  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.
