Efficient emulation of reaction-diffusion systems using SciML

This thesis will focus on the development of efficient emulators of reaction-diffusion systems defined on a network, using Scientific Machine Learning (SciML) architectures, and with specific applications to computational neuroscience. In the first part of the thesis, we will test the viability of a SciML approach in capturing the evolution of a biologically realistic neural network under external perturbations. Previous work has verified the effectiveness of operator neural networks in reproducing the output of a single neuron under electrical stimulation in the FitzHugh-Nagumo model. This setup will be extended by including a whole network of interacting neurons between the input (stimulated) and the output (recorded) neuron. Upon characterizing the accuracy of prediction as a function of the model's parameters, we will test the method on a real data set. In the second part of the thesis, we will exploit ML-based tools to approximate the evolution of a diffusion-reaction system on a network. In particular, we will test the accuracy improvements afforded by re-formulating the problem in the eigenbasis of the network Laplacian, and apply the method to describe the evolution of a biologically realistic neural network model.