Geometric RO-HNN: Reduced-Order Hamiltonian Networks

• Geometric RO-HNN is a machine learning framework that fuses Hamiltonian mechanics with neural networks to build low-dimensional, physically consistent surrogate models.
• It employs a symplectically-constrained autoencoder and a Hamiltonian neural network to ensure conservation laws and accurate phase-space dynamics.
• The approach enables long-term stable simulations and scalable performance for high-dimensional systems in engineering and scientific applications.

A Geometric Reduced-order Hamiltonian Neural Network (RO-HNN) is a machine learning framework that fuses the geometric structure-preserving principles of Hamiltonian mechanics with deep neural architectures designed to learn low-dimensional, physically-consistent surrogate models of high-dimensional dynamical systems. RO-HNNs leverage advanced concepts from symplectic geometry, model order reduction, and neural network theory to construct data-driven surrogates that maintain conservation laws and the qualitative integrity of phase-space evolution—even in the reduced latent spaces typical of practical scientific computing, engineering, and control applications.

1. Geometric Foundations and Components

The RO-HNN is constructed upon two key structure-preserving modules:

A. Geometrically-Constrained Symplectic Autoencoder:

This component discovers a low-dimensional latent symplectic submanifold $\mathcal{M}^{\checkmark}$ of the full phase space $\mathcal{M}$. The encoder $\rho: \mathcal{M} \to \mathcal{M}^{\checkmark}$ and decoder (embedding) $\phi: \mathcal{M}^{\checkmark} \to \mathcal{M}$ are learned so that

$d\phi^T \mathbb{J}_{2n} d\phi = \mathbb{J}_{2d}$

where $\mathbb{J}_{2n}$ and $\mathbb{J}_{2d}$ are canonical symplectic forms in the full and reduced spaces, respectively. This ensures the latent space is a valid symplectic manifold.

A cotangent-lift construction allows the encoder-decoder pair to be lifted from configuration to phase space, resulting in reduced coordinates $(q^{\checkmark}, p^{\checkmark})$ in which the symplectic structure and projection property $(\rho \circ \phi)(z) = z$ are preserved exactly (Friedl et al., 29 Sep 2025).

B. Geometric Hamiltonian Neural Network:

On the reduced manifold, a neural network models the latent dynamics via a Hamiltonian function

$$\mathcal{H}^{\checkmark}(q^{\checkmark}, p^{\checkmark}) = \frac{1}{2} (p^{\checkmark})^T M^{-1}_{\checkmark}(q^{\checkmark}) p^{\checkmark} + V^{\checkmark}(q^{\checkmark})$$

where $M^{-1}_{\checkmark}(\cdot)$ (inverse inertia/mass matrix) is enforced to be symmetric positive definite (SPD) via geometry-aware (Riemannian) parameterization—for example, using the affine-invariant metric exponential map (Aboussalah et al., 21 Jul 2025). The vector field is defined by Hamilton's equations in latent space. For dissipative or forced systems, additional terms (Rayleigh dissipation, external forces) are incorporated as extra learned fields, respecting the geometric construction.

2. Mathematical and Algorithmic Structure

Symplecticity and Consistency Conditions:

RO-HNN's architecture and training are constrained to enforce:

• Symplecticity: $d\phi^T \mathbb{J}_{2n} d\phi = \mathbb{J}_{2d}$ (encoder/decoder Jacobians),
• Projection: $\rho \circ \phi = \mathrm{id}$, and $d\rho|_{\phi(z)} = (d\phi)^+$ (the symplectic inverse),
• Cotangent-lift for mapping configuration to phase space projections (Friedl et al., 29 Sep 2025).

Losses and Training:

Training involves minimizing the reconstruction error

$$\ell_{\text{rec}} = \frac{1}{N}\sum_i \left\| \phi \circ \rho(x_i) - x_i \right\|^2$$

and additional penalties for projection and symplectic constraints (if not built-in). The latent Hamiltonian is trained by enforcing that numerical integration in latent space (using a structure-preserving, e.g., Strang–split symplectic integrator) closely matches projected full-order trajectories or available observations. In high-dimensional, parametric, or nonlinear settings, joint training of autoencoder and Hamiltonian network is typical, with possible inclusion of energy conservation and stability losses (Franck et al., 18 Jun 2025, Côte et al., 2023). Non-standard gradient descent algorithms (e.g., Riemannian manifold gradient flows) are used to ensure parameter updates remain on the relevant geometric manifolds (Brantner et al., 2023, Aboussalah et al., 21 Jul 2025).

Symplectic Neural Architectural Elements:

Recent approaches utilize symplectic neural architectures (e.g., HénonNets, constrained autoencoders with biorthogonal layers) to guarantee exact symplecticity of both the dimension reduction and flow map (Chen et al., 16 Aug 2025, Aboussalah et al., 21 Jul 2025).

3. Error Analysis, Stability, and Performance

RO-HNNs aim to bound the numerical error in predicted trajectories by controlling the network's loss relative to the symplectic and reconstruction constraints. Analytical error bounds may be derived showing that, by ensuring a small maximum training loss and preserving the minimum singular value of the latent Hessian, the maximum error in phase space is limited (Mattheakis et al., 2020). Symplectic integrators (used both in simulation in the latent space and for training) are essential in ensuring long-term conservation of physical invariants, such as the Hamiltonian, and in preventing secular energy drift—particularly in nonlinear and chaotic systems.

In benchmark experiments on high-dimensional mechanical, fluid, and plasma systems, RO-HNNs demonstrate:

• An ability to produce long-term stable, physically consistent predictions, even far beyond the timescales seen during training (Friedl et al., 29 Sep 2025, Côte et al., 2023).
• Lower trajectory and energy errors compared to traditional black-box or linearly-reduced models, especially for highly nonlinear or chaotic dynamics (Aboussalah et al., 21 Jul 2025, Chen et al., 16 Aug 2025).
• Strong scalability: models with tens to hundreds of latent degrees of freedom can capture the essential dynamics of systems with thousands of full-order variables (e.g., cloth simulations, 1D–1V Vlasov–Poisson particle-in-cell models).

4. Algorithmic Extensions and Generalizations

Nonlinear and Adaptive Reduction:

In transport- or wave-dominated systems where global linear reduction fails, time-varying (dynamical low-rank) reductions or adaptive rank hyper-reduction can be used, with error indicators such as

$r(w) = \|X_H(w) - \pi_w X_H(w)\|_F$

monitoring the projection fidelity (Pagliantini et al., 2023). Autoencoders may be convolutional or graph-based to exploit system structure (Côte et al., 2023, Lepri et al., 2023).

Generalized and Non-separable Hamiltonians:

Extensions exist for generalized (non-separable) Hamiltonians, with predictor-corrector symplectic integrators and robust adjoint sensitivity training to handle more complex or noisy observations while still enforcing geometric structure (Choudhary et al., 17 Sep 2024).

Symmetry and Nonholonomic Constraints:

RO-HNN concepts can be applied to model systems with symmetries (using embedded Lie algebra actions and explicit symmetry regularization in the loss) (Dierkes et al., 2023) or to systems under holonomic and nonholonomic constraints (using pseudo-Hamiltonian formulations and multi-network architectures to learn Hamiltonians, constraints, and Lagrange multipliers simultaneously) (T. et al., 4 Dec 2024).

Geometry-driven Neural Construction:

For some applications, the entire neural network—including weights, nonlinearities, and activation functions—may be constructed explicitly from the Hamiltonian, Lie group actions, and symplectic structure of the underlying statistical manifold (e.g., on the lognormal/Poincaré disk manifold), enabling fully interpretable, geometry-aware architectures (Assandje et al., 30 Sep 2025).

5. Applications and Impact

RO-HNNs have demonstrated efficacy on a wide variety of physical and engineering systems:

• High-dimensional mass-spring-damper and soft robotics models (Lepri et al., 2023, Friedl et al., 29 Sep 2025),
• Compressible/incompressible fluid and wave models, including nonlinear Schrödinger, shallow water, and Vlasov–Poisson plasma systems (Franck et al., 18 Jun 2025, Côte et al., 2023, Pagliantini et al., 2023),
• Multi-body dynamics with geometric structure and shape constraints (cloth, deformable body simulation) (Friedl et al., 29 Sep 2025, Aboussalah et al., 21 Jul 2025),
• Parametric and many-query scenarios in control, optimization, and uncertainty quantification, where rapid evaluation and global geometric fidelity are required (Hesthaven et al., 2021, Brantner et al., 2023).

Their main advantages include strong long-term qualitative behavior, robust energy and invariant conservation, scalable performance to very high dimensions, and applicability to real-time control and scientific modeling tasks. RO-HNNs also provide a blueprint for embedding hard physical priors as architectural constraints, a theme gaining prominence in physics-informed neural modeling.

6. Limitations and Current Challenges

Despite their strengths, several limitations persist:

• Training can be complex due to the need to satisfy differential-geometric constraints (e.g., symplecticity, SPD conditions) at every step. Manifold-aware optimization methods and architectural constraints are still being refined for scalability and generality (Brantner et al., 2023, Aboussalah et al., 21 Jul 2025).
• Construction and optimization of the reduced basis or autoencoder required for symplecticity may be computationally intensive, especially in the adaptive or fully nonlinear setting.
• For high-dimensional and pathological systems (e.g., those lacking a clear separation of scales or exhibiting stiff chaotic behavior), network expressivity or sample complexity can become a bottleneck (Franck et al., 18 Jun 2025).
• Current approaches work optimally for conservative Hamiltonian systems; dissipative or stochastic extensions require further conceptual development.
• The “backpropagation-free” (single-step) training methods (Rahma et al., 26 Nov 2024) circumvent iterative optimization but may face scalability issues due to the linear system solve in high dimensions.

7. Future Directions

The trajectory of research in RO-HNNs includes:

• Expanding to non-canonical and time-dependent symplectic forms,
• Integrating control and optimal decision-making in the reduced Hamiltonian latent space,
• Developing online and adaptive learning algorithms for systems with time-varying or parametric complexity (Pagliantini et al., 2023),
• Applying geometric RO-HNNs to manifold-valued data (e.g., in information geometry, as with the lognormal manifold neural network (Assandje et al., 30 Sep 2025)),
• Leveraging interpretable group-theoretic and symmetry-based priors for scientific discovery in complex dynamical systems.

These directions promise to further bridge the gap between differential-geometric model reduction and modern machine learning, providing a general-purpose toolset for learning, control, and analysis of complex physical systems at scale.