We study shallow and deep neural networks whose inputs range over a general topological space. The model is built from a prescribed family of continuous feature maps and reduces to multilayer feedforward networks in the Euclidean case. We focus on the universal approximation property and establish general conditions under which such networks are dense in spaces of continuous vector-valued functions on arbitrary topological spaces and, in particular, locally convex spaces. Universality results obtained in the arbitrary-width case extend classical approximation theorems to non-Euclidean spaces. We also consider the deep narrow setting, in which the width of each hidden layer is uniformly bounded while the depth is allowed to grow. We identify conditions under which such networks retain the universal approximation property. As a concrete example, we employ Ostrand's extension of the Kolmogorov superposition theorem to derive an explicit universality result for products of compact metric spaces, with width bounds expressed in terms of topological dimension.

翻译：我们研究输入范围为一般拓扑空间的浅层和深度神经网络。该模型由一组给定的连续特征映射构建，并在欧几里得情形下退化为多层前馈网络。我们聚焦于普适逼近性质，建立了此类网络在任意拓扑空间（特别是局部凸空间）上的连续向量值函数空间中稠密的通用条件。在任意宽度情形下得到的普适性结果将经典逼近定理推广至非欧几里得空间。我们还考虑了深度窄网络设置，其中每个隐藏层的宽度一致有界而深度允许增长。我们识别出此类网络保持普适逼近性质的条件。作为具体实例，我们利用科尔莫戈罗夫叠加定理的奥斯特兰德推广，为紧度量空间的乘积导出了一个显式普适性结果，其宽度界限以拓扑维数表示。