Generating Physical Dynamics under Priors

Generating physically feasible dynamics in a data-driven context is challenging, especially when adhering to physical priors expressed in specific equations or formulas. Existing methodologies often overlook the integration of physical priors, resulting in violation of basic physical laws and suboptimal performance. In this paper, we introduce a novel framework that seamlessly incorporates physical priors into diffusion-based generative models to address this limitation. Our approach leverages two categories of priors: 1) distributional priors, such as roto-translational invariance, and 2) physical feasibility priors, including energy and momentum conservation laws and PDE constraints. By embedding these priors into the generative process, our method can efficiently generate physically realistic dynamics, encompassing trajectories and flows. Empirical evaluations demonstrate that our method produces high-quality dynamics across a diverse array of physical phenomena with remarkable robustness, underscoring its potential to advance data-driven studies in AI4Physics. Our contributions signify a substantial advancement in the field of generative modeling, offering a robust solution to generate accurate and physically consistent dynamics.

💡 Research Summary

The paper addresses a fundamental challenge in data‑driven physics: generating high‑dimensional dynamical trajectories that obey the underlying physical laws. While diffusion‑based generative models have demonstrated impressive ability to capture complex data distributions, they typically ignore explicit physical priors, leading to samples that violate conservation laws or PDE constraints. To close this gap, the authors propose a unified framework that embeds two categories of priors into the diffusion generation process.

Distributional priors capture symmetries inherent in physical systems, such as SE(n) (rigid‑body rotation‑translation) invariance and permutation invariance of indistinguishable particles. The authors prove that if the initial data distribution is G‑invariant and the forward SDE respects volume‑preserving, isometric transformations, then all intermediate marginal distributions remain G‑invariant (Theorem 1). Consequently, the score function ∇ₓ log qₜ(xₜ) is (G, ∇⁻¹)‑equivariant. They therefore prescribe the use of G‑equivariant neural architectures (e.g., SE(3)‑equivariant GNNs) for score estimation. Moreover, they introduce the concept of an Equivalence‑Class Manifold (ECM): a minimal set of representatives that spans the orbit of the symmetry group. By training the score network only on ECM points, the model automatically generalizes to the full orbit, improving sample efficiency and robustness (Theorem 2).

Physical feasibility priors encode explicit laws such as energy conservation, momentum conservation, and PDE constraints (e.g., Navier‑Stokes, continuity equation). Using Tweedie’s formula, the authors show that both the noise predictor ϵ_θ and the data predictor x_θ are ultimately learning the conditional expectation E