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.

翻译：机器学习已成为求解微分方程(DEs)的变革性工具，但主流方法仍受双重局限：数据驱动方法需要昂贵的标注数据集，而模型驱动技术面临效率与精度的权衡。本文提出数学人工数据(MAD)框架——一种融合物理定律与数据驱动学习的新范式，旨在促进大规模算子发现。通过利用微分方程固有数学结构生成物理嵌入的解析解及相关合成数据，MAD从根本上消除了对实验或模拟训练数据的依赖。这使得跨多参数系统的算子学习在保持数学严谨性的同时实现计算高效性。通过涵盖边界值和源项均为函数的二维参数化问题的数值演示，我们展示了MAD在不同微分方程场景中的泛化能力及优越的效率/精度表现。这种物理嵌入-数据驱动的框架及其处理复杂参数空间的能力，使其有望成为科学计算中物理信息机器智能的通用范式。