Physics Informed Neural Networks and Neural Tangent Kernel: preliminary results for parametric Optimal Control Problems

Physics-Informed Neural Networks (PINNs) offer a promising framework for solving differential problems, including Partial Differential Equations (PDEs) and Optimal Control Problems (OCPs). They have also been explored in parametric settings, which involves PDEs that depend on a set of parameters. In this context, the objective is to create a framework capable of efficiently generating numerical approximations of the solution when the parameter input of the differential problem changes. In fact, standard numerical methods can be too time-consuming to solve the problem in real-time and for many parameters. This thesis focuses on applying PINNs parametric OCPs. The first contribution is the improvement of the performances of standard PINNs on two test cases already investigated in the literature: a parametric Elliptic OCP and a parametric Stokes OCP. Improvements have been achieved by studying different aspects such as the sampling techniques, the use of an alternative architecture, named PIARCH, and the strong enforcing of the Dirichlet boundary conditions. Although results should still be improved, we enhanced the performance, for both the problems. To better study the training phase capabilities, we focused on the theory of Neural Tangent Kernel (NTK), which is a matrix that describes how the training of a generic neural network evolves. The theory behind NTK helps explain why neural networks, and specifically PINNs, often fail to train. It has been proven that, while neural networks can easily learn low-frequency components of the training data, they struggle to learn higher-frequency components, a challenge known as Spectral Bias. To address this issue, we study two approaches: augmenting the input of the PINN and applying an adaptive balancing of the loss function. After evaluating these approaches on one-dimensional uncontrolled problems, we applied them to two parametric OCPs we investigated to improve their performances.