用数学构造数据，让机器学习高效求解微分方程
Mathematical artificial data for operator learning
- 基于微分方程的数学结构生成含物理规律的合成数据
- 在2D多参数问题中实现高精度、高效率的算子学习
- 适合需要大规模物理信息建模的科学计算场景
机器学习已成求解微分方程（DEs）的关键工具，但现有方法受限于双重瓶颈：数据驱动方法依赖昂贵标注数据，模型驱动方法存在效率与精度权衡。本文提出数学人工数据（MAD）框架，融合物理定律与数据驱动学习，实现大规模算子发现。通过利用微分方程内在数学结构生成嵌入物理规律的解析解及合成数据，MAD彻底摆脱对实验或仿真训练数据的依赖。该方法在2D参数化问题中表现出优异泛化能力，即使边界值和源项均为函数时，仍能保持高计算效率与高精度。此物理嵌入式数据驱动框架具备处理复杂参数空间的能力，有望成为科学计算中物理信息机器智能的通用范式。
原文摘要 · Abstract (English)
Machine learning has emerged as a transformative tool for solving differential equations (DEs), yet prevailing methodologies remain constrained by dual limitations: data-driven methods demand costly labeled datasets while model-driven techniques face efficiency-accuracy trade-offs. We present the Mathematical Artificial Data (MAD) framework, a new paradigm that integrates physical laws with data-driven learning to facilitate large-scale operator discovery. By exploiting DEs' intrinsic mathematical structure to generate physics-embedded analytical solutions and associated synthetic data, MAD fundamentally eliminates dependence on experimental or simulated training data. This enables computationally efficient operator learning across multi-parameter systems while maintaining mathematical rigor. Through numerical demonstrations spanning 2D parametric problems where both the boundary values and source term are functions, we showcase MAD's generalizability and superior efficiency/accuracy across various DE scenarios. This physics-embedded-data-driven framework and its capacity to handle complex parameter spaces gives it the potential to become a universal paradigm for physics-informed machine intelligence in scientific computing.
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