Learning Stable and Passive Neural Differential Equations

TL;DR

This work addresses learning stable neural differential equations by enforcing Lyapunov-based stability and passivity. It introduces stable Hamiltonian neural dynamics (SHND), where a PLNet-based Hamiltonian $H$ guides the flow through $\,\dot{x}=[J(x)-R(x)]\nabla H(x)$, with $J$ skew-symmetric and $R$ positive semi-definite; when $R\succeq \epsilon I$, the system achieves global exponential stability with rate $\lambda=\epsilon\mu^2$ and controllable overshoot. A passive port-Hamiltonian extension adds input-output ports that preserve passivity via $\dot H\le u^T y$, enabling passivity-based control insights. The approach leverages a bi-Lipschitz network to construct $H$ with quadratic bounds, ensuring both lower and upper bounds on the Lyapunov function and robust stability guarantees. Empirical results on a damped double pendulum show SHND outperforms Lyapunov-projection baselines in fitting, simulation accuracy, and robustness, while providing explicit stability guarantees that are valuable for control-oriented modeling and safe deployment.

Abstract

In this paper, we introduce a novel class of neural differential equation, which are intrinsically Lyapunov stable, exponentially stable or passive. We take a recently proposed Polyak Lojasiewicz network (PLNet) as an Lyapunov function and then parameterize the vector field as the descent directions of the Lyapunov function. The resulting models have a same structure as the general Hamiltonian dynamics, where the Hamiltonian is lower- and upper-bounded by quadratic functions. Moreover, it is also positive definite w.r.t. either a known or learnable equilibrium. We illustrate the effectiveness of the proposed model on a damped double pendulum system.

Figures (5)

• Figure 1: Training and test loss \ref{['eq:l2loss']} versus epochs. The proposed SHND model achieves better performance than SD-ICNN and SD-MLP.
• Figure 2: Batch averaged (solid) and maximum (dashed) simulation error between the trajectories generated by the learned model and the original pendulum dynamics w.r.t. a batch of initial conditions. Our model has much smaller simulation error and converges to 0 as time increases. This is mainly due to the stability guarantee w.r.t. the known equilibrium $x^\star$. The SD-ICCN has a steady error due to the fact that it converges to another equilibrium for some initial conditions. The SD-MLP produces oscillating trajectories.
• Figure 3: Training and test loss \ref{['eq:l2loss']} versus training data size. We use the same test data set with 500 samples. For our model, the performance improves as the size increases while the other two have some variations.
• Figure 4: Effect of the ratio $\nu/\mu$ on the train/test loss \ref{['eq:l2loss']}. We set $\mu = 0.1$ and trained the SHND models with different $\nu$. The ratio $\nu/\mu$ can be serve as a regularizer for the proposed SHND model.
• Figure 5: Simulation trajectories of different models with two initial states. The SD-MLP model generates oscillating behaviors. Both SD-ICNN and the proposed SHND converge. But the SD-ICNN converges to a non-zero equilibrium point despite a small change in the initial condition.

Theorems & Definitions (11)

• Definition 1
• Definition 2: khalil2002nonlinear
• Definition 3
• Definition 4
• Theorem 1
• proof
• Remark 1
• Remark 2
• Theorem 2
• proof