Stochastic Port-Hamiltonian Neural Networks

• Stochastic pHNNs are neural parameterizations of port-Hamiltonian systems that learn the Hamiltonian and associated maps while preserving skew symmetry and dissipation constraints.
• They leverage dedicated network architectures to enforce structural properties, ensuring positive semidefiniteness and energy consistency amid stochastic excitation.
• The framework provides theoretical guarantees, including weak passivity in expectation and a universal approximation theorem that bounds trajectory error on compact sets.

Stochastic port-Hamiltonian neural networks, denoted SPH-NNs in one formulation, are neural parameterizations of stochastic port-Hamiltonian systems: open dynamical systems with dissipation, inputs, and stochastic forcing represented in an energy-based form. Their defining objective is to learn the Hamiltonian and associated coefficient maps while preserving structural constraints such as skew symmetry of the interconnection operator and positive semidefiniteness of the dissipation operator, thereby retaining passivity-oriented properties under stochastic excitation. In the 2026 formulation, SPH-NNs are introduced together with a weak passivity inequality in expectation and a structured universal approximation theorem on compact sets and finite horizons (Persio et al., 10 Mar 2026). Earlier work positioned stochastic pHNNs within a broader extension of deterministic port-Hamiltonian learning to stochastic, non-autonomous, and interconnected systems (Persio et al., 2024, Persio et al., 8 Sep 2025).

1. Deterministic antecedents and the extension to stochasticity

The deterministic port-Hamiltonian system (PHS) is written in input-state-output form as

$$\dot x = [J(x)-R(x)]\nabla H(x) + g(x)u,\qquad y = g(x)^\top \nabla H(x),$$

with state $x\in\mathbb R^n$, Hamiltonian $H(x)$, skew-symmetric interconnection $J(x)=-J(x)^\top$, symmetric positive semidefinite dissipation $R(x)=R(x)^\top\succeq 0$, input map $g(x)$, control $u$, and output $y$ (Persio et al., 2024). This structure yields the deterministic power balance

$$\frac{dH}{dt} = -\nabla H^\top R \nabla H + y^\top u \le y^\top u,$$

so the stored energy changes through dissipation and input-output exchange.

Deterministic port-Hamiltonian neural networks were introduced as structured learners for unknown or partially known PHS. In the 2024 account, one parameterizes a scalar Hamiltonian $H_\theta(x)$, a dissipation term $x\in\mathbb R^n$0 in simple mechanical problems, and an external forcing term $x\in\mathbb R^n$1, while keeping the canonical interconnection structure fixed in the simplest $x\in\mathbb R^n$2-D mechanical case (Persio et al., 2024). This deterministic pHNN formulation is explicitly contrasted with pure HNNs, which lack dissipation terms and therefore fail on damped systems in the reported mass-spring benchmark.

The stochastic extension promotes the deterministic energy-based model to an SDE with diffusion or, in the more geometric 2024 presentation, to a Stratonovich semimartingale system with noise ports (Persio et al., 2024). This extension is motivated by uncertainty, environmental noise, and random perturbations, while retaining the port-Hamiltonian emphasis on energy storage, dissipation, and interconnection (Persio et al., 8 Sep 2025). A central consequence is that deterministic passivity statements generally become mean-energy or expectation-based statements in the stochastic regime rather than pathwise monotonicity statements.

2. Mathematical formulations of stochastic port-Hamiltonian systems

A stochastic port-Hamiltonian system (SPHS) is given in Itô form by

$x\in\mathbb R^n$3

where $x\in\mathbb R^n$4 is the state, $x\in\mathbb R^n$5 the input, and $x\in\mathbb R^n$6 a Wiener process; the corresponding output is

$x\in\mathbb R^n$7

In this formulation, $x\in\mathbb R^n$8 is the total stored energy, $x\in\mathbb R^n$9 is skew-symmetric, $H(x)$0 is positive semidefinite, $H(x)$1 is the port map, and $H(x)$2 is the diffusion map (Persio et al., 10 Mar 2026).

A closely related 2025 presentation uses

$H(x)$3

with $H(x)$4 playing the role of the diffusion map (Persio et al., 8 Sep 2025). The drift $H(x)$5 and the diffusion $H(x)$6 together define an Itô SDE on $H(x)$7.

The 2024 work also gives a Stratonovich semimartingale formulation,

$H(x)$8

where $H(x)$9 are independent semimartingales, often Brownian motions, and $J(x)=-J(x)^\top$0 is a noise-port mapping (Persio et al., 2024). Under vanishing quadratic covariations, this can be converted to an Itô form. The coexistence of Itô and Stratonovich formulations reflects two emphases already present in the literature: direct SDE learning and likelihood-based training on the one hand, and a more explicitly geometric port interpretation on the other.

3. Architectural parameterizations and structural enforcement

In the 2026 SPH-NN architecture, each coefficient map is represented by a small feed-forward neural network. The Hamiltonian is learned by a scalar network $J(x)=-J(x)^\top$1; the interconnection is learned by a matrix network $J(x)=-J(x)^\top$2 whose skew part

$J(x)=-J(x)^\top$3

enforces $J(x)=-J(x)^\top$4; the dissipation is learned by a “square root” network $J(x)=-J(x)^\top$5 whose Gram matrix

$J(x)=-J(x)^\top$6

enforces positive semidefiniteness; and the diffusion is either fixed and known or learned by a network $J(x)=-J(x)^\top$7 (Persio et al., 10 Mar 2026). The resulting learned drift and diffusion are

$J(x)=-J(x)^\top$8

The 2025 framework uses an analogous architectural pattern but adds an explicit energy-consistency construction for the diffusion. There, $J(x)=-J(x)^\top$9, $R(x)=R(x)^\top\succeq 0$0, and a raw diffusion network $R(x)=R(x)^\top\succeq 0$1 is projected according to

$R(x)=R(x)^\top\succeq 0$2

so that $R(x)=R(x)^\top\succeq 0$3 by construction (Persio et al., 8 Sep 2025). The input map $R(x)=R(x)^\top\succeq 0$4 may also be learned if it is not known a priori.

Component	2026 SPH-NN construction	2025 pHNN construction
Hamiltonian	Scalar net $R(x)=R(x)^\top\succeq 0$5	Scalar feed-forward net $R(x)=R(x)^\top\succeq 0$6
Interconnection	$R(x)=R(x)^\top\succeq 0$7	$R(x)=R(x)^\top\succeq 0$8
Dissipation	$R(x)=R(x)^\top\succeq 0$9	$g(x)$0
Diffusion	Fixed known $g(x)$1 or learned $g(x)$2	Projected $g(x)$3 from raw $g(x)$4

These constructions are important because the structural constraints are not left to the optimizer alone. Skew symmetry and positive semidefiniteness are exact architectural properties in both formulations, while the 2025 diffusion projection imposes an additional orthogonality constraint. A common misunderstanding is that port-Hamiltonian learning is merely a penalty-based regularization scheme; the cited formulations instead emphasize structural enforcement by design, with optional penalties treated as secondary or unnecessary in the 2026 construction (Persio et al., 10 Mar 2026).

4. Stochastic passivity, mean-energy balance, and stability claims

For Itô dynamics, the 2026 SPH-NN analysis introduces the stochastic generator $g(x)$5 associated with the SPHS. For any $g(x)$6 function $g(x)$7,

$g(x)$8

Taking $g(x)$9 yields

$u$0

If, on a compact set $u$1, one assumes $u$2 for all $u$3, and defines the exit time $u$4, then Itô’s formula gives

$u$5

In particular, if $u$6, then

$u$7

which is identified as the weak passivity inequality in expectation (Persio et al., 10 Mar 2026).

The 2025 framework gives a different sufficient condition. There, if the diffusion is “energy-consistent,” meaning

$u$8

then the diffusion term is removed from the generator calculation, leaving

$u$9

Dynkin’s formula then yields

$y$0

and the text states that the closed-loop system is stable in expectation whenever the net supplied energy $y$1 is bounded (Persio et al., 8 Sep 2025).

These results clarify a recurrent point of confusion. In the cited stochastic formulations, passivity is not presented as a pathwise conservation or decay law. The 2026 result is a stopped-process inequality on a compact set under an explicit generator condition, whereas the 2025 result is a mean-passivity statement under an energy-consistency constraint on the diffusion. The two statements are compatible in spirit but not identical in hypothesis.

5. Structured universal approximation and trajectory closeness

The 2026 SPH-NN paper establishes a structured universal approximation theorem for stochastic port-Hamiltonian systems. Let the target SPHS

$y$2

have $y$3 coefficients and satisfy $y$4 and $y$5. Then, for any compact $y$6, horizon $y$7, and accuracy $y$8, $y$9, there exist networks $$\frac{dH}{dt} = -\nabla H^\top R \nabla H + y^\top u \le y^\top u,$$0 of SPH-NN form such that, uniformly on $$\frac{dH}{dt} = -\nabla H^\top R \nabla H + y^\top u \le y^\top u,$$1,

$$\frac{dH}{dt} = -\nabla H^\top R \nabla H + y^\top u \le y^\top u,$$2

with $$\frac{dH}{dt} = -\nabla H^\top R \nabla H + y^\top u \le y^\top u,$$3 and $$\frac{dH}{dt} = -\nabla H^\top R \nabla H + y^\top u \le y^\top u,$$4 exactly. Moreover, if $$\frac{dH}{dt} = -\nabla H^\top R \nabla H + y^\top u \le y^\top u,$$5 solves the learned SPH-NN SDE with the same initial data and Wiener path, then up to the exit time $$\frac{dH}{dt} = -\nabla H^\top R \nabla H + y^\top u \le y^\top u,$$6 from $$\frac{dH}{dt} = -\nabla H^\top R \nabla H + y^\top u \le y^\top u,$$7,

$$\frac{dH}{dt} = -\nabla H^\top R \nabla H + y^\top u \le y^\top u,$$8

(Persio et al., 10 Mar 2026).

The proof strategy, as outlined in the paper, combines classical universal approximation of continuous functions and their derivatives with structure-preserving parameterizations. $$\frac{dH}{dt} = -\nabla H^\top R \nabla H + y^\top u \le y^\top u,$$9, $H_\theta(x)$0, $H_\theta(x)$1, and $H_\theta(x)$2 are chosen so that $H_\theta(x)$3, $H_\theta(x)$4, $H_\theta(x)$5, and $H_\theta(x)$6, while skew symmetry and positive semidefiniteness hold by construction. A Grönwall-type SDE-stability lemma is then used to show that coefficient errors of order $H_\theta(x)$7 imply solution paths that remain $H_\theta(x)$8 close in mean square, hence with high probability, up to the exit time.

This theorem is stronger than a purely function-approximation statement. It does not merely assert that a neural network can represent the coefficient fields of an SPHS on a compact set; it also ties that approximation to coupled sample-path behavior over a finite horizon. A plausible implication is that the structural constraints are not only inductive biases for training but also part of the approximation class in which theorem-level guarantees are proved.

6. Training objectives, implementation patterns, and reported benchmarks

The 2026 SPH-NN formulation presents three training losses, all implemented via automatic differentiation plus an Adam optimizer. The increment-based loss is

$H_\theta(x)$9

The conditional-expectation loss replaces the empirical velocity with a pre-estimated conditional mean obtained by local regression. The negative log-likelihood loss uses an Euler–Maruyama Gaussian transition model with covariance $x\in\mathbb R^n$00, where $x\in\mathbb R^n$01 (Persio et al., 10 Mar 2026). Optional penalties such as $x\in\mathbb R^n$02 or $x\in\mathbb R^n$03 can be added, but the paper states that they are not needed by the construction.

The 2025 framework proposes a broader training stack for stochastic and interconnected settings. Training data consist of sampled trajectories $x\in\mathbb R^n$04, possibly noisy, and, if available, estimated increments of $x\in\mathbb R^n$05. Three complementary loss terms are combined: a drift-diffusion moment-matching loss $x\in\mathbb R^n$06, a vector-field + energy + rollout loss $x\in\mathbb R^n$07, and $x\in\mathbb R^n$08-penalties on $x\in\mathbb R^n$09, $x\in\mathbb R^n$10, and $x\in\mathbb R^n$11. Backpropagation is performed through an SDE integrator such as Euler–Maruyama or implicit midpoint, with PyTorch + torchsde listed as the implementation framework. Reported computational details include complexity per batch $x\in\mathbb R^n$12 for Jacobians $x\in\mathbb R^n$13, $x\in\mathbb R^n$14 for diffusion, and typical training with $x\in\mathbb R^n$15–$x\in\mathbb R^n$16, $x\in\mathbb R^n$17–$x\in\mathbb R^n$18, $x\in\mathbb R^n$19, converging in $x\in\mathbb R^n$20–$x\in\mathbb R^n$21 hours on a single GPU (Persio et al., 8 Sep 2025).

For the 2026 experiments, three canonical benchmarks with additive Brownian noise are reported (Persio et al., 10 Mar 2026):

Benchmark	Reported result	Baseline comparison
Mass–spring oscillator	SPH-NN-IB/CE/NLL reduce mean position/momentum error by factor $x\in\mathbb R^n$22–$x\in\mathbb R^n$23	Baseline MLP rollout error grows quickly
Mass–spring oscillator	Energy drift: baseline $x\in\mathbb R^n$24, SPH-NN-CE $x\in\mathbb R^n$25	Energy error strongly reduced
Duffing oscillator	SPH-NN-IB one-step MSE $x\in\mathbb R^n$26	Baseline $x\in\mathbb R^n$27
Duffing oscillator	Mean energy error reduced from $x\in\mathbb R^n$28 to $x\in\mathbb R^n$29	Lower energy error than baseline
Stochastic Van der Pol	Rollout MSE: baseline $x\in\mathbb R^n$30, SPH-NN-IB $x\in\mathbb R^n$31	Baseline phase-space spirals distort
Stochastic Van der Pol	SPH-NN captures the limit cycle and keeps energy oscillations correct	Non-canonical benchmark

The 2025 simulation studies extend the benchmark palette to damped mass-spring systems, chaotic Duffing oscillators, and a $x\in\mathbb R^n$32-DOF robot manipulator under time-varying torque inputs (Persio et al., 8 Sep 2025). The reported metrics are prediction MSE on states and derivatives, energy drift $x\in\mathbb R^n$33 versus $x\in\mathbb R^n$34, robustness to measurement noise, and comparisons with plain MLP, HNN, and TDHNN. The key findings are that pHNN achieves one-to-two orders of magnitude lower long-horizon MSE than MLP or unconstrained HNN, that energy remains bounded and follows the true dissipation exactly in expectation while baselines exhibit unphysical energy drift, and that under additive observation noise the pHNN retains predictive fidelity whereas unconstrained nets degrade rapidly.

7. Conceptual distinctions, recurrent misconceptions, and the research trajectory

Several distinctions are central to the interpretation of stochastic pHNNs. First, they are not restricted to conservative systems. The foundational state equations include dissipation through $x\in\mathbb R^n$35, input ports through $x\in\mathbb R^n$36, and stochastic forcing through $x\in\mathbb R^n$37 or $x\in\mathbb R^n$38 (Persio et al., 10 Mar 2026, Persio et al., 8 Sep 2025). This is consistent with the deterministic 2024 account, where pHNNs were explicitly introduced to handle damping and external forcing in settings where pure HNNs are inadequate (Persio et al., 2024).

Second, they are not limited to canonical Hamiltonian coordinates. The 2026 experiments include a stochastic Van der Pol oscillator described as non-canonical, with storage $x\in\mathbb R^n$39, damping parameter $x\in\mathbb R^n$40, and noise on $x\in\mathbb R^n$41 (Persio et al., 10 Mar 2026). The 2024 paper likewise discusses stochastic motion of interacting agents on a ring and discrete-time SPHS constructions, indicating that the framework is intended for broader geometric and networked settings than simple separable mechanics (Persio et al., 2024).

Third, the literature does not adopt a single stochastic passivity mechanism. One line imposes an explicit generator bound $x\in\mathbb R^n$42 on a compact set and proves a stopped-process weak passivity inequality; another enforces $x\in\mathbb R^n$43 so that diffusion is energy-consistent and mean passivity follows directly from Dynkin’s formula (Persio et al., 10 Mar 2026, Persio et al., 8 Sep 2025). These should not be conflated. A common misconception is that any stochastic pHNN automatically satisfies the same form of energy balance regardless of diffusion modeling; the cited formulations instead provide different sufficient conditions.

Taken together, the sequence of works suggests a clear research trajectory. The 2024 paper formulates the deterministic-to-stochastic extension and identifies how drift and diffusion can be learned while preserving interconnection geometry. The 2025 paper systematizes the architecture for non-autonomous and interconnected stochastic systems and emphasizes simulation, rollout, and control-oriented benchmarks. The 2026 paper sharpens the theory by giving a weak passivity inequality in expectation under an explicit generator condition and a structured universal approximation theorem with coupled trajectory guarantees on compact sets and finite horizons (Persio et al., 2024, Persio et al., 8 Sep 2025, Persio et al., 10 Mar 2026). This suggests an emerging division of labor within the literature: geometric formulation, architecture-level constraint enforcement, and theorem-level approximation and stability analysis are increasingly being treated as distinct but complementary components of stochastic port-Hamiltonian learning.