分段多项式有限元函数的两隐层多层感知机精确表示

中文摘要 AI 辅助

我们构造了任意空间维数单纯形网格上任意多项式次数的有限元函数的两隐层多层感知机（MLP）的精确表示。第一隐层使用 $\u0000mathrm{ReLU}^0+\u0000mathrm{ReLU}$，第二隐层使用依赖于次数的激活函数 $\u0000mathrm{ReLU}^k$。该神经网络可以实现不连续有限元函数的零骨架和指定骨架表示，其中前者在网格骨架上消失，后者在每个相对开放的低维网格子单纯形上独立指定多项式数据；特别地，连续有限元函数可以在闭域上逐点表示。在连续情形下，半开分解通过将每个子单纯形分配给一个单一关联单元来减少第二隐层的宽度。对于固定的空间维数、多项式次数和输出维数，非零参数的数量随单元数量线性增长，并且在网格形状正则的连续情形下，其数量级与有限元空间的全局自由度数量相同。神经网络中的所有参数都可以通过网格和有限元函数的信息显式计算，无需训练。

英文摘要

We construct exact representations of finite element functions of arbitrary polynomial degree on simplicial meshes in any spatial dimension by two-hidden-layer multilayer perceptrons (MLPs). The first hidden layer uses $\mathrm{ReLU}^0+\mathrm{ReLU}$, and the second uses the degree-dependent activation $\mathrm{ReLU}^k$. The neural networks can realize both zero-skeleton and prescribed-skeleton representatives of discontinuous finite element functions, where the former vanish on the mesh skeleton and the latter admit independently prescribed polynomial data on each relatively open lower-dimensional mesh subsimplex; in particular, continuous finite element functions can be represented pointwise on the closed domain. In the continuous case, a half-open decomposition reduces the second-hidden-layer width by assigning each subsimplex to a single incident element. The number of nonzero parameters grows linearly with the number of elements for fixed spatial dimension, polynomial degree, and output dimension, and it is of the same order as the number of global degrees of freedom of the finite element space in the continuous case, provided that the mesh is shape-regular. All parameters in the neural network can be explicitly computed via the information of the mesh and finite element function without training.

发表机构

• Peking University（北京大学）
• Chongqing Research Institute of Big Data, Peking University（北京大学重庆大数据研究院）

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