A Particle-Field Algorithm with Neural Interpolation for a Parabolic-Hyperbolic Chemotaxis System in 3D

Abstract: Tumor angiogenesis involves a collection of tumor cells moving towards blood vessels for nutrients to grow. Angiogenesis, and in general chemo- taxis, systems have been modeled using partial differential equations (PDEs) and as such require numerical methods to approximate their solutions. Here we study a Parabolic-Hyperbolic Keller-Segel (PHKS) system in three space dimensions. The model arises in the angiogenesis literature. To compute solutions to the PHKS system, we develop a neural stochastic interacting particle-field (NSIPF) method where the density variable is represented as empirical measures of particles and the field variable (concentration of chemoattractant) approximated by a convolutional neural network (CNN). We discuss the performance of NSIPF in computing multi-bump solutions to the system.