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Pi-fusion: Physics-informed diffusion model for learning fluid dynamics

Jing Qiu, Jiancheng Huang, Xiangdong Zhang, Zeng Lin, Minglei Pan, Zengding Liu, Fen Miao

TL;DR

Experimental results show that the proposed Pi-fusion significantly outperforms existing methods for predicting temporal evolution of velocity and pressure field, confirming its strong generalization by drawing probabilistic inference of forward process and physics-informed guidance sampling.

Abstract

Physics-informed deep learning has been developed as a novel paradigm for learning physical dynamics recently. While general physics-informed deep learning methods have shown early promise in learning fluid dynamics, they are difficult to generalize in arbitrary time instants in real-world scenario, where the fluid motion can be considered as a time-variant trajectory involved large-scale particles. Inspired by the advantage of diffusion model in learning the distribution of data, we first propose Pi-fusion, a physics-informed diffusion model for predicting the temporal evolution of velocity and pressure field in fluid dynamics. Physics-informed guidance sampling is proposed in the inference procedure of Pi-fusion to improve the accuracy and interpretability of learning fluid dynamics. Furthermore, we introduce a training strategy based on reciprocal learning to learn the quasiperiodical pattern of fluid motion and thus improve the generalizability of the model. The proposed approach are then evaluated on both synthetic and real-world dataset, by comparing it with state-of-the-art physics-informed deep learning methods. Experimental results show that the proposed approach significantly outperforms existing methods for predicting temporal evolution of velocity and pressure field, confirming its strong generalization by drawing probabilistic inference of forward process and physics-informed guidance sampling. The proposed Pi-fusion can also be generalized in learning other physical dynamics governed by partial differential equations.

Pi-fusion: Physics-informed diffusion model for learning fluid dynamics

TL;DR

Experimental results show that the proposed Pi-fusion significantly outperforms existing methods for predicting temporal evolution of velocity and pressure field, confirming its strong generalization by drawing probabilistic inference of forward process and physics-informed guidance sampling.

Abstract

Physics-informed deep learning has been developed as a novel paradigm for learning physical dynamics recently. While general physics-informed deep learning methods have shown early promise in learning fluid dynamics, they are difficult to generalize in arbitrary time instants in real-world scenario, where the fluid motion can be considered as a time-variant trajectory involved large-scale particles. Inspired by the advantage of diffusion model in learning the distribution of data, we first propose Pi-fusion, a physics-informed diffusion model for predicting the temporal evolution of velocity and pressure field in fluid dynamics. Physics-informed guidance sampling is proposed in the inference procedure of Pi-fusion to improve the accuracy and interpretability of learning fluid dynamics. Furthermore, we introduce a training strategy based on reciprocal learning to learn the quasiperiodical pattern of fluid motion and thus improve the generalizability of the model. The proposed approach are then evaluated on both synthetic and real-world dataset, by comparing it with state-of-the-art physics-informed deep learning methods. Experimental results show that the proposed approach significantly outperforms existing methods for predicting temporal evolution of velocity and pressure field, confirming its strong generalization by drawing probabilistic inference of forward process and physics-informed guidance sampling. The proposed Pi-fusion can also be generalized in learning other physical dynamics governed by partial differential equations.
Paper Structure (16 sections, 33 equations, 9 figures, 3 tables, 2 algorithms)

This paper contains 16 sections, 33 equations, 9 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Incompressible fluid flow
  4. Learning-based methods for modeling physical dynamics
  5. Diffusion models
  6. Denoising diffusion probabilistic models
  7. Deterministic implicit sampling
  8. Method
  9. Problem formulation
  10. Training strategy based on reciprocal learning
  11. Physics-informed guidance sampling for model inference
  12. Experiment
  13. Synthetic data
  14. Real-World data
  15. Ablation studies
  16. ...and 1 more sections

Figures (9)

  • Figure 1: Visualization of time evolution of the velocity in the case of 2D Compressible NS equations.
  • Figure 2: (a) The architecture of diffusion model in Pi-fusion. (b) The reciprocal learning strategy in the training of Pi-fusion. (c) The physics-informed guidance sampling in the inference of Pi-fusion.
  • Figure 3: Comparison of the accuracy between Pi-fusion and other methods for the synthetic data in terms of x-direction velocity. Four representative time snapshots are chosen as the example ($t = 2.0s, 2.5s, 3.0s, 3.5s$). Err refers to the difference in the entire domain between ground truth (using direct numerical simulation) and prediction by the approach.
  • Figure 4: Comparison of the accuracy between Pi-fusion and other methods for the synthetic data in terms of y-direction velocity. Four representative time snapshots are chosen as the example ($t = 2.0s, 2.5s, 3.0s, 3.5s$). Err refers to the difference in the entire domain between ground truth (using direct numerical simulation) and prediction by the approach.
  • Figure 5: Comparison of the accuracy between Pi-fusion and other methods for the synthetic data in terms of pressure. Four representative time snapshots are chosen as the example ($t = 2.0s, 2.5s, 3.0s, 3.5s$). Err refers to the difference in the entire domain between ground truth (using direct numerical simulation) and prediction by the approach.
  • ...and 4 more figures