We propose a boundary neuron method with random features (BNM-RF) for solving partial differential equations. The method approximates the unknown boundary function by a shallow network within the boundary integral formulation. With randomly sampled and fixed hidden parameters, the computation reduces to a linear least squares problem for the output coefficients, which avoids gradient based nonconvex optimization. This construction retains the dimensionality reduction of boundary integral equations and the linear solution structure of the random feature method. For elliptic problems, we establish convergence analysis by combining kernel-based method with random feature approximation, and obtain error bounds on both the boundary and the interior solution. Numerical experiments on Laplace and Helmholtz problems, including interior and exterior cases, show that the proposed method achieves competitive accuracy relative to the boundary element method and favorable performance relative to boundary integral neural networks in the tested settings with only few neurons. Overall, the proposed method provides a practical framework for combining boundary integral equations with neural network for problems on complex geometries and unbounded domains.
Moving sample method for solving time-dependent partial differential equations
The Institute of Mathematical Sciences, ShanghaiTech University（上海科技大学数学科学学院）；Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学与科学工程计算研究所）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）；Department of Mathematics, School of Sciences, Great Bay University（大湾区大学理学院数学系）；Department of Mathematics, Center for Computation and Technology, Louisiana State University（路易斯安那州立大学计算与技术中心数学系）
Robust Parameter and State Estimation in Multiscale Neuronal Systems Using Physics-Informed Neural Networks
Department of Mathematics, University of Iowa（数学系，爱荷华大学）；Department of Mathematics, Volen National Center for Complex Systems, Brandeis University（数学系，布兰迪大学复杂系统国家中心）
Beijing International Center for Mathematical Research, Peking University（北京大学北京国际数学研究中心）；Center for Quantitative Biology, Center for Machine Learning Research, Peking University（北京大学定量生物学中心、机器学习研究中心）；Academy for Multidisciplinary Studies, Capital Normal University（首都师范大学跨学科学院）；Beijing National Center for Applied Mathematics（北京应用数学中心）
Chebyshev Spectral Neural Networks for Solving Partial Differential Equations
School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）；Institute of Natural Sciences, Shanghai Jiao Tong University（上海交通大学自然科学研究院）
