In this paper, we introduce the Spectral Coefficient Learning via Operator Network (SCLON), a novel operator learning-based approach for solving parametric partial differential equations (PDEs) without the need for data harnessing. The cornerstone of our method is the spectral methodology that employs expansions using orthogonal functions, such as Fourier series and Legendre polynomials, enabling accurate PDE solutions with fewer grid points. By merging the merits of spectral methods - encompassing high accuracy, efficiency, generalization, and the exact fulfillment of boundary conditions - with the prowess of deep neural networks, SCLON offers a transformative strategy. Our approach not only eliminates the need for paired input-output training data, which typically requires extensive numerical computations, but also effectively learns and predicts solutions of complex parametric PDEs, ranging from singularly perturbed convection-diffusion equations to the Navier-Stokes equations. The proposed framework demonstrates superior performance compared to existing scientific machine learning techniques, offering solutions for multiple instances of parametric PDEs without harnessing data. The mathematical framework is robust and reliable, with a well-developed loss function derived from the weak formulation, ensuring accurate approximation of solutions while exactly satisfying boundary conditions. The method's efficacy is further illustrated through its ability to accurately predict intricate natural behaviors like the Kolmogorov flow and boundary layers. In essence, our work pioneers a compelling avenue for parametric PDE solutions, serving as a bridge between traditional numerical methodologies and cutting-edge machine learning techniques in the realm of scientific computation.

翻译：本文介绍了一种基于算子网络学习的谱系数学习方法（SCLON），这是一种无需数据驱动的参数偏微分方程（PDE）求解新范式。该方法的核心在于采用正交函数展开的谱方法，如傅里叶级数和勒让德多项式，从而以更少的网格点实现高精度PDE求解。通过融合谱方法的高精度、高效性、泛化能力及严格满足边界条件等优势与深度神经网络的强大性能，SCLON提出了一种变革性策略。该方法不仅免除了需要大量数值计算才能生成的成对输入-输出训练数据，还能有效学习并预测奇异摄动对流扩散方程至纳维-斯托克斯方程等复杂参数PDE的解。与现有科学机器学习技术相比，所提框架展现了卓越性能，可在无需数据驱动的情况下求解多实例参数PDE。其数学框架稳健可靠，基于弱形式推导的损失函数兼具良好理论性质，能在精确满足边界条件的同时保证解的高精度近似。通过准确预测科尔莫戈罗夫流与边界层等复杂自然现象，进一步验证了该方法的有效性。本质上，本研究开创了参数PDE求解的新路径，构建了传统数值方法与前沿机器学习技术在科学计算领域的桥梁。