Geometric Hamiltonian Neural Networks

• Geometric Hamiltonian Neural Networks are neural architectures that integrate symplectic, Riemannian, and information-geometric properties to simulate Hamiltonian dynamics accurately.
• They employ structure-preserving integrators and manifold-constrained autoencoders to ensure energy conservation and robust long-term predictions in complex, high-dimensional systems.
• GeoHNNs enable applications in N-body simulations, material science, and molecular dynamics while addressing challenges like hyperparameter sensitivity and computational overhead.

Geometric Hamiltonian Neural Networks (GeoHNNs) are a class of neural network architectures specifically designed to learn or simulate the dynamics of systems governed by Hamiltonian mechanics while encoding the underlying geometric structure of the physical laws. By embedding symplectic, Riemannian, and information-geometric properties directly into the model architecture, these networks enable robust, accurate long-term prediction and efficient modeling of complex and high-dimensional dynamics.

1. Geometric Structure in Hamiltonian System Learning

GeoHNNs inherit fundamental concepts from classical mechanics, where the evolution of a system with state $z = (q, p)$ is described by Hamilton's equations: $$\dot{q} = \frac{\partial H}{\partial p} ; \quad \dot{p} = -\frac{\partial H}{\partial q}$$ with $H(q, p)$ denoting the Hamiltonian (total energy). The essential geometric properties of such systems are:

• Symplectic structure of phase space: The dynamics preserve the canonical symplectic two-form $\omega = dq \wedge dp$, enforcing volume preservation in phase space (Liouville's theorem).
• Riemannian geometry in configuration space: Kinetic energy and inertia tensors are functions on the manifold of configurations, often structured as symmetric positive-definite (SPD) matrices that live on a non-Euclidean manifold.
• Conservation laws and symmetry: Physical invariants (energy, momentum, angular momentum) follow from underlying symmetries as captured by Lie group actions and the corresponding infinitesimal generators.

GeoHNNs aim to encode these properties at the architectural level, avoiding ad hoc or purely data-driven parameterizations that can violate physics or lead to physically inconsistent predictions, particularly over long time horizons or in high-dimensional settings (Aboussalah et al., 21 Jul 2025, Friedl et al., 29 Sep 2025).

2. Architectural Principles and Parameterizations

Symplectic Neural Network Modules

Symplecticity is ensured either by constructing models whose outputs are gradients of scalar Hamiltonians and whose evolution is integrated using symplectic (area or volume-preserving) integrators, or by designing the network map as a composition of exact symplectic (Hamiltonian flow) layers (Tapley, 19 Aug 2024, David et al., 2021). For example, a symplectic layer may be implemented as the exact solution to a Hamiltonian vector field corresponding to a specific basis function, guaranteeing that each layer is a symplectic diffeomorphism. The generic construction can be written as: $$\Phi_h^{\bar{H}^\theta}(x) = \phi_h^{\bar{H}_k^\theta} \circ \dots \circ \phi_h^{\bar{H}_1^\theta}(x)$$ where each $\phi_h^{\bar{H}_i}$ is the exact flow of a Hamiltonian vector field generated by $\bar{H}_i^\theta$.

Riemannian Parameterization of Physical Quantities

For systems with configuration-dependent inertia (or mass) matrices $M(q)$, GeoHNNs parameterize $M(q)$ directly on the SPD manifold $\mathcal{S}_{++}^n$ using the Affine Invariant Metric. This is realized as

$M(q)^{-1} = \mathrm{Exp}_{M_0}(N_M(q; \theta_M))$

where $\mathrm{Exp}_{M_0}(\cdot)$ is the matrix exponential map at some reference point $M_0$ and $N_M$ is a neural network predicting tangent vectors (Aboussalah et al., 21 Jul 2025, Friedl et al., 29 Sep 2025). This guarantees positive-definiteness and adherence to the natural Riemannian metric.

Manifold-Constrained Autoencoders

For high-dimensional systems, dimensionality reduction is accomplished through a symplectic autoencoder—a nonlinear map $(\phi, \rho)$ such that the encoder $\rho$ and decoder $\phi$ roughly satisfy $\rho \circ \phi = \mathrm{id}$ on the latent space and, crucially,

$d\phi^\top J_{2n} d\phi = J_{2d}$

meaning that the pullback symplectic structure is exactly preserved (Friedl et al., 29 Sep 2025, Aboussalah et al., 21 Jul 2025). Biorthogonal or other Riemannian constraints on the layer weights ensure that reduced latent dynamics remain Hamiltonian.

3. Mathematical Formulations and Integration Schemes

GeoHNNs employ structure-preserving integrators—either exact or high-order symplectic numerical methods—which are crucial for long-term stability and energy conservation. Examples include symplectic Euler, implicit midpoint, Strang splitting, and pseudo-symplectic explicit Runge-Kutta schemes with guaranteed error bounds (Cheng et al., 27 Feb 2025, David et al., 2021, Tapley, 19 Aug 2024). This choice of integrator avoids the artificial energy drift common in standard neural ODEs or unconstrained recurrent architectures.

In certain frameworks, the Hamiltonian function is not learned by a generic MLP but decomposed using the Kolmogorov–Arnold representation: $$H(z) = \sum_{r=1}^{2d+1} \Phi_r \left( \sum_{s=1}^d \phi_{r,s}(z_s) \right)$$ where $\phi_{r,s}$, $\Phi_r$ are univariate functions (Wu et al., 26 Aug 2025). This localized representation enables the network to capture multi-scale, high-frequency dynamics more effectively than fully connected MLPs.

Other networks operate intrinsically on statistical or non-Euclidean manifolds (e.g., the lognormal statistical manifold), with architectural components (weights, activations) explicitly derived from the manifold’s differential geometry and associated Hamiltonian flows (Assandje et al., 30 Sep 2025).

4. Symmetry, Invariance, and Conservation

Automatic detection and enforcement of symmetries via Lie group or Lie algebra actions ensure the conservation of associated physical quantities through Noether’s theorem (Dierkes et al., 2023). This is accomplished either by:

• Loss terms penalizing deviation from invariance under the symmetry group’s infinitesimal generators,
• Architectural constraints (as in pooling, message passing, or weight sharing) enforcing invariances to permutation, translation, or rotation, particularly in the context of N-body or graph-based systems (Rahma et al., 6 Jun 2025, Su et al., 2022),
• Explicit parameterization of group actions in geometric modules, such as rotation matrices derived from SU(1,1) acting on the Poincaré disk in information-geometric architectures (Assandje et al., 30 Sep 2025).

These design choices are essential not only for physical consistency but also for robust generalization and transfer to larger or different system configurations.

5. High-Dimensional and Graph-Structured Systems

Scaling geometric Hamiltonian neural networks to high-dimensional regimes requires careful integration of geometric reduction and graph-structured parameterizations. Key strategies include:

• Symplectic autoencoder + geometric HNN stack: Enables learning from extremely high-dimensional systems (hundreds or thousands of degrees of freedom, e.g., cloth simulation or particle vortices) by mapping to a learned low-dimensional, structure-preserving latent submanifold (Friedl et al., 29 Sep 2025, Aboussalah et al., 21 Jul 2025).
• Geometric Graph Neural Networks: For N-body or materials systems, GNNs parameterize the Hamiltonian based on local environments, ensuring invariance under graph isomorphisms and geometric symmetries (e.g., complete local coordinate representations, LC layers, rotationally equivariant features) (Su et al., 2022, Rahma et al., 6 Jun 2025).
• Zero-shot generalization: Invariant architectures can transfer from small to large-scale systems (e.g., 8-node to 4096-node mass-spring graphs) without retraining, owing to symmetry and locality in model parametrizations (Rahma et al., 6 Jun 2025).

6. Practical Performance and Limitations

Extensive experimental benchmarks demonstrate that GeoHNNs and related models outperform traditional HNNs, unconstrained MLPs, and standard autoencoders in terms of energy conservation, trajectory accuracy, and long-term predictive stability across a variety of tasks: mechanical oscillators, coupled high-dimensional systems, chaotic n-body problems, graph-structured materials, and electronic property prediction (Aboussalah et al., 21 Jul 2025, Friedl et al., 29 Sep 2025, Wu et al., 26 Aug 2025, Cheng et al., 27 Feb 2025, Su et al., 2022).

Recent advances—such as explicit pseudo-symplectic integrators with learnable Padé-type activation functions (Cheng et al., 27 Feb 2025), random-feature-based parameter construction to eliminate slow gradient descent (Rahma et al., 6 Jun 2025), and symbolic Hamiltonian regression via backward error analysis (Tapley, 19 Aug 2024)—have improved computational efficiency, reduced training times, and enabled interpretable science-oriented modeling.

However, certain limitations remain. Modeling systems under nonholonomic constraints requires three-network architectures to separately recover the Hamiltonian, constraints, and associated multipliers, and the geometric structure only approximately holds due to "almost Poisson" properties of the brackets (T. et al., 4 Dec 2024). Hyperparameter sensitivity, computational overhead from Riemannian or symplectic regularization, and memory costs of complex metric parameterizations also present ongoing challenges.

7. Broader Implications and Future Directions

Geometric Hamiltonian Neural Networks represent an overview of deep learning, geometric mechanics, and differential geometry, advancing the state-of-the-art in scientific machine learning, physics-informed control, and statistical modeling. They enable physically rigorous, structure-aware modeling for applications ranging from molecular and celestial mechanics, robotics, and continuum mechanics to quantum materials and information geometry (Aboussalah et al., 21 Jul 2025, Friedl et al., 29 Sep 2025, Assandje et al., 30 Sep 2025, Su et al., 2022).

Key future directions include:

• Extension to stochastic, dissipative, or driven-dissipative Hamiltonian systems,
• Unified learning of multiple conservation laws and geometric structures in complex multi-physics environments,
• Symbolic discovery and interpretable regression of governing laws in unknown or partially known systems,
• Integration with fast training methods and scalable distributed architectures for real-time control and large-scale simulation.

By fully embedding the manifold structure, symplecticity, and symmetry of physical law, geometric Hamiltonian neural networks offer a mathematically grounded paradigm with demonstrable advantages for robust, stable, and physically consistent AI in science and engineering.