A mathematical framework of physics-informed neural networks for the solution of parabolic PDEs

Abstract

Partial differential equations are some of the most useful mathematical tools to describe physical phenomena. Yet useful, many partial differential equations are difficult to solve in a classical way, even the solutions are difficult to find just by using analytical methods. For this reason many numerical methods are used in order to find numerical approximations. In the last years, the interest for developing numerical methods for solving PDEs using artificial neural networks have risen, following the accessibility to more user friendly frameworks for their development using home computers. Many methods have been documented that solve PDE in its strong and variational form for parabolic problems, however, there is not many progress in using neural networks for the solution of parabolic partial differential equations. Here we make use recent ideas for the development of variational physics informed neural networks for the solution of parabolic PDEs and expand on the mathematical background that supports their robustness. We obtained an neural network architecture that via time discretization, minimizes a loss function which achieves good approximations of the solution of time dependent problems in each time-step of the approximation maintaining coherence with the main idea behind classical VPINNs and the deep Fourier method. We show the effectiveness of our method in the solution of the freezing of coffee extracts in a industrial context and make use of measurements data for validation. We anticipate our work as a introductory material for any mathematician who wants to begin working on physics informed neural networks, by including an exhaustive and rigorous treatment on all the functional analysis topics necessary to lay the foundations in a mathematical way of the subject and also for any machine learning enthusiast who needs to make a bridge between the main ideas of data driven learning and the mathematical machinery behind residual minimization methods or partial differential equations in general.