Port-Hamiltonian Neural Networks for Systems with Multiple Asymptotically Stable Equilibria

Stable port-Hamiltonian neural networks certify asymptotic stability by construction, yet their global Lyapunov function structure restricts them to systems with only one attractor. This work overcomes this limitation by proposing a multiplicative Hamiltonian structure within port-Hamiltonian neural networks (ms-PHNNs), enabling the modeling of systems possessing multiple asymptotically stable equilibria.

Mismatch in Inductive Biases

The paper identifies a mismatch between the inductive bias of standard stable port-Hamiltonian neural networks (s-PHNNs) and the dynamics of many physical systems, such as the Duffing oscillator, which exhibit bistability. The s-PHNN's global asymptotic stability guarantee forces it to learn only one equilibrium, failing to capture the system's true behavior. This necessitates a model that preserves the port-Hamiltonian inductive bias while allowing for multiple stable equilibria.

Port-Hamiltonian Neural Networks with Multiple Stable Equilibria

The authors propose a multiplicative Hamiltonian structure for the ms-PHNN dynamics:

1. The dynamics are defined as: x˙ = fPH(x) = [Jθ(x) − Rθ(x)] ∇xHθ(x), where Jθ and Rθ are positive semidefinite matrices.

2. The Hamiltonian is parametrized as the product of factors, one for each equilibrium: Hθ(x) = Ym k=1 Dθ,k(x).

3. Each factor Dθ,k follows a Bregman divergence construction generated by a single fully input-convex neural network (FICNN) fθ. This shared FICNN generates all factors, avoiding the need for separate networks for each minimum-energy equilibrium.

Stability Analysis

The theoretical framework establishes rigorous stability guarantees for the ms-PHNN under specific assumptions:

1. Assumption 1 requires that every Bregman divergence Dθ,k satisfies second-order sufficient conditions at its minimizer x¯k (i.e., ∇2xDθ,k(x¯k) ≻ 0). This ensures each minimum-energy equilibrium is locally stable and asymptotically stable under strict dissipation (Rθ(x) ≻ 0 for all x).

2. Assumption 2 requires that the set of other equilibria Xs is finite, and every xs in Xs is hyperbolic with at least one unstable direction (maxi Re λi > 0). This implies that the stable manifold of every other equilibrium has Lebesgue measure zero.

Almost Global Asymptotic Stability

Under Assumptions 1 and 2, combined with strict dissipation (Rθ(x) ≻ 0 for all x), the authors prove almost global asymptotic stability:

1. Theorem 2 states that for all initial conditions x0 not in the region of attraction Ws(Xs) of the other equilibria, the solution converges to one minimum-energy equilibrium x¯i ∈ X¯.

2. The region of attraction Ws(Xs) has Lebesgue measure zero because it is a finite union of nullsets (Theorem 9). This demonstrates that almost every trajectory converges to one of the target minimum-energy equilibria.

Separable Canonical ms-PHNN

For mechanical systems where the state splits into generalized coordinates q and momenta p, the Hamiltonian is decomposed into kinetic energy Tθ(p) and potential energy Vθ(q), where Vθ is modeled by a product of Bregman divergences Dθ,k(q). The kinetic energy Tθ is modeled by an s-PHNN. By imposing a canonical symplectic structure (J = [0 I; -I 0]) and restricting dissipation to the momenta (Rp,θ(x) ≻ 0), the authors show that asymptotic stability can be guaranteed under this weaker condition, confirming convergence to a minimum-energy equilibrium for almost every initial condition outside the measure-zero set Ws(Xs).

Experiments

The approach was validated on three systems: the Duffing oscillator, an asymmetric Duffing oscillator variant, and a four-magnets pendulum. Numerical results show that the ms-PHNN consistently outperforms both unconstrained Neural ODEs (NODE) and standard s-PHNNs in terms of derivative loss. Furthermore, the ms-PHNN demonstrates significant convergence speedups (ranging from 1.8× to 8.5×) compared to NODE, confirming its effectiveness in capturing complex dynamics with multiple stable equilibria while maintaining rigorous stability guarantees.

Conclusion

The work successfully extends asymptotically stable port-Hamiltonian neural networks to model systems with multiple stable equilibria using a multiplicative Hamiltonian structure derived from a shared FICNN. The theoretical contributions establish local stability for minimum-energy equilibria and almost global asymptotic stability under appropriate assumptions, while experimental results confirm superior performance and convergence speed compared to existing methods.

Based on the scientific paper Port-Hamiltonian Neural Networks for Systems with Multiple Asymptotically Stable Equilibria, here are specific, high-value improvements that can be implemented in AI systems:

)AI System Improvements and Capabilities Derived from ms-PHNNs:

1. Acknowledge and Model Complex, Bistable Dynamics (The Core Improvement):

2. Model Systems with Multiple Attractors (e.g., Physical or Control Systems exhibiting hysteresis, switching behavior, or multiple stable states):

3. Enhance Robustness Against Initial Conditions:

4. Improve Convergence Speed in Non-Convex Energy Landscapes:

)Specific Technical Implementation Details for Each Improvement:

1. Acknowledge and Model Complex, Bistable Dynamics (The Core Improvement):

2. The system can now be trained to learn dynamics that exhibit multiple stable equilibria (e.g., a physical system like a buckled beam settling into two different shapes, or a control system with distinct on/off states). Current standard models (like s-PHNNs) are restricted to systems with only one attractor.

3. Model Systems with Multiple Attractors:

4. The model structure is extended from the simple global Lyapunov function (s-PHNN) to a multiplicative Hamiltonian structure (ms-PHNN). This allows the network to capture the energy landscape of multiple minima simultaneously, effectively learning which attractor an initial condition belongs to.

5. Enhance Robustness Against Initial Conditions:

6. The theoretical framework proves that for almost every initial condition, the system will converge to one of the minimum-energy equilibria (Theorem 2). This means the model provides a strong probabilistic guarantee on long-term behavior across the entire state space, rather than just local stability around one point.

7. Improve Convergence Speed in Non-Convex Energy Landscapes:

8. The use of a shared Fully Input Convex Neural Network (FICNN) to generate factors for each equilibrium minimizes the number of parameters needed compared to training separate networks for every possible state, leading to faster convergence (reported speedups up to 1.8×–8.5× over standard Node ODEs).

9. Acknowledge and Model Complex, Bistable Dynamics (The Core Improvement):

10. The system can now handle systems where the energy landscape is not a single bowl but a complex topography, such as those found in mechanical oscillators or multi-state control problems. The model explicitly learns the structure of this energy surface through its multiplicative components.

)Summary of Improved AI System Capabilities:

The improved AI system, utilizing the ms-PHNN architecture, can be deployed for tasks requiring accurate modeling and prediction of systems with complex, multi-stable behavior where traditional single-attractor models fail. This includes:

1. Predicting the long-term state of bistable physical systems (e.g., structural stability analysis).

2. Developing robust controllers for switching systems where the system must choose between distinct stable operating modes.

3. Improving the efficiency and speed of learning dynamics in complex, non-convex environments by leveraging a theoretically sound, structure-preserving Hamiltonian formulation instead of generic vector field fitting (NODE).

Stable port-Hamiltonian neural networks certify asymptotic stability by construction. Yet, their Hamiltonian is a global Lyapunov function with a single global minimum, so they can represent only dynamic systems with one attractor. We demonstrate that this excludes even simple systems with energy landscapes forming a double well, and we overcome the restriction by parametrising the Hamiltonian as a product of Bregman divergences generated by one input-convex network. We prove that the resulting model is locally Lyapunov stable, that the coexistence of stable equilibria forces additional non-asymptotically-stable equilibria to exist, that all equilibria lie in a bounded region, and under a hyperbolicity assumption that almost-everywhere stability holds. On three systems our approach is able to recover the energy surface characteristics and improve the convergence speed by 1.8 times-8.5 times.