摘要：We propose a neural network method inspired by the tailored finite point method (TFPM) for singularly perturbed eigenvalue problems (SPEPs). A TFPM-informed loss enforces continuity of the solution and its derivative across interior interfaces, as well as the boundary conditions and the normalization constraint. We also derive an error bound for the approximation, establish a stability estimate, and prove a convergence theorem for the proposed approach. Numerical experiments on representative potentials demonstrate the accuracy of the computed eigenpairs in the small-regime and the method’s ability to resolve near-degenerate spectra. Comparisons with several neural-network-based eigenvalue solvers further show the robustness of the proposed method in the singularly perturbed regime.

个人简介：黄忠亿，清华大学长聘教授，清华大学数学科学系党委副书记。1999年博士毕业于清华大学数学科学系。2026年当选为中国工业与应用数学学会会士。主要从事科学与工程计算领域中多尺度建模、分析与数值方法研究，在传统科学计算理论与机器学习相融合的计算方法研究方面取得一系列原创性成果。曾获北京市科学技术二等奖（2002）、Pieter Hemker奖（2008）、CSIAM应用数学落地成果认证（2025），获国家优秀青年基金（2013）、国家杰出青年基金（2020）资助。