Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Generating Physical Dynamics under Priors

Zihan Zhou, Xiaoxue Wang, Tianshu Yu

TL;DR

A novel framework that seamlessly incorporates physical priors into diffusion-based generative models to address this limitation and produces high-quality dynamics across a diverse array of physical phenomena with remarkable robustness, underscoring its potential to advance data-driven studies in AI4Physics.

Abstract

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.

Generating Physical Dynamics under Priors

TL;DR

A novel framework that seamlessly incorporates physical priors into diffusion-based generative models to address this limitation and produces high-quality dynamics across a diverse array of physical phenomena with remarkable robustness, underscoring its potential to advance data-driven studies in AI4Physics.

Abstract

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.
Paper Structure (75 sections, 10 theorems, 39 equations, 16 figures, 13 tables)

This paper contains 75 sections, 10 theorems, 39 equations, 16 figures, 13 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Diffusion models
  4. Invariant distributions
  5. Method
  6. Incorporating distributional priors
  7. Property of score functions.
  8. Equivalence class manifold for invariant distributions.
  9. Objective for fitting the score function.
  10. Incorporating physical feasibility priors
  11. Linear cases.
  12. Multilinear cases.
  13. Convex cases.
  14. Reducible nonlinear cases.
  15. General nonlinear cases.
  16. ...and 60 more sections

Key Result

Theorem 1

Let $q_0$ be a $\mathcal{G}$-invariant distribution. If for all $\mathbf{G} \in \mathcal{G}$, $\mathbf{G}$ is volume-preserving diffeomorphism and isometry, and for all $0 < a < 1$, there exists $\mathbf{H} \in \mathcal{G}$ such that $\mathbf{H}(a\boldsymbol{x}) = a \mathbf{G}(\boldsymbol{x})$, then

Figures (16)

  • Figure 1: Animated visualization of generated samples of shallow water dynamics, showcasing the variations over time. Use the latest version of Adobe Acrobat Reader to view.
  • Figure 2: Visualization of generated samples from the three-body (first row) and five-spring (second row) datasets. The leftmost figures in each row represent methods without priors, the middle figures correspond to our proposed methods, and the rightmost figures illustrate the physical properties as they evolve over time. Both total momentum and total energy should remain conserved. The samples generated by our methods demonstrate stronger adherence to physical feasibility.
  • Figure 3: Samples from the advection dataset with varying initial conditions. The horizontal axis represents the spatial coordinate $x$, while the vertical axis represents the parameter $t$.
  • Figure 4: Results of an ablation study comparing the effects of data matching and noise matching techniques. The findings show that incorporating a distributional prior improves model performance. We use the mean of trajectory error and velocity error as the metric for the three-body dataset.
  • Figure 4: The figure illustrates representative training samples from the Darcy flow dataset. The first row displays the values for the function $a(x)$, while the second row shows the values of $u(x, t)$ at time $t=1$.
  • ...and 11 more figures

Theorems & Definitions (24)

  • Theorem 1: Sufficient conditions for the invariance of $q_0$ to imply the invariance of $q_t$
  • Example 1
  • Example 2
  • Theorem 2: Equivalence class manifold representation
  • Remark 1
  • Theorem 3: Multilinear Jensen's gap
  • Remark 2
  • Theorem 4: Equivalence class manifold of SE(3)-invariant distribution dokmanic2015euclideanhoffmann2019generatingzhou2024diffusion
  • Definition 5: volume-preserving
  • Definition 6: isomorphism
  • ...and 14 more