Denoising Hamiltonian Network for Physical Reasoning

TL;DR

This work addresses the limitations of existing physics-informed neural methods that rely on local, next-step updates and forward-only simulation. It introduces the Denoising Hamiltonian Network (DHN), which generalizes Hamiltonian mechanics into neural operators by using block-wise right Hamiltonians with block size $b$ and stride $s$, a denoising objective across noise levels $\alpha_0,\dots,\alpha_N$, and a global latent code $z$ for multi-system conditioning. The architecture employs a decoder-only transformer to map stacked state tokens and the latent code to a Hamiltonian value $\mathcal{H}$, with an autodecoder for per-trajectory adaptation and denoising steps that refine predictions toward physically consistent trajectories. Across single and double pendulums, the authors demonstrate improved long-horizon stability, parameter inference from partial observations, and robust trajectory interpolation via progressive super-resolution, illustrating broader physical reasoning capabilities beyond forward simulation.

Abstract

Machine learning frameworks for physical problems must capture and enforce physical constraints that preserve the structure of dynamical systems. Many existing approaches achieve this by integrating physical operators into neural networks. While these methods offer theoretical guarantees, they face two key limitations: (i) they primarily model local relations between adjacent time steps, overlooking longer-range or higher-level physical interactions, and (ii) they focus on forward simulation while neglecting broader physical reasoning tasks. We propose the Denoising Hamiltonian Network (DHN), a novel framework that generalizes Hamiltonian mechanics operators into more flexible neural operators. DHN captures non-local temporal relationships and mitigates numerical integration errors through a denoising mechanism. DHN also supports multi-system modeling with a global conditioning mechanism. We demonstrate its effectiveness and flexibility across three diverse physical reasoning tasks with distinct inputs and outputs.

Figures (18)

• Figure 1: Denoising Hamiltonian Network (DHN) generalizes Hamiltonian mechanics into neural operators. It enforces physical constraints while leveraging the flexibility of neural networks, opening pathways for broader applications in physical reasoning.
• Figure 2: How can we solve for a physical state? (I) Closed-form analytical solutions for simple systems. (II) For more complex physical systems, most physical PDEs only model local relations of close-by time steps. (III) For certain physical systems, states can be directly related even if they are not close by temporally.
• Figure 3: Discrete (right) Hamiltonian neural network. Dark blue and dark red indicate network inputs and outputs. Light colors illustrate the adjacent time steps.
• Figure 4: Block-wise Hamiltonian. Left: Classical HNN viewed as a special case of block size $b=1$ and stride $s=1$. Right: A discrete (right) Hamiltonian block with $b=4, s=2$. Dark blue and dark red indicate network inputs and outputs. Light colors illustrate the adjacent time steps.
• Figure 5: Denoising Hamiltonian block. Left: Random masking on input states. Right: Random noise sampling on input states. Different states have different sampled noise scales.