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Hamiltonian Score Matching and Generative Flows
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Introduces Hamiltonian velocity predictors and Hamiltonian score matching/discrepancy, and Hamiltonian Generative Flows (HGFs) that unify diffusion and flow matching as zero-force-field cases, with Oscillation HGFs evaluated on CIFAR-10 and FFHQ image generation.
Problem
Score matching and generative modeling typically rely on denoising score matching for noised data or fixed diffusion/flow dynamics; the paper asks whether deliberately designed Hamiltonian force fields can give new score-matching objectives and generative models that learn the unnoised data distribu
Approach
Train a velocity predictor on the location marginal of a parameterized Hamiltonian ODE; the optimal predictor generates the data probability path, so diffusion models and OT flow matching are HGFs with zero force, and a harmonic-oscillator force gives Oscillation HGFs. Score matching is obtained by minimizing a Hamiltonian score discrepancy that is zero iff the force equals the score.
Key results
- Oscillation HGF reaches FID 2.12 on unconditional CIFAR-10 at 35 NFE, beating VP-SDE (3.01) and DDPM (3.17) but behind EDM (1.98).
- On class-conditional CIFAR-10, Oscillation HGF achieves FID 1.97 versus EDM 1.79 and VP-SDE 2.48.
- On FFHQ 64x64 unconditional, Oscillation HGF achieves FID 2.86 versus EDM 2.39 and VP-SDE 3.39 at 79 NFE.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use an oscillation force field instead of zero-force VP-SDE | Hamiltonian Generative Flows · CIFAR-10 unconditional 32x32 | FID: 3.01 -> 2.12 | Helped |
| Use an oscillation force field instead of zero-force VP-SDE | Hamiltonian Generative Flows · CIFAR-10 class conditional 32x32 | FID: 2.48 -> 1.97 | Helped |
| Use an oscillation force field instead of zero-force VP-SDE | Hamiltonian Generative Flows · FFHQ unconditional 64x64 | FID: 3.39 -> 2.86 | Helped |
| Use an oscillation force field instead of EDM | Hamiltonian Generative Flows · CIFAR-10 unconditional 32x32 | FID: 1.98 -> 2.12 | Hurt |
| Use an oscillation force field instead of EDM | Hamiltonian Generative Flows · CIFAR-10 class conditional 32x32 | FID: 1.79 -> 1.97 | Hurt |
| Use an oscillation force field instead of EDM | Hamiltonian Generative Flows · FFHQ unconditional 64x64 | FID: 2.39 -> 2.86 | Hurt |
| Train a score network by minimizing Hamiltonian score discrepancy rather than denoising score matching | Score matching · Gaussian mixtures, low noise levels | gradient-estimation signal-to-noise ratio | Helped |
| Use HSM trajectories for supervision across locations of the same data points | Hamiltonian Score Matching · Gaussian mixtures | gradient-estimation variance | Helped |
| Remove diffusion-optimized input/output scaling and skip connections | Oscillation HGF · Image generation | FID | No clear effect |
Surprising findings
- Oscillation HGFs are reported to work out of the box, without the diffusion-specific preconditioning or hyperparameter tuning, and still come close to EDM.
Limitations
- Hamiltonian score discrepancy minimization requires a min-max (adversarial) optimization, which is less stable.
- HGFs with more complex force fields may not converge to a known distribution.
- Oscillation HGFs still trail EDM on CIFAR-10 and FFHQ.
Experimental setup
Datasets: CIFAR-10 FFHQ
Benchmarks: CIFAR-10 FFHQ 64x64
Metrics: FID NFE
Baselines: StyleGAN2-ADA DDPM LSGM PFGM VE-SDE VP-SDE EDM FM-OT (BNS)
Builds on: Hamiltonian Monte Carlo denoising score matching flow matching EDM critically-damped Langevin diffusion
Closest papers by meaning
- Hamiltonian Generative Networks (ICLR 2020 · similarity 0.91)
- Hamiltonian Dynamics with Non-Newtonian Momentum for Rapid Sampling (NeurIPS 2021 · similarity 0.88)
- Generating Physical Dynamics under Priors (ICLR 2025 · similarity 0.86)
- Hamiltonian Neural PDE Solvers through Functional Approximation (NeurIPS 2025 · similarity 0.86)
- Towards Cross Domain Generalization of Hamiltonian Representation via Meta Learning (ICLR 2024 · similarity 0.86)
- Physics-Constrained Flow Matching: Sampling Generative Models with Hard Constraints (NeurIPS 2025 · similarity 0.86)
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