Neural Signature Kernels

近年来，微分方程组和深度学习之间的共生成为一个活跃的研究领域，特别是通过引入名为神经微分方程组的混合模型。事实上，许多标准的神经网络结构可以解释为对某些微分方程式的近似。在序列数据深度学习的背景下，有限宽度随机神经网络被非正式地识别为神经控制微分方程的近似，其灵感来源于粗分析中研究的同名动力系统。我的研究项目的主要目标是为随机初始化的受控ResNet在大宽度和大深度区域的行为提供严格的数学分析。

The symbiosis between differential equations and deep learning has become an active research area in recent years, notably through the introduction of hybrid models named neural differential equations. In fact, many standard neural network architectures may be interpreted as approximations to some differential equations. This approximation has been treated rigorously for finite-width ResNets, which in the infinite-depth limit converge in distribution to zero-drift neural stochastic differential equations (Neural SDEs) with diffusion depending on the choice of activation function.In the context of deep learning for sequential data, finite-width RNNs have been informally identified as approximations to neural controlled differential equations (Neural CDEs) inspired from the homonymous class of dynamical systems studied in rough analysis. The main goal of my research project is to provide a rigorous mathematical analysis of the behaviour of randomly initialised controlled ResNets in the large width and depth regimes.