Hamiltonian Velocity Predictors (HVPs)

• Hamiltonian Velocity Predictors (HVPs) are data-driven frameworks that model Hamiltonian vector fields using neural, kernel, and variational methods.
• They combine partial or noisy observations with Hamiltonian constraints to produce surrogate models that ensure short-term accuracy and long-term conservation of invariants.
• HVPs find applications in generative modeling and inverse quantum problems, demonstrating robust performance in sample efficiency and dynamic prediction.

Hamiltonian velocity predictors (HVPs) are a class of data-driven methods and modeling frameworks for learning, predicting, and analyzing the velocity (vector field) structure generated by Hamiltonian dynamics from partial or noisy observations. By explicitly exploiting (or, in some cases, remaining agnostic to) the underlying Hamiltonian structure, HVPs span neural, kernel-based, Gaussian process, and variational approaches across physics-based modeling, inverse quantum problems, and modern generative modeling. HVPs unify the prediction of Hamiltonian vector fields, surrogate dynamics, and the design of sample-efficient learning architectures and metrics for conservative systems.

1. Foundations of Hamiltonian Velocity Predictors

Hamiltonian systems are governed by Hamilton's equations over phase space $z = (q, p) \in \mathbb{R}^{2n}$ for a scalar Hamiltonian $H(q, p)\in C^1$, yielding the ODE

$$\dot q = +\frac{\partial H}{\partial p}, \qquad \dot p = -\frac{\partial H}{\partial q}$$

and collectively,

$F(q, p) = (\partial_p H, -\partial_q H)$

where $F$ denotes the Hamiltonian vector field. The central task in HVP frameworks is to approximate $F$ based on observed data (possibly partial or noisy) and, in many applications, to construct a surrogate $\hat F$ or $\hat H$ that lends itself both to accurate short-term velocity prediction and to stable long-term integration with conservation of invariants.

HVPs generally refer to learned predictors—functions, neural networks, or kernel surrogates—that recover either $F$ directly or, more typically, the instantaneous velocity field or its data-driven conditional mean, given the available measurements or states. This architecture underpins both classical system identification and advanced generative modeling via learned Hamiltonian flows (Khoo et al., 2023, Holderrieth et al., 2024).

2. Methodological Architectures

Modern HVP frameworks incorporate varied learning architectures, all designed to estimate the velocity field from pointwise or trajectory data. The four principal approaches are (Khoo et al., 2023):

• Uninformed neural network (NN): A feedforward MLP, $$NN_\theta : \mathbb{R}^{2n} \rightarrow \mathbb{R}^{2n}$$, trained directly to regress velocity observations without Hamiltonian constraints.
• Uninformed Gaussian process (GP): A vector-valued GP prior over $F$, trained using log-marginal likelihood criteria to infer the posterior mean surrogate.
• Hamiltonian-informed neural network (PINN): A neural network $H_\phi: \mathbb{R}^{2n} \to \mathbb{R}$ whose derivatives yield $$\hat F_\phi(z) = (\partial_p H_\phi, -\partial_q H_\phi)$$. This imposes structural Hamiltonian symmetries.
• Hamiltonian-informed GP (PIGP): A GP prior on $H$, subject to derivative constraints matching observed velocities, solved exactly in kernel space.

Each architecture defines a corresponding loss or training objective: for PINN, this includes both Hamilton's equations (velocity matching), a pinning term on the Hamiltonian value, and weighted penalties. For GP-based surrogates, the fitting criteria are either likelihood-based or arise from constrained optimization in function spaces (Khoo et al., 2023, Herkert et al., 26 Jan 2026). A distinct neural HVP approach is the velocity-inferred HNN (VI-HNN), which leverages only position-time measurements by transforming $H(q,p)$ into $H(q,v)$ with $v$ inferred from finite differences under suitable invertibility assumptions on the mass matrix (Xu et al., 5 May 2025).

3. Hamiltonian Velocity Predictors in Probabilistic and Generative Models

Beyond traditional ODE learning, HVPs enable a unified framework for score estimation and flow-based generative modeling (Holderrieth et al., 2024). In this context, Hamiltonian dynamics is interpreted over $(x,v)$ space, where $v$ plays the role of velocity or momentum.

Let $H(x, v) = U(x) + \tfrac{1}{2} \|v\|^2$, where $U(x) = -\log \pi(x)$ for a target distribution $\pi$. The canonical ODE,

$\dot x = v, \qquad \dot v = -\nabla U(x)$

induces a deterministic flow $\varphi_t$ preserving the Boltzmann–Gibbs density $\pi_{BG}(x, v) \propto \pi(x)\mathcal{N}(v; 0, I)$. In the HVP framework, one introduces:

• Parameterized force fields $F_\theta(x, t)$, replacing $\nabla U(x)$, allowing expressive, learnable dynamics.
• Velocity predictor $V_\phi(x, t)$, trained to minimize $\mathbb{E}\|V_\phi(x_t, t) - v_t\|^2$. The optimal $V^*$ is the conditional mean velocity, $\mathbb{E}[v_t | x_t = x]$.

This construction yields:

• Hamiltonian Score Matching (HSM): Using HVPs to match scores, one designs a loss whose minimizer drives $F_\theta$ towards the true data score, generalizing classical denoising objectives to Hamiltonian-augmented flows.
• Hamiltonian Generative Flows (HGFs): Training an HVP on general PH-ODE flows yields a sampler for $\pi(x, t)$, with specializations (zero-force case) recovering both diffusion models and flow-matching as limiting cases.
• Oscillation HGFs: By imposing an oscillator potential $U(x) = \frac{1}{2}\alpha^2 \|x\|^2$, the PH-ODE produces a trajectory space for rich, tractable generative modeling (Holderrieth et al., 2024).

4. Training Objectives, Loss Functions, and Data Regimes

Loss functions in HVP frameworks are dictated by the available observables, the physical or probabilistic interpretation, and whether the design is Hamiltonian-informed. For supervised vector field prediction (Khoo et al., 2023):

• NN loss: $$L_\mathrm{NN}(\theta) = \frac{1}{N}\sum_{i=1}^N \|NN_\theta(z_i) - v_i\|^2$$.
• PINN loss: Incorporates penalties on Hamilton's equations residuals at sample points, alongside a pinning term for $H$.
• GP/PIGP: Maximize the marginal likelihood or solve for the posterior mean enforcing derivative constraints.

For VI-HNN (Xu et al., 5 May 2025): $$\mathcal L(\theta) = \sum_k \|\dot q^{(k)} - \nabla_v \widetilde H_\theta(q^{(k)}, \dot q^{(k)})\|^2 + \sum_k\|\ddot q^{(k)} + \nabla_q \widetilde H_\theta(q^{(k)}, \dot q^{(k)})\|^2$$

For probabilistic models (Holderrieth et al., 2024), the loss supports both estimation and marginal velocity matching, e.g.,

$$L_V(\phi \mid \theta, t) = \mathbb{E}_{(x_0, v_0) \sim \Pi} \|V_\phi(x_t, t) - v_t\|^2$$

and the HSM loss $L_\mathrm{hsm}$ as above.

Data requirements and regime selection are tightly coupled to the architecture: kernel and GP-based HVPs achieve high sample efficiency with low $N$, PINNs and large NNs scale better with increased data, and VI-HNN targets position-only settings prevalent in real-world sensing (Khoo et al., 2023, Xu et al., 5 May 2025, Herkert et al., 26 Jan 2026).

5. Empirical Performance, Efficiency, and Trade-offs

Benchmarks indicate that the empirical speed-accuracy trade-off varies markedly with method and task (Khoo et al., 2023, Herkert et al., 26 Jan 2026). Key findings:

Architecture	Data regime / Speed	Short-term accuracy	Long-term conservation
Uninformed NN	Effective at large $N$	Best one-step MSE	Poor energy conservation
Uninformed GP	Fastest at small $N$	High MSE	Does not conserve invariants
PINN (NN+Hamilton)	Slower than GP, faster than uninformed NN	Lower MSE than GP	Excellent with symplectic integrator
PIGP (GP+Hamilton)	Most efficient across systems at small $N$	Moderate MSE	Fast, but invariants drift unless symplectic integrator used

For long-horizon prediction and conservation of energy or phase-volume, Hamiltonian-informed methods with symplectic integration (e.g., PINN + symplectic Euler) are strongly preferred. Kernel-based surrogates with Hermite–Birkhoff interpolation in RKHS achieve algebraic convergence of both vector field and trajectory errors, with a two-to-three order-of-magnitude gain in long-term accuracy relative to implicit midpoint schemes, notably for large macro–time steps (Herkert et al., 26 Jan 2026).

6. Extensions: Applications in Inverse Quantum Problems and Generative Modeling

HVP techniques extend to quantum-mechanical inverse problems, as exemplified by the reconstruction of the Iwatsuka Hamiltonian from current (velocity) observations. In such a setting, the velocity operator $v_y = i[H, y]$ tracks quantum currents, and its spectral properties enable unique recovery of the underlying magnetic field from edge current data and band function derivatives using the Feynman–Hellmann theorem (Choulli et al., 2023).

In generative modeling, HVPs unify and generalize diffusion models and flow-matching via learned force fields in phase-space ODEs. Zero-force HGF recovers both denoising score-matching and OT-based flow matching, while oscillator-force HGFs enable new families of scale-stable, generative flows. Empirical studies show that oscillation HGFs match or exceed the performance of leading diffusion models in terms of Fréchet inception distance (FID) and sample quality, even without preconditioning (Holderrieth et al., 2024).

7. Theoretical Guarantees, Limitations, and Open Directions

HVP frameworks enjoy rigorous guarantees in several aspects:

• Energy conservation: For Hamiltonian-informed surrogates, the learned flow preserves the Hamiltonian and symplectic structure by construction in the absence of model or integration error (Xu et al., 5 May 2025, Herkert et al., 26 Jan 2026).
• Existence and convergence: Kernel-based (RKHS) HVPs permit rigorous interpolation, error control, and rates depending on the approximation residual and training center selection (Herkert et al., 26 Jan 2026).
• Score matching optimality: In HSM, the score discrepancy vanishes if and only if the predicted force is the true data score; small-$t$ expansion bridges to Fisher information (Holderrieth et al., 2024).

Principal limitations include singular or ill-conditioned mass matrices in velocity inference (VI-HNN), sensitivity to noise in finite-difference estimates of velocities, and computational scaling for kernel or GP surrogates as dataset size increases. Non-canonical coordinates, high-dimensional continuum systems, and compensation for measurement noise remain open challenges (Xu et al., 5 May 2025, Herkert et al., 26 Jan 2026).

• (Khoo et al., 2023) "What's Next? Predicting Hamiltonian Dynamics from Discrete Observations of a Vector Field"
• (Xu et al., 5 May 2025) "Velocity-Inferred Hamiltonian Neural Networks: Learning Energy-Conserving Dynamics from Position-Only Data"
• (Choulli et al., 2023) "Determining an Iwatsuka Hamiltonian through quantum velocity measurement"
• (Holderrieth et al., 2024) "Hamiltonian Score Matching and Generative Flows"
• (Herkert et al., 26 Jan 2026) "Symplecticity-Preserving Prediction of Hamiltonian Dynamics by Generalized Kernel Interpolation"