Solving high dimensional partial differential equations (PDEs) has historically posed a considerable challenge when utilizing conventional numerical methods, such as those involving domain meshes. Recent advancements in the field have seen the emergence of neural PDE solvers, leveraging deep networks to effectively tackle high dimensional PDE problems. This study introduces Inf-SupNet, a model-based unsupervised learning approach designed to acquire solutions for a specific category of elliptic PDEs. The fundamental concept behind Inf-SupNet involves incorporating the inf-sup formulation of the underlying PDE into the loss function. The analysis reveals that the global solution error can be bounded by the sum of three distinct errors: the numerical integration error, the duality gap of the loss function (training error), and the neural network approximation error for functions within Sobolev spaces. To validate the efficacy of the proposed method, numerical experiments conducted in high dimensions demonstrate its stability and accuracy across various boundary conditions, as well as for both semi-linear and nonlinear PDEs.

翻译：求解高维偏微分方程（PDEs）历来对基于域网格的传统数值方法构成重大挑战。近年来，该领域涌现出利用深度网络有效处理高维PDE问题的神经PDE求解器。本研究提出Inf-SupNet，一种基于模型的无监督学习方法，专门用于求解特定类别椭圆型PDE的解。其核心思想在于将底层PDE的inf-sup公式嵌入损失函数。理论分析表明，全局解误差可由三项不同误差之和界定：数值积分误差、损失函数的对偶间隙（训练误差）以及Sobolev空间中函数逼近的神经网络近似误差。为验证所提方法的有效性，在高维条件下开展的数值实验证实，该方法在多种边界条件、半线性及非线性PDE场景下均展现出良好的稳定性和精度。