  Solutions of evolution equation generally lies in certain Bochner-Sobolev
spaces, in which the solution may has regularity and integrability properties
for the time variable that can be different for the space variables. Therefore,
in this paper, we develop a framework shows that deep neural networks can
approximate Sobolev-regular functions with respect to Bochner-Sobolev spaces.
In our work we use the so-called Rectified Cubic Unit (ReCU) as an activation
function in our networks, which allows us to deduce approximation results of
the neural networks while avoiding issues caused by the non regularity of the
most commonly used Rectivied Linear Unit (ReLU) activation function.
