We show how continuous-depth neural ODE models can be framed as single-layer, infinite-width nets using the Chen--Fliess series expansion for nonlinear ODEs. In this net, the output "weights" are taken from the signature of the control input -- a tool used to represent infinite-dimensional paths as a sequence of tensors -- which comprises iterated integrals of the control input over a simplex. The "features" are taken to be iterated Lie derivatives of the output function with respect to the vector fields in the controlled ODE model. The main result of this work applies this framework to derive compact expressions for the Rademacher complexity of ODE models that map an initial condition to a scalar output at some terminal time. The result leverages the straightforward analysis afforded by single-layer architectures. We conclude with some examples instantiating the bound for some specific systems and discuss potential follow-up work.

翻译：本文展示了如何利用非线性ODE的Chen-Fliess级数展开，将连续深度神经ODE模型重构为单层无限宽网络。在该网络中，输出"权重"取自控制输入的签名（一种将无限维路径表示为张量序列的工具），该签名包含控制输入在单纯形上的迭代积分。"特征"则取为受控ODE模型中输出函数关于向量场的迭代李导数。本研究的主要成果是应用该框架推导出将初始条件映射至终端时刻标量输出的ODE模型的Rademacher复杂度的紧凑表达式。该结果利用了单层架构所赋予的简洁分析优势。最后，我们通过若干具体系统实例验证该界限的实用性，并讨论潜在的后续研究方向。