**Soufiane Abid, Khalid Atifi, El-Hassan Essoufi**

##
An Inverse Backward Problem for a Heat Equation with a Memory Term, Solved by a
Deep Learning Method

**abstract:**

Our goal of this paper is the identification of an initial condition in a heat
equation that contains a memory term from final data. To this aim, we first
establish the well-posedness of the direct problem. Then we prove the continuity
and the G-derivability of the cost function. Finally, we validate the results
numerically by using a deep neural network. Our algorithm is meshfree.

**Mathematics Subject Classification:**
35R30, 35Q93, 35R09, 68T07

**Key words and phrases:** Inverse problem, memory term, optimization, deep
learning