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Port-Hamiltonian Neural Networks for Systems with Multiple Asymptotically Stable Equilibria
Simon Heilig, Jens Püttschneider, Mohammad Itani, Asja Fischer, Timm Faulwasser
TL;DR
Stable port-Hamiltonian networks' single-attractor Hamiltonian cannot represent systems with multiple stable equilibria. This paper uses a product Hamiltonian to represent multiple minima and establishes stability guarantees under stated assumptions. Experiments on three systems report faster convergence than unconstrained NODEs.
Problem
Method
The ms-PHNN forms its Hamiltonian as a product of Bregman divergences generated by a shared input-convex network, with one factor per target equilibrium.
Source: §3 · pp. 2–3Results
Takeaways & Limitations
The experiments confirm the extended capabilities of ms-PHNNs, and the paper reports faster convergence than unconstrained NODEs with a matched inductive bias.
Source: §5 · p. 4The local-stability result assumes second-order sufficient conditions at each minimum-energy equilibrium.
Source: §3.1 · p. 3
Abstract
from arXiv · showhide
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$\times$.
1 Introduction
Port-Hamiltonian neural networks encode physical structure and dissipation, which supports stability guarantees by construction. But many dissipative systems have multiple attractors, motivating an extension that preserves those guarantees while representing multiple stable equilibria.
1 Introduction
- Port-Hamiltonian neural networks add dissipation and external ports to physics-informed dynamics, providing architecture-level stability guarantees.The passage also notes that matching the hypothesis class to the data-generating process can reduce sample and optimization needs.Source: §1 · p. 1
- Many dissipative systems have finitely many isolated attractors separated by saddles, with basins partitioning state space up to a null set.Examples include a buckled beam settling into either of two shapes and a pendulum settling onto one of several magnets.Source: §1 · p. 1
- The paper proposes a multiplicative Hamiltonian for multiple stable equilibria and gives theoretical guarantees for stability and boundedness of other equilibria.Its contributions also include an almost-global stability result under assumptions about the other equilibria.Source: §1 · p. 2
2 Mismatch in Inductive Biases
The Duffing oscillator exposes a mismatch between a single-attractor stability guarantee and a system with two stable equilibria. The fitted s-PHNN learns only one of them.
2 Mismatch in Inductive Biases
- The Duffing oscillator has stable equilibria at [±1, 0] and a saddle at [0, 0] whose stable manifold divides the basins; convergence depends on the initial condition.The system's energy has global minima at the two stable equilibria.Source: §2 · p. 2
- Because an s-PHNN's global asymptotic stability guarantee permits only one attractor, it learns only one of the Duffing oscillator's two stable equilibria.The passage identifies this as the model's inductive-bias mismatch with bistable systems.Source: §2 · p. 2
- The mismatch motivates a model that preserves the port-Hamiltonian inductive bias and stability guarantees while allowing multiple stable equilibria.Source: §2 · p. 2
3 Port-Hamiltonian Neural Networks with Multiple Stable Equilibria
The ms-PHNN represents multiple stable equilibria with a product Hamiltonian, and its stability results depend on explicit regularity and hyperbolicity conditions. It also uses a separable Hamiltonian for mechanical systems with multiple stable positions.
3 Port-Hamiltonian Neural Networks with Multiple Stable Equilibria
- A shared input-convex network generates one Bregman-divergence factor per target equilibrium, whose product forms the ms-PHNN Hamiltonian.The factors share hidden layers and differ in their final convex read-out; target minima may be fixed or learned jointly with network parameters.Source: §3 · pp. 2–3
- For separable mechanical systems, the Hamiltonian combines kinetic energy with a product-structured potential, making each target position an equilibrium paired with the kinetic-energy minimum.The passage says explicitly modeling this separable structure helps the network converge faster.Source: §3 · p. 3
3.1 Stability Analysis
- If each divergence satisfies second-order sufficient conditions at its minimizer, the target equilibria are locally stable and become locally asymptotically stable under strict dissipation and the stated nonsingularity condition.Positive definiteness can be enforced by adding a strongly convex regularization term to each divergence.Source: §3.1 · p. 3
- With at least two strict energy minima, the mountain pass theorem guarantees another equilibrium; the additional equilibria lie in a bounded state-space region.These other equilibria prevent a global convergence guarantee without further assumptions about their stability.Source: §3.1 · p. 3
- If the other equilibria are finite and hyperbolic with unstable directions, and dissipation is positive definite, almost every trajectory converges to a target minimum-energy equilibrium.For the separable canonical model, the same guarantee holds when dissipation acts on momenta only and is positive definite there.Source: §3.1 · pp. 3–4
4 Experiments
Across three port-Hamiltonian systems, the ms-PHNN outperforms the s-PHNN on Duffing derivative RMSE and beats an unconstrained NODE on both Duffing variants. It also shows a reported training-convergence speedup over NODE.
4 Experiments
- The evaluation used the Duffing oscillator, an asymmetric Duffing variant, and a pendulum with four magnets, comparing NODE, s-PHNN, and a separable-Hamiltonian ms-PHNN.Models were trained by derivative fitting on 50 trajectories over 10 seeds.Source: §4 · p. 4
- 1.8×–2.7× training convergence speedup over NODE was recorded for the asymmetric Duffing and Four-Magnets systems.For the Duffing system, the ms-PHNN consistently outperformed the s-PHNN in derivative RMSE and outperformed NODE on both Duffing variants, while performing slightly worse on the harder Four-Magnets task.Source: §4 · p. 4
5 Conclusion
The paper extends asymptotically stable port-Hamiltonian neural networks to systems with multiple stable equilibria, using a multiplicative Hamiltonian whose components model individual energy minima. Its theory establishes local stability and, under appropriate assumptions, almost-global stability; experiments on three systems report faster convergence than unconstrained NODEs with a matched inductive bias.
5 Conclusion
- The multiplicative extension models multiple stable equilibria, and experiments on three systems report faster convergence than unconstrained NODEs with a matched inductive bias.Each component, derived from a shared FICNN, models one energy minimum.Source: §5 · p. 4
- The theory establishes local stability and shows almost-global stability under appropriate assumptions on the systems’ qualitative behavior.Source: §5 · p. 4, B Stability Analysis · p. 6
B Stability Analysis
- The stability analysis also considers a separable canonical ms-PHNN, where dissipation on momenta alone can suffice for global asymptotic stability.Source: B Stability Analysis · p. 6
B.1 Properties of the Bregman Divergence and the Product Hamiltonian
The section derives properties of the FICNN-generated Bregman divergences and their product Hamiltonian. These include bounded gradients, positive-definite Hessians at designed minima, radial unboundedness, and boundedness of all equilibria.
B.1 Properties of the Bregman Divergence and the Product Hamiltonian
- The Bregman divergence has a globally bounded gradient, with a bound depending on the FICNN’s weight matrices.Source: §B.1 · p. 7
- At each designed minimum, the product Hamiltonian has a positive-definite Hessian under Assumption 1, while its gradient and Hessian simplify there.The corresponding divergence and its gradient vanish at that minimum, leaving the matching factor in the Hessian.Source: §B.1 · pp. 7–8
- Each convex Bregman divergence is radially unbounded, so their product Hamiltonian is also radially unbounded.Source: §B.1 · p. 8
- All equilibria lie in a compact region, as radial growth of the divergences makes the Hamiltonian gradient point outward beyond a finite radius.The proof uses bounded gradients of the divergences and the centroid of the designed minima.Source: §B.1 · pp. 8–9
B.2 Local stability
Each designed minimum-energy equilibrium is locally stable under the stated regularity and non-singularity conditions. Strictly positive dissipation strengthens this to local asymptotic stability.
B.2 Local stability
- Under Assumption 1 and nonsingularity of Jθ(x) − Rθ(x), each minimum-energy equilibrium is stable by a local Lyapunov argument.Source: §B.2 · p. 9
- Assumption 1 requires each designed minimum to satisfy second-order sufficient conditions, giving the Hamiltonian a positive-definite Hessian there.Source: §B.2 · p. 9
- Strict dissipation, Rθ(x) ≻ 0 everywhere, makes the Hamiltonian decrease strictly nearby and yields local asymptotic stability.Source: §B.2 · p. 9
B.3 Analysis of other equilibria
With at least two designed minimum-energy equilibria, the product Hamiltonian has at least one additional equilibrium that is not a local minimum and is not asymptotically stable. The other equilibria are bounded under the stated nonsingularity condition.
B.3 Analysis of other equilibria
- At least two designed minima force a third equilibrium by the mountain pass argument, and that equilibrium is not a local minimum of the Hamiltonian.Source: §B.3 · p. 10
- The additional equilibrium is not asymptotically stable: every neighborhood contains initial states whose trajectories do not converge to it.Source: §B.3 · p. 10
- When Jθ(x) − Rθ(x) is nonsingular everywhere, all other equilibria lie in a bounded region.Under nonsingularity, equilibria coincide with critical points of the Hamiltonian, which are bounded by Theorem 6.Source: §B.3 · p. 11
B.4 Almost global asymptotic stability
Under the stated hyperbolicity and dissipation conditions, almost every trajectory converges to a minimum-energy equilibrium. The other equilibria may attract trajectories, but their regions of attraction have Lebesgue measure zero.
B.4 Almost global asymptotic stability
- Assumption 2 makes each other equilibrium unstable in at least one direction, so it is a saddle or a maximum of the Hamiltonian.For a hyperbolic equilibrium, the linearization has at least one eigenvalue with positive real part.Source: §B.4 · p. 11
- The other equilibria’s regions of attraction have Lebesgue measure zero under Assumption 2, which makes almost-global convergence to the minimum-energy equilibria possible.The result also requires Assumption 1 and strictly positive dissipation throughout the state space.Source: §B.4 · pp. 11–12
- With strictly positive dissipation, LaSalle’s invariance principle gives convergence to the set of minimum-energy and other equilibria, and each trajectory converges to a single equilibrium.The finite, isolated equilibrium set and the measure-zero attraction regions of the other equilibria yield convergence to a minimum-energy equilibrium for almost every initial condition.Source: §B.4 · p. 12
B.5 Separable Canonical ms-PHNN
The separable canonical ms-PHNN encodes multiple stable positions in its product potential while using a kinetic-energy model and canonical structure. Under the stated assumptions, dissipation on momentum alone suffices for almost-global convergence to a minimum-energy equilibrium.
B.5 Separable Canonical ms-PHNN
- The model encodes multiple stable positions through a product potential and combines it with a kinetic-energy s-PHNN built from a second FICNN.The kinetic energy has zero momentum as its unique minimizer.Source: §B.5 · p. 12
- The separable canonical model uses a canonical symplectic structure and restricts dissipation to the momenta.Its Hamiltonian decomposes into kinetic and potential energies over generalized coordinates and conjugate momenta.Source: §B.5 · p. 12
- With Assumptions 1 and 2 and strictly positive momentum dissipation, almost every initial condition converges to a minimum-energy equilibrium in the separable canonical ms-PHNN.The other equilibria’s regions of attraction have Lebesgue measure zero.Source: §B.5 · p. 13
- The separable-model argument uses radially unbounded kinetic and potential energies to make Hamiltonian sublevel sets compact, then applies LaSalle’s invariance principle.With momentum dissipation, vanishing dissipation implies zero momentum; the largest invariant set consists of states where the potential gradient also vanishes.Source: §B.5 · p. 13
C.1 Systems
- The Duffing oscillator has two minimum-energy equilibria and a saddle; the asymmetric Duffing variant retains two equal-depth minima but gives them different stiffnesses.The four-magnet pendulum has four minima, four saddles, and one maximum; its parameters separately control anisotropy and barrier scale.Source: §C.1 · p. 14
- For the four-magnet pendulum, the critical set comprises four minima, four saddles on the diagonals, and one maximum at the origin.The potential has four positional coordinates’ minima in a two-coordinate position space.Source: §C.1 · p. 14
C.2 Data Generation
Trajectories were generated with a specified solver and batch-covering initial-condition scheme across three systems. Numerical checks over fitted models found no additional minima and verified the stated critical-point condition for Assumption 2.
C.2 Data Generation
- Trajectories were integrated with Tsit5 and a PID step-size controller at rtol = atol = 10^-6, using the closed-form ground-truth gradient.The train, validation, and test sets contain 50, 50, and 200 initial conditions, respectively.Source: §C.2 · p. 14
C.3 Numerical Verification of Assumption
- Across 10 seeds per system, no fitted instance produced another minimum, and every other critical point met the stated eigenvalue condition for Assumption 2.Critical points were located by damped Newton iteration from uniform-grid starts covering the sampling window.Source: §C.3 · pp. 14–15