Structure- and Stability-Preserving Learning of Port-Hamiltonian Systems

1. Problem Statement

The paper addresses the challenge of data-driven modeling of Port-Hamiltonian Systems (PHS) using neural networks. While Port-Hamiltonian Neural Networks (PHNNs) are effective for learning physical dynamics while preserving energy structures, existing methods face two critical limitations:

1. Convexity Constraint: Most current approaches rely on Input Convex Neural Networks (ICNNs) to approximate the Hamiltonian function. This forces the Hamiltonian to be convex, which is physically inaccurate for many real-world systems (e.g., pendulums, Toda lattices) where the energy landscape is non-convex.
2. Single Equilibrium Limitation: ICNN-based methods typically enforce stability at a single equilibrium point (usually the origin). They struggle to model systems with multiple isolated stable equilibria, often forcing the system to converge to the wrong equilibrium or failing to capture the correct local stability properties.

The goal is to develop a learning framework that preserves the intrinsic PHS structure, guarantees stability at multiple known stable equilibria, and relaxes the convexity requirement on the Hamiltonian function.

2. Methodology

The authors propose a novel neural network architecture and learning procedure designed to enforce stability by construction rather than through regularization penalties.

A. Novel Neural Hamiltonian Architecture

The core innovation is a specially designed Hamiltonian function $\hat{H}(x)$ that ensures specific points $x_{eq}$ are strict minima, thereby guaranteeing stability.

• Structure: The Hamiltonian is defined as:
  $$\hat{H}(x) = \text{NN}_H(x) \cdot h(\sigma(\|x - x_{eq}\|^2))$$
  Where:
  • $\text{NN}_H(x)$ is a standard neural network outputting positive values.
  • $\sigma(\cdot)$ is a smooth, differentiable smoothing function (e.g., $\sqrt{\|x-x_{eq}\|^2 + \delta^2} - \delta$) to handle non-differentiability at the equilibrium.
  • $h(\cdot)$ is a smooth step function (based on a polynomial construction) that transitions from 0 to 1.
• Mechanism:
  • Inside the region $\Omega_b$ (near $x_{eq}$): The function $h$ and its derivative are designed such that $\nabla_x \hat{H}(x_{eq}) = 0$ and $x_{eq}$ becomes a strict local minimum. This satisfies the Lyapunov stability conditions without requiring global convexity.
  • Outside the region $\Omega_b$: The function $h$ becomes 1, and $\hat{H}(x)$ reverts to the standard $\text{NN}_H(x)$, allowing the network to learn complex, non-convex dynamics freely.

B. Handling Multiple Equilibria

To model systems with $n_{eq}$ isolated stable equilibria, the authors extend the architecture:
$$\hat{H}(x) = \text{NN}_H(x) \cdot h_{\Sigma}(x; x^{(1)}_{eq}, \dots, x^{(n_{eq})}_{eq})$$
Where $h_{\Sigma}$ is a sum of individual step functions centered at each equilibrium. The active regions (balls of radius $b$) around each equilibrium are chosen to be non-overlapping. This allows the stability of each equilibrium to be analyzed and guaranteed independently.

C. Learning Procedure

• Structure Preservation: The interconnection matrix $\hat{J}(x)$ and dissipation matrix $\hat{R}(x)$ are parameterized using neural networks with structural constraints (skew-symmetric for $J$, symmetric positive semi-definite for $R$) to maintain the PHS form.
• Training: The model is trained by minimizing the error between the predicted state trajectory (obtained via symplectic integration) and the ground truth data.
• Stability Guarantee: Unlike previous methods that add a regularization term $\lambda \|\nabla H(x_{eq})\|^2$ (which only approximates zero gradient), this method hard-codes the zero-gradient and strict minimum properties into the network architecture.

3. Key Contributions

1. Relaxed Convexity: The method removes the restrictive convexity assumption on the Hamiltonian, enabling the modeling of non-convex energy landscapes common in physical systems.
2. Multi-Equilibrium Stability: It provides a rigorous framework for preserving multiple isolated stable equilibria simultaneously, a capability lacking in standard ICNN-based PHNNs.
3. Theoretical Guarantees: The authors provide a rigorous stability analysis (Theorem 3 and Proposition 4) proving that the learned model preserves local asymptotic stability at the specified equilibria, provided the dissipation matrix is positive definite in the neighborhood.
4. Region of Attraction Estimation: The paper derives conditions to estimate the region of attraction for each equilibrium, ensuring that trajectories starting within a specific bound converge to the correct equilibrium.

4. Experimental Results

The method was validated on two numerical benchmarks and compared against a baseline PH-ICNN (Port-Hamiltonian Input Convex Neural Network).

• Case 1: Toda Lattice System (Single Equilibrium)

  • Setup: A nonlinear chain of particles with a non-convex Hamiltonian.
  • Result: The proposed model achieved higher accuracy in predicting state trajectories, output, and Hamiltonian values compared to PH-ICNN under both pulse and sinusoidal inputs. It correctly maintained the system at the origin equilibrium when unperturbed.

• Case 2: Double Pendulum System (Multiple Equilibria)

  • Setup: An autonomous double pendulum with infinitely many equilibria (e.g., hanging down at $0, 2\pi, -2\pi$).
  • Result:
    • PH-ICNN Failure: The ICNN-based model failed to converge to non-origin equilibria (e.g., $2\pi$), forcing the system to the origin due to its structural limitations. It also produced incorrect transient trajectories even when converging to the origin.
    • Proposed Method Success: The proposed model successfully converged to the correct equilibrium points (including $2\pi$ and $-2\pi$) and accurately reproduced the transient dynamics, matching the ground truth significantly better than the baseline.

5. Significance and Conclusion

This work represents a significant advancement in physics-informed machine learning for dynamical systems. By decoupling stability preservation from global convexity, the proposed method bridges the gap between the theoretical requirements of Port-Hamiltonian systems and the practical complexity of real-world physical models.

• Impact: It enables the use of neural networks for learning complex, multi-stable physical systems (like robotics, power grids, and mechanical linkages) where multiple operating points exist.
• Future Work: The authors plan to extend this framework to control applications and validate it on real-world physical hardware.

In summary, the paper offers a robust, theoretically grounded solution for learning PHS that is both structurally faithful and stability-preserving across multiple operating points, overcoming the fundamental limitations of convexity-based approaches.