DiffPhyCon: A Generative Approach to Control Complex Physical Systems

TL;DR

Diffusion Physical systems Control (DiffPhyCon), a new class of method to address the physical systems control problem, is introduced and an intriguing fast-close-slow-open pattern observed in the jellyfish is unveiled, aligning with established findings in the field of fluid dynamics.

Abstract

Controlling the evolution of complex physical systems is a fundamental task across science and engineering. Classical techniques suffer from limited applicability or huge computational costs. On the other hand, recent deep learning and reinforcement learning-based approaches often struggle to optimize long-term control sequences under the constraints of system dynamics. In this work, we introduce Diffusion Physical systems Control (DiffPhyCon), a new class of method to address the physical systems control problem. DiffPhyCon excels by simultaneously minimizing both the learned generative energy function and the predefined control objectives across the entire trajectory and control sequence. Thus, it can explore globally and plan near-optimal control sequences. Moreover, we enhance DiffPhyCon with prior reweighting, enabling the discovery of control sequences that significantly deviate from the training distribution. We test our method on three tasks: 1D Burgers' equation, 2D jellyfish movement control, and 2D high-dimensional smoke control, where our generated jellyfish dataset is released as a benchmark for complex physical system control research. Our method outperforms widely applied classical approaches and state-of-the-art deep learning and reinforcement learning methods. Notably, DiffPhyCon unveils an intriguing fast-close-slow-open pattern observed in the jellyfish, aligning with established findings in the field of fluid dynamics. The project website, jellyfish dataset, and code can be found at https://github.com/AI4Science-WestlakeU/diffphycon.

Figures (19)

• Figure 1: Overview of DiffPhyCon. The figure depicts the training (top), inference (bottom left), and evaluation (bottom right) of DiffPhyCon. Orange and blue colors respectively represent models learning the joint distribution $p_{\theta}(\mathbf{u},\mathbf{w})$ and the prior distribution $p_{\phi}(\mathbf{w})$. Through prior reweighting and guidance, DiffPhyCon is capable of generating superior control sequences.
• Figure 2: Intuition of Prior Reweighting. The top surface illustrates the landscape of $\mathcal{J}(\mathbf{u},\mathbf{w})$, where the high-dimensional variables $\mathbf{u}$ and $\mathbf{w}$ are represented using one dimension. The middle and lower planes depict probability heatmaps for the reweighted distribution $p^\gamma(\mathbf{w})p(\mathbf{u}|\mathbf{w})/Z$. Adjusting $\gamma$ from $\gamma=1$ (middle plane) to $0<\gamma<1$ (lower plane), a better minimal of $\mathcal{J}$ (red dot in the lower plane) gains the chance to be sampled. This contrasts with the suboptimal red point in the middle plane highly influenced by the prior $p(\mathbf{w})$.
• Figure 3: Pareto frontier of $\mathcal{J}_\text{energy}$ vs. $\mathcal{J}_{\text{actual}}$ of different methods for 1D Burgers' equation.
• Figure 4: Comparison of generated control curves of three test jellyfish. The resulting control objective $\mathcal{J}$ for each curve is presented.
• Figure 5: Visualization of jellyfish movement and fluid field controlled by DiffPhyCon as in the middle subfigure of Figure \ref{['fig:2d_control_curve']}.

Theorems & Definitions (2)

• Theorem 3.1
• proof : Proof of Theorem \ref{['prop:1']}