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P$^2$C$^2$Net: PDE-Preserved Coarse Correction Network for efficient prediction of spatiotemporal dynamics

Qi Wang, Pu Ren, Hao Zhou, Xin-Yang Liu, Zhiwen Deng, Yi Zhang, Ruizhi Chengze, Hongsheng Liu, Zidong Wang, Jian-Xun Wang, Ji-Rong_Wen, Hao Sun, Yang Liu

TL;DR

A new PDE-Preserved Coarse Correction Network (P$^2$C$^2$Net) to efficiently solve spatiotemporal PDE problems on coarse mesh grids in small data regimes and accelerates the prediction of PDE solutions on coarse spatiotemporal grids while maintaining a high accuracy.

Abstract

When solving partial differential equations (PDEs), classical numerical methods often require fine mesh grids and small time stepping to meet stability, consistency, and convergence conditions, leading to high computational cost. Recently, machine learning has been increasingly utilized to solve PDE problems, but they often encounter challenges related to interpretability, generalizability, and strong dependency on rich labeled data. Hence, we introduce a new PDE-Preserved Coarse Correction Network (P$^2$C$^2$Net) to efficiently solve spatiotemporal PDE problems on coarse mesh grids in small data regimes. The model consists of two synergistic modules: (1) a trainable PDE block that learns to update the coarse solution (i.e., the system state), based on a high-order numerical scheme with boundary condition encoding, and (2) a neural network block that consistently corrects the solution on the fly. In particular, we propose a learnable symmetric Conv filter, with weights shared over the entire model, to accurately estimate the spatial derivatives of PDE based on the neural-corrected system state. The resulting physics-encoded model is capable of handling limited training data (e.g., 3--5 trajectories) and accelerates the prediction of PDE solutions on coarse spatiotemporal grids while maintaining a high accuracy. P$^2$C$^2$Net achieves consistent state-of-the-art performance with over 50\% gain (e.g., in terms of relative prediction error) across four datasets covering complex reaction-diffusion processes and turbulent flows.

P$^2$C$^2$Net: PDE-Preserved Coarse Correction Network for efficient prediction of spatiotemporal dynamics

TL;DR

A new PDE-Preserved Coarse Correction Network (PCNet) to efficiently solve spatiotemporal PDE problems on coarse mesh grids in small data regimes and accelerates the prediction of PDE solutions on coarse spatiotemporal grids while maintaining a high accuracy.

Abstract

When solving partial differential equations (PDEs), classical numerical methods often require fine mesh grids and small time stepping to meet stability, consistency, and convergence conditions, leading to high computational cost. Recently, machine learning has been increasingly utilized to solve PDE problems, but they often encounter challenges related to interpretability, generalizability, and strong dependency on rich labeled data. Hence, we introduce a new PDE-Preserved Coarse Correction Network (PCNet) to efficiently solve spatiotemporal PDE problems on coarse mesh grids in small data regimes. The model consists of two synergistic modules: (1) a trainable PDE block that learns to update the coarse solution (i.e., the system state), based on a high-order numerical scheme with boundary condition encoding, and (2) a neural network block that consistently corrects the solution on the fly. In particular, we propose a learnable symmetric Conv filter, with weights shared over the entire model, to accurately estimate the spatial derivatives of PDE based on the neural-corrected system state. The resulting physics-encoded model is capable of handling limited training data (e.g., 3--5 trajectories) and accelerates the prediction of PDE solutions on coarse spatiotemporal grids while maintaining a high accuracy. PCNet achieves consistent state-of-the-art performance with over 50\% gain (e.g., in terms of relative prediction error) across four datasets covering complex reaction-diffusion processes and turbulent flows.

Paper Structure

This paper contains 41 sections, 33 equations, 10 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Related work
  3. Numerical methods.
  4. Deep learning methods.
  5. Physics-aware learning methods.
  6. Hybrid learning methods.
  7. Methodology
  8. Problem formulation
  9. Network architecture
  10. The flow of data
  11. RK4 integration scheme
  12. Learnable PDE Block
  13. Correction block:
  14. Conv filter with symmetric constraint:
  15. BC encoding:
  16. ...and 26 more sections

Figures (10)

  • Figure 1: Schematic of P$^2$C$^2$Net for learning Navier-Stokes flows. (a), Overall model architecture. (b), Poisson block. (c), learnable PDE block. (d), NN block. (e), Poisson solver. (f), Symbol notations. (g), Conv filter with symmetric constraint.
  • Figure 2: Periodic BC padding.
  • Figure 3: An overview of the comparison between our P$^2$C$^2$Net and baseline models, including error distributions (left), error propagation curves (middle), and predicted solutions (right). (a)-(d) show the qualitative results on Burgers, GS, FN, and NS equations, respectively. These PDE systems are trained with grid sizes of 25$\times$25, 32$\times$32, 64$\times$64, and 64$\times$64 accordingly.
  • Figure 4: Energy spectra.
  • Figure 5: The error distribution and propagation of P$^2$C$^2$Net for generalization over different external forces (a, b) and Reynolds numbers (c, d).
  • ...and 5 more figures