Solving Dynamic Inverse Problems with Physics-Informed Neural Networks.

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This research relies on two big mathematical areas: inverse problems and deep learning. In inverse problems, one is typically interested in the reconstruction of some quantity from noisy measurements. Some examples are deblurring or denoising images, for instance, in the context of microscopes, or reconstructing images from indirect measurements such as in computed tomography in the context of medical imaging. However, most of the processes in the world are in motion. This is the case, for instance, of imaging the heart during the cardiac cycle, where the collected images are affected by the motion of the myocardium but also by the breathing of the patient, or in weather forecasting where the measurements of the atmosphere are related to the movement and dynamics of the wind and clouds.This research is focused on inverse problems where the quantity of interest is dynamic and changes over time. As a consequence, the desired quantity needs to be reconstructed at several time steps. However, we are not only interested in the images, but also in the underlying motion of the process, which can be modelled through a velocity field. In consequence, both quantities need to be reconstructed jointly. In inverse problems, it is always the case that a priori information about these quantities needs to be incorporated through regularisers. Our formulation will allow imposing not only commonly used regularisers in the context of images but also physical constraints through the velocity field that can endorse the final reconstruction of both the image and the dynamics.For solving this problem numerically, it is proposed to use Physics-Informed Neural Networks (PINNs), a new framework that connects the worlds of non-linear Partial Differential Equations and Deep Learning by taking advantage of automatic differentiation to compute differential operators, but also they might overcome the curse of dimensionality which becomes critical for large scale problems.The main goals of this research are twofold: from a theoretical point of view, we want to prove the existence of solutions for the inverse problem considering both a three-dimensional domain and different properties of the forward operator. It would be also desirable to prove its stability with respect to the presence of noise. Stability is relevant as it can help to quantify the error in our reconstruction in the presence of inaccurate measurements. From a numerical point of view, the objective is to exploit the idea of PINNs by developing new architectures for neural networks inspired by our mathematical knowledge such that the architecture can act as a regulariser by itself, which can lead to more accurate predictions.