The aim of the project is to propose a measure of the degree of observability of PDEs without resorting to an explicit model reduction technique (i.e. by constructing a model leading to finite-dimensional approximations) and by restricting ourselves to linear or nonlinear parabolic PDEs. Parabolic PDEs cover a wide variety of physics problems, such as the heat equation, the advection-diffusion equation, or the Burgers equation (fluid mechanics, gas dynamics, road traffic, etc.). For this purpose, we will explore the use of PINNs (Physics-Informed Neural Networks [4]). The PINN approach is a new class of neural networks that hybridizes machine learning and physical laws. We will here focus on the advection-diffusion equation (with a source estimation problem) and the Burgers equation with kinematic viscosity (with a state estimation problem).

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