Fixed-Kinetic Neural Hamiltonian Flows

• Fixed-Kinetic Neural Hamiltonian Flows are generative models that fix kinetic energy to a quadratic form while learning the potential via a neural network, ensuring volume preservation and invertibility.
• They offer high interpretability and parameter efficiency by dedicating all learnable capacity to the potential, reducing parameter counts by over 80% compared to traditional methods.
• The approach supports robust density estimation, surrogate modeling for kinetic PDEs, Bayesian inference, and control through symplectic integrators like the leapfrog scheme.

Fixed-Kinetic Neural Hamiltonian Flows (FKNHF) are a specialized class of generative models that employ the formalism of classical Hamiltonian dynamics but with the kinetic energy ("kinetic" term) fixed to a prescribed quadratic structure—typically corresponding to canonical physical forms. The only learnable component is the potential energy, usually parameterized by a neural network. This architectural choice yields volume-preserving, invertible, and highly interpretable flows that dramatically reduce parameter complexity relative to previous neural Hamiltonian flow architectures, while retaining favorable robustness and sample quality properties. Applications span normalizing flows, surrogate modeling for kinetic partial differential equations (PDEs), Bayesian inference, symmetry-exploiting generative modeling, and model-based control.

1. Mathematical Definition and Core Structure

The central object in FKNHF is a Hamiltonian of the form

$H(q, p; \theta) = T(p) + V(q; \theta)$

where $q \in \mathbb{R}^D$ are generalized positions (e.g., configuration variables, physical coordinates, or data), $p \in \mathbb{R}^D$ are momenta (auxiliary latent variables or physical momenta), and $\theta$ are the learnable parameters of the potential network. The kinetic term $T(p)$ is fixed in advance to a quadratic form, such as

$T(p) = \frac{1}{2}p^T \mathcal{M} p$

with constant, symmetric positive definite "mass" matrix $\mathcal{M}$, or, in scalar-mass regimes, $T(p)=\frac{a^2}{2} \sum_{j=1}^N \|p_j\|^2$ for a learned scalar $a$. The potential $V(q; \theta)$ is a neural network, often designed with symmetry constraints (e.g., permutation and translation invariance via DeepSet architectures).

Hamilton's equations govern a continuous-time, volume-preserving dynamical system: $q \in \mathbb{R}^D$0 This ODE system enables a deterministic mapping ("flow") from initial states $q \in \mathbb{R}^D$1 to final states $q \in \mathbb{R}^D$2 via integration.

2. Symplectic Volume-Preserving Flow and Discretization

To practically implement the flow and guarantee volume preservation, FKNHF utilizes explicit symplectic integrators. The canonical choice is the leapfrog (Störmer-Verlet) scheme, which for step size $q \in \mathbb{R}^D$3 evolves the state as

$q \in \mathbb{R}^D$4

Stacking $q \in \mathbb{R}^D$5 such steps yields a discrete-time flow map with unit Jacobian determinant (exact volume preservation). This property enables tractable likelihood estimation and invertibility without explicit Jacobian computation, a major advantage over general normalizing flows (Souveton et al., 2023, Souveton et al., 7 May 2025).

3. Interpretability, Parameter Efficiency, and Robustness

Fixing the kinetic energy to the quadratic canonical form transfers all expressiveness to the potential $q \in \mathbb{R}^D$6, resulting in highly interpretable models. In the context of probabilistic modeling or density estimation, the learned $q \in \mathbb{R}^D$7 mimics $q \in \mathbb{R}^D$8, directly reflecting the multimodal structure of the target distribution (Souveton et al., 2023). Table 1 summarizes key structural differences compared to more general neural flows:

Architecture	Kinetic Term $q \in \mathbb{R}^D$9	Learnable Potential $p \in \mathbb{R}^D$0
General Neural H. Flow	Neural network (MLP)	Neural network (MLP)
Fixed-Kinetic NHF	Fixed quadratic: $p \in \mathbb{R}^D$1	Neural network (MLP, DeepSet, etc.)

Because the kinetic term's parameter count scales as $p \in \mathbb{R}^D$2 (or even $p \in \mathbb{R}^D$3 in isotropic cases), the total parameter count can be reduced by over 80% in typical scenarios (Souveton et al., 2023). Empirically, fixed-kinetic flows also demonstrate greater robustness to hyperparameter settings such as number of integration steps, step size, and network width. For example, reducing the hidden layer width from 128 to 32 alters final loss by ≤4%, and variation with leapfrog step count beyond $p \in \mathbb{R}^D$4 is minor compared to original NHF (Souveton et al., 2023). This suggests that architectural rigidity in the kinetic sector confers both sample efficiency and stability.

4. Training Procedures and Objective Functions

The generic training pipeline for FKNHF involves the following:

• Sampling from a base distribution (e.g., Gaussian prior in $p \in \mathbb{R}^D$5).
• Integrating $p \in \mathbb{R}^D$6 (forward or backward) leapfrog steps to obtain $p \in \mathbb{R}^D$7.
• Evaluating negative log-likelihood or forward-reconstructed initial density, enabled by the volume preservation property:

$p \in \mathbb{R}^D$8

• Loss is typically the forward KL divergence or ELBO (for Bayesian posterior sampling tasks).
• Gradients are backpropagated through the sequence of symplectic maps (using autodiff) and parameters updated (e.g., via Adam).

In physical system surrogates (e.g., kinetic PDEs), the model is trained on input-output pairs generated by numerical simulation, and the flow learns an implicit surrogate for the map from initial to final phase-space densities (Souveton et al., 7 May 2025). In Bayesian inference, one minimizes KL divergence between transformed densities and desired posteriors (Souveton et al., 2023).

5. Applications: Density Estimation, Surrogate Modeling, Inference, and Control

Normalizing flows and generative modeling: FKNHF are natural candidates for multimodal density estimation, sampling, and Bayesian inference. On tasks such as 2D Gaussian mixture modeling, both original and fixed-kinetic NHF recover all modes, but fixed-kinetic NHF achieves equivalent or lower training loss with approximately 10× fewer parameters (Souveton et al., 2023). In supernova cosmology, NHF match Hamiltonian Monte Carlo within 1–2σ and produce independent posterior samples in a single pass.

Kinetic PDE surrogate modeling: In contexts such as the 1D Vlasov–Poisson equations, FKNHF provides a fast alternative to traditional simulators (Particle-in-Cell + Poisson solver + leapfrog integration). The model learns to map initial Gaussian phase-space distributions directly to final states, achieves low Wasserstein distances (W₁ ≈ 0.057 for positions, 0.079 for momenta at T=1.0), and generalizes to intermediate times by integrating the same learned potential for fewer steps (Souveton et al., 7 May 2025).

Hamiltonian ODEs for robotics and control: On SE(3), fixed-kinetic Hamiltonian neural ODEs enable energy- and symmetry-preserving modeling of rigid body dynamics. The kinetic energy is fixed as a quadratic in generalized velocities with block-diagonal mass matrix. Only the mass/inertia parameters and potential are learned via neural networks, guaranteeing exact energy conservation and enabling port-Hamiltonian control protocols such as IDA-PBC. Applications are demonstrated on pendulums, fully actuated rigid bodies, and quadrotor drones, with key metrics such as <0.01 rollout error for pose/velocity and RMS position error ≈0.05 m (Duong et al., 2021).

Latent symplectic flows and canonical transformations: FKNHF enters as a prior-imposing device in symplectic neural normalizing flows, e.g., mapping data to latent variables with independent harmonic Hamiltonians (fixed quadratic kinetic term). This structure has been shown to drive clean disentanglement of slow and fast collective modes in molecular systems and vision datasets (Li et al., 2019).

6. Implementation and Algorithmic Details

Algorithmic implementations rely on symplectic integration for both stability and geometric fidelity of the learned flow. Pseudocode typically follows:

1. Initialize mass matrix or kinetic scalar(s), potential network parameters, and encoder parameters if applicable.
2. For each minibatch:
   • Sample initial states from the prior.
   • Evolve states through $p \in \mathbb{R}^D$9 leapfrog steps.
   • Compute loss by pulling back to the base density or matching target samples.
   • Backpropagate and update parameters using an optimizer like Adam (Souveton et al., 2023, Souveton et al., 7 May 2025). When symmetry or invariance is required (e.g., particle permutation or translation invariance), the potential network uses DeepSet constructions: $\theta$0 where $\theta$1 and $\theta$2 are parameter-shared MLPs (Souveton et al., 7 May 2025).

In normalizing flows built as nonlinear canonical transformations, pointwise coordinate maps $\theta$3 are constructed from compositions of volume-preserving coupling layers (e.g., RealNVP) and associated momentum updates via autodifferentiation, with the prior density (and thus the latent Hamiltonian) fixed to a simple decoupled quadratic (Li et al., 2019).

7. Theoretical Significance and Empirical Outcomes

The fixed-kinetic approach yields several key benefits:

• Interpretability: All nontrivial structural or multimodal features are learned solely in the potential $\theta$4, mirroring physical intuition (Souveton et al., 2023, Souveton et al., 7 May 2025).
• Parameter efficiency: By eliminating the need to learn the kinetic metric, parameter count is minimized (e.g., to three unknowns in 2D for $\theta$5), yielding >80% reductions in small to moderate dimensions.
• Robustness: Models exhibit insensitivity to hyperparameters (integration steps, step size, network width) and maintain performance under parameter downsizing.
• Generalization: The architectural constraints enable extrapolation to regimes or trajectories not encountered during training, as the model cannot overfit to idiosyncratic kinetic metric artifacts (Duong et al., 2021).
• Sampling and computational benefits: Volume preservation and invertibility remove the need for Jacobian computation, support exact sampling and density tracking, and accelerate inference.

Empirically, FKNHF achieves state-of-the-art performance on mixture density modeling, outperforms non-Hamiltonian normalizing flows in sample quality, and provides interpretable surrogates in scientific simulation contexts (Souveton et al., 2023, Souveton et al., 7 May 2025). In control, the exact energy preservation of the SE(3) neural ODE enables model-based tracking and stabilization with compact and physically meaningful parameterization (Duong et al., 2021). In latent symplectic flows, the fixed kinetic prior facilitates conceptual compression and mode disentanglement in molecular and vision data (Li et al., 2019).