems.

Abstract

In this paper, we present a novel sufficient condition for the stability of discrete-time linear systems that can be represented as a set of piecewise linear constraints, which make them suitable for quadratic programming optimization problems. More specifically, we tackle the problem of imposing asymptotic stability to a Koopman matrix learned from data during iterative gradient descent optimization processes. We show that this sufficient condition can be decoupled by rows of the system matrix, and propose a control barrier function-based projected gradient descent to enforce gradual evolution towards the stability set by running an optimization-in-the-loop during the iterative learning process. We compare the performance of our algorithm with other two recent approaches in the literature, and show that we get close to state-of-the-art performance while providing the added flexibility of allowing the optimization problem to be further customized for specific applications.

Learning deep Koopman operators with convex stability constraints

TL;DR

The paper tackles stability guarantees for data-driven Koopman models by introducing a new sufficient, piecewise-linear stability condition that decouples across matrix rows and can be enforced with a control barrier function–based projected gradient descent during learning. By combining a neural lifting of observables with a learnable basis change, the approach yields convex, row-wise constraints that are suitable for quadratic programming projections and can be integrated into an iterative optimization loop. Empirically, the KoopmanQP method achieves competitive predictive performance on LASA handwriting trajectories while offering faster training than SOC baselines and providing flexible incorporation of application-specific constraints. This work enables stable, learnable Koopman representations with practical optimization tooling for nonlinear dynamical systems.

Abstract

In this paper, we present a novel sufficient condition for the stability of discrete-time linear systems that can be represented as a set of piecewise linear constraints, which make them suitable for quadratic programming optimization problems. More specifically, we tackle the problem of imposing asymptotic stability to a Koopman matrix learned from data during iterative gradient descent optimization processes. We show that this sufficient condition can be decoupled by rows of the system matrix, and propose a control barrier function-based projected gradient descent to enforce gradual evolution towards the stability set by running an optimization-in-the-loop during the iterative learning process. We compare the performance of our algorithm with other two recent approaches in the literature, and show that we get close to state-of-the-art performance while providing the added flexibility of allowing the optimization problem to be further customized for specific applications.
Paper Structure (12 sections, 8 theorems, 23 equations, 3 figures, 1 table)

This paper contains 12 sections, 8 theorems, 23 equations, 3 figures, 1 table.

Table of Contents

  1. INTRODUCTION
  2. PROBLEM OVERVIEW AND PRELIMINARIES
  3. The Koopman operator
  4. Data-driven approximations of the Koopman operator
  5. Control barrier functions
  6. A novel set-based stability condition for discrete-time linear systems
  7. CBF-BASED PGD FOR STABILITY GUARANTEES
  8. CBF-based projected gradient descent with stability constraints
  9. Expanding the set of learnable dynamics
  10. KOOPMAN LEARNING FRAMEWORK
  11. NUMERICAL PERFORMANCE EVALUATION
  12. CONCLUSIONS AND FUTURE WORK

Key Result

Lemma 1

If the constraint is satisfied $\forall k$ with $\alpha$ being a class-$K$ function, the system $f(x)$ is guaranteed to never enter the infeasible set $\bar{\mathcal{C}}$ if $h(\mathbf{x}_0) \geq 0$, and to asymptotically approach the feasible set if $h(\mathbf{x}_0) < 0$.

Figures (3)

  • Figure 1: Comparison of the vector fields of the same system in two different coordinate frames, where the bounding hypercube is depicted in dashed black lines. On the left, the original system clearly does not satisfy the constraints imposed by \ref{['lemma:unit-hypercube']}. On the right, after applying the right change of basis $\mathbf{S}$, the same system is now compliant with the constraints.
  • Figure 2: Boxplot comparison of training times (orange) and final training loss value (blue) for the three different architectures.
  • Figure 3: Comparison of the test trajectories for 9 different shapes of the LASA dataset predicted by each one of the 3 compared architectures. In solid blue lines, the ground truth trajectories. In dashed red lines, the predicted Koopman trajectories.

Theorems & Definitions (16)

  • Lemma 1: based on agrawal2017discrete
  • Lemma 2
  • proof
  • Corollary 1
  • proof
  • Definition 1
  • Corollary 2
  • proof
  • Lemma 3
  • proof
  • ...and 6 more