We present a comprehensive framework for deriving rigorous and efficient bounds on the approximation error of deep neural networks in PDE models characterized by branching mechanisms, such as waves, Schr\"odinger equations, and other dispersive models. This framework utilizes the probabilistic setting established by Henry-Labord\`ere and Touzi. We illustrate this approach by providing rigorous bounds on the approximation error for both linear and nonlinear waves in physical dimensions $d=1,2,3$, and analyze their respective computational costs starting from time zero. We investigate two key scenarios: one involving a linear perturbative source term, and another focusing on pure nonlinear internal interactions.

翻译：本文提出了一个全面的框架，用于推导深度神经网络在具有分支机制的偏微分方程模型（如波动方程、薛定谔方程及其他色散模型）中逼近误差的严格且高效的界。该框架利用了Henry-Labordère和Touzi建立的概率论设定。我们通过为物理维度$d=1,2,3$中的线性和非线性波提供严格的逼近误差界来阐明此方法，并从初始时刻开始分析其相应的计算成本。我们研究了两种关键情形：一种涉及线性微扰源项，另一种则聚焦于纯非线性内部相互作用。