用于高维偏微分方程的下确界-上确界神经网络
Inf-Sup Neural Networks for High Dimensional PDEs
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中文总结 AI 辅助
针对高维偏微分方程求解难题，提出基于神经网络框架，将其转化为下确界-上确界优化问题，通过两个网络参数化并经迭代鞍点优化计算，证明理论等价性并推导误差估计，数值实验验证了该方法求解高维PDEs的性能。
中文摘要 AI 辅助
由于维度诅咒，求解高维偏微分方程（PDEs）仍然具有挑战性。我们提出了一个基于神经网络的框架，通过引入拉格朗日乘数将PDEs重新表述为下确界-上确界优化问题。原始解和相关的拉格朗日乘数由两个网络参数化，并通过迭代鞍点优化过程计算。我们证明了所提出的优化公式与原始PDE问题之间的理论等价性，并推导了严格的误差估计，以网络近似误差、统计（采样）误差和优化误差来量化总近似误差。数值实验证明了该方法求解高维PDEs的准确性、稳定性和效率。
英文摘要
Solving partial differential equations (PDEs) in high dimensions remains challenging due to the curse of dimensionality. We propose a neural-network-based framework that reformulates PDEs as inf--sup optimization problems through the introduction of a Lagrange multiplier. The primal solution and the associated Lagrange multiplier are parameterized by two networks and are computed via an iterative saddle-point optimization procedure. We prove the theoretical equivalence between the proposed optimization formulation and the original PDE problem, and we derive rigorous error estimates that quantify the total approximation error in terms of the network approximation error, statistical (sampling) error, and optimization error. Numerical experiments demonstrate the accuracy, stability, and efficiency of the proposed method for solving high-dimensional PDEs.
