Solving Dynamic Inverse Problems with Physics-Informed Neural Networks.

这项研究依赖于两大数学领域：逆问题和深度学习。在反问题中，人们通常对从噪声测量中重建某些量感兴趣。一些例子是去模糊或去噪图像，例如，在显微镜的背景下，或从间接测量中重建图像，例如在医学成像的背景下的计算机断层扫描。然而，世界上的大多数进程都在进行中。例如，在心脏周期期间对心脏进行成像，收集到的图像受到心肌运动的影响，但也受到患者呼吸的影响，或者在天气预报中，对大气的测量与风和云的运动和动力学有关。本研究的重点是逆问题，其中兴趣的数量是动态的，并随着时间的推移而变化。因此，需要在几个时间步骤中重建所需的数量。然而，我们不仅对图像感兴趣，而且对过程的潜在运动也感兴趣，这可以通过速度场来建模。因此，这两个量需要共同重构。在逆问题中，通常需要通过正则子来包含关于这些量的先验信息。我们的公式不仅允许在图像上下文中施加常用的正则子，还允许通过速度场施加物理约束，从而支持图像和动力学的最终重建。为了在数值上解决这一问题，提出使用物理信息神经网络（pinn），这是一种新的框架，通过利用自动微分计算微分算子，将非线性偏微分方程和深度学习的世界联系起来，但它们也可能克服维数的诅咒，这对大规模问题至关重要。本研究的主要目标有两个：从理论的角度来看，我们想要证明考虑三维域和正演算子的不同性质的逆问题解的存在性。还需要证明其在噪声存在下的稳定性。稳定性是相关的，因为它可以帮助我们在不准确测量的情况下量化重建的误差。从数值的角度来看，目标是通过开发受数学知识启发的神经网络新架构来利用pinn的思想，这样架构本身就可以作为正则器，从而可以产生更准确的预测。

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.