用模块化神经算子分解流体方程，提升模型适应性与可解释性。
Compositional Neural Operators for Multi-Dimensional Fluid Dynamics
- 将复杂方程拆解为预训练的物理模块（对流、扩散等）组合使用。
- 在三个流体方程上实现高精度求解，且可快速适配新系统。
- 适合需要可解释性和跨场景复用的物理建模研究者。
偏微分方程（PDEs）描述多种物理现象，但高精度数值解计算成本高昂，而机器学习方法泛化能力有限。尽管科学基础模型（SFMs）旨在提供通用代理，传统编码-解码方法存在预训练成本高、可解释性差的问题。本文提出二维系统的组合神经算子（CompNO），将复杂PDE分解为一组基础模块库。每个模块是针对基本物理现象预训练的专用神经算子，包含对流、扩散、非线性对流块及泊松求解器，可处理速度-压力耦合问题。通过带有聚合器的自适应块组装专家模块，聚合器通过最小化数据损失和基于控制方程的物理残差来学习非线性交互。该方法在对流-扩散方程、Burgers方程和不可压缩纳维-斯托克斯方程上验证有效，结果表明，从基础算子学习显著提升模型适应性、增强可解释性，并支持预训练模块在新物理系统中的复用。
原文摘要 · Abstract (English)
Partial differential equations (PDEs) govern diverse physical phenomena, yet high-fidelity numerical solutions are computationally expensive and Machine Learning approaches lack generalization. While Scientific Foundation Models (SFMs) aim to provide universal surrogates, typical encoding-decoding approaches suffer from high pretraining costs and limited interpretability. In this paper, we propose Compositional Neural Operators (CompNO) for 2D systems, a framework that decomposes complex PDEs into a library of Foundation Blocks. Each block is a specialized Neural Operator pretrained on elementary physics. This modular library contains convection, diffusion, and nonlinear convection blocks as well as a Poisson Solver, enabling the framework to address the pressure-velocity coupling. These experts are assembled via an Adaptation Block featuring an Aggregator. This aggregator learns nonlinear interactions by minimizing data loss and physics-based residuals driven from governing equations. The proposed approach has been evaluated on the Convection-Diffusion equation, the Burgers' equation, and the Incompressible Navier-Stokes equation. Our results demonstrate that learning from elementary operators significantly improves adaptability, enhances model interpretability and facilitates the reuse of pretrained blocks when adapting to new physical systems.
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