用基础物理知识提升神经算子的数据效率和泛化能力
Learning Data-Efficient and Generalizable Neural Operators via Fundamental Physics Knowledge
- 联合学习原PDE及其简化形式，融合基本物理规律
- 在多种维度的PDE问题上降低预测误差，提升跨参数泛化性
- 适合需要少样本、强泛化性的物理模拟场景
科学机器学习（SciML）的进展使神经算子（NOs）成为建模由偏微分方程（PDEs）支配的物理系统动态演化的强大代理。现有方法主要关注从目标PDE学习仿真，却忽视了其背后的更基础物理原理。受数值求解器对不同设定下PDE仿真兼容性的启发，我们提出一种多物理训练框架，联合学习原始PDE及其简化基本形式。该框架提升了数据效率，降低了预测误差，并增强了分布外（OOD）泛化能力，尤其在物理参数变化及合成到真实迁移场景中表现突出。方法架构无关，且在一系列1D/2D/3D PDE问题上均实现归一化均方根误差（nRMSE）的一致改进。大量实验表明，显式引入基础物理知识显著增强了神经算子的泛化能力。代码与模型将公开于https://sites.google.com/view/sciml-fundemental-pde。
原文摘要 · Abstract (English)
Recent advances in scientific machine learning (SciML) have enabled neural operators (NOs) to serve as powerful surrogates for modeling the dynamic evolution of physical systems governed by partial differential equations (PDEs). While existing approaches focus primarily on learning simulations from the target PDE, they often overlook more fundamental physical principles underlying these equations. Inspired by how numerical solvers are compatible with simulations of different settings of PDEs, we propose a multiphysics training framework that jointly learns from both the original PDEs and their simplified basic forms. Our framework enhances data efficiency, reduces predictive errors, and improves out-of-distribution (OOD) generalization, particularly in scenarios involving shifts of physical parameters and synthetic-to-real transfer. Our method is architecture-agnostic and demonstrates consistent improvements in normalized root mean square error (nRMSE) across a wide range of 1D/2D/3D PDE problems. Through extensive experiments, we show that explicit incorporation of fundamental physics knowledge significantly strengthens the generalization ability of neural operators. We will release models and codes at https://sites.google.com/view/sciml-fundemental-pde.
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