Inverse Problems for Tumour Growth Models and Neural ODEs

Sammanfattning: This thesis concerns the application of methods and techniques from the theory of inverse problems and differential equations to study models arising in the areas of mathematical oncology and deep learning. The first problem studied is to develop methods to perform numerical simulations with full 3-dimensional brain imaging data of reaction-diffusion models for tumour growth forwards as well as backwards in time with the goal of enabling the numerical reconstruction of the source of the tumour given an image (or similar data) at a later stage in time of the tumour. This inverse ill-posed problem is solved as a sequence of well-posed forward problems using the nonlinear Landweber regularization method. Such models and method allow to generate realistic synthetic medical images that can be used for data augmentation. Mathematical analysis of the problems solved as well as establishing uniqueness of the source are presented. The second problem includes a novel method allowing training self-contained neural ordinary differential equation networks (termed standalone NODEs) via a nonlinear conjugate gradient method, where the Sobolev gradient can be incorporated to improve smoothness of model weights. Relevant functions spaces are introduced, the adjoint problems with the needed gradients are calculated and the robustness is studied. The developed framework has many advantages in that it can incorporate relevant dynamics from physical models as well as help to understand more on how neural networks actually work and how sensitive they are to natural and adversarial perturbations. Combination of the two main problems will allow for example the training of neural networks to identify tumours in real imaging data.