这篇论文很硬核：用一个固定 RNN 就能逼近任意连续函数，运行越长越准，像图灵机一样。
本文证明：对于 [-1,1] 上的任意连续函数，存在一个固定的 ReLU RNN（隐层维度固定、权重固定），通过延长运行时间即可实现一致逼近。核心创新在于引入中间模型 TMNU（Turing machine with neural units），它保留了实现多项式逼近方案的算法自由度，同时能被隐维度和权重大小有明确上界的 RNN 模拟。得到的收敛速率与底层多项式逼近率对应。本文还给出了极小极大下界，证明运行时间是该固定网络逼近范式中不可避免的资源。
Recurrent neural networks approximate continuous functions
Classical approximation theorems ask for a new neural network whenever the target accuracy is improved. This paper studies the opposite possibility: can the network be chosen once and for all, and can accuracy be bought only by letting it run longer? We prove that this is possible for every continuous function on [-1,1]. More precisely, each such function is uniformly approximated by the time evolution of a single ReLU recurrent neural network with fixed weights and fixed hidden dimension. The mechanism behind the construction is a new intermediate model, the Turing machine with neural units (TMNU). This model retains the algorithmic freedom needed to implement polynomial approximation schemes, while remaining rigid enough to be simulated by RNNs with explicit bounds on hidden dimension and weight magnitude. The resulting convergence rates reflect the underlying polynomial approximation rates. We complement the construction with minimax lower bounds showing that runtime is not merely a proof artifact, but an unavoidable resource in this fixed-network approximation paradigm.