PAPM: A Physics-aware Proxy Model for Process Systems

TL;DR

PAPM proposes a physics-aware proxy framework that fully leverages partial prior physics, including multiple input conditions and conservation laws, to improve out-of-sample generalization and data efficiency for process systems. It introduces a holistic Temporal-Spatial Stepping Module (TSSM) with three structural paths (Localized, Spectral, Hybrid) and an ODE-based update, enabling flexible adaptation across diverse 2D PDE benchmarks. Across five datasets and nine generalization tasks, PAPM achieves an average improvement of $6.7\%$ with dramatically fewer FLOPs and only $1\%$ of the parameters of prior leading methods, while maintaining robust coefficient extrapolation and time extrapolation performance. The approach demonstrates meaningful practical impact by offering a scalable, efficient surrogate capable of handling complex conservation relations in real-world process systems.

Abstract

In the context of proxy modeling for process systems, traditional data-driven deep learning approaches frequently encounter significant challenges, such as substantial training costs induced by large amounts of data, and limited generalization capabilities. As a promising alternative, physics-aware models incorporate partial physics knowledge to ameliorate these challenges. Although demonstrating efficacy, they fall short in terms of exploration depth and universality. To address these shortcomings, we introduce a physics-aware proxy model (PAPM) that fully incorporates partial prior physics of process systems, which includes multiple input conditions and the general form of conservation relations, resulting in better out-of-sample generalization. Additionally, PAPM contains a holistic temporal-spatial stepping module for flexible adaptation across various process systems. Through systematic comparisons with state-of-the-art pure data-driven and physics-aware models across five two-dimensional benchmarks in nine generalization tasks, PAPM notably achieves an average performance improvement of 6.7%, while requiring fewer FLOPs, and just 1% of the parameters compared to the prior leading method. The code is available at https://github.com/pengwei07/PAPM.

Figures (10)

• Figure 1: Overview of the PAPM's pipeline. The model takes the multiple conditions of process systems for time extrapolation and outputs solutions at an arbitrary time point. The core is the temporal-spatial stepping module (TSSM) $(\boldsymbol{U}^{t=i}\rightarrow\boldsymbol{U}^{t=i+1})$. Spatially, a structure-preserved operation aligns with the specific equation characteristics of different process systems. Temporally, it utilizes a continuous-time modeling framework through an ODE solver.
• Figure 2: A detailed structure of PAPM at time $t$. Here, $v$, $h_O$, and $h_F$ are the corresponding unknown mapping, and the neural networks are needed for learning. We propose a temporal-spatial stepping module (TSSM) for DF, CF, IST, and EST in section \ref{['tssm_all']}, which aligns with the distinct equation characteristics of different process systems.
• Figure 3: Left: Pre-defined convolutional kernels, where fixed and trainable correspond to the matrices at the top and bottom, respectively. The bottom kernels approximate the unidirectional convection (upwind scheme) and directionless diffusion (central scheme). Symbols $\bm{\ast}$ and $\bm{\star}$ indicate trainable parameters corresponding to the upper triangular and symmetric matrices. Mid: Structure-preserved localized operator. Right: Structure-preserved spatial operator.
• Figure 4: $\epsilon$ (Eq. 4) of predicted each time step on Burgers2d, where in is the same as the training, out is the time extrapolation.
• Figure 5: Predicted flow velocity ($|\boldsymbol{u}|_2$) snapshots by FNO, PPNN, and PAPM (Ours) vs. Ground Truth (GT) on NSM2d dataset in T Ext. task.