Approximations with deep neural networks in Sobolev time-space
This work addresses the problem of approximating solutions of evolution equations for researchers working with Bochner-Sobolev spaces, offering an incremental improvement in the theoretical foundation of neural network approximations.
This paper develops a framework demonstrating that deep neural networks can approximate Sobolev-regular functions within Bochner-Sobolev spaces. By utilizing the Rectified Cubic Unit (ReCU) as an activation function, the authors achieve approximation results while circumventing regularity issues associated with the commonly used Rectified Linear Unit (ReLU).
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.