An Efficient Particle-Field Algorithm with Neural Interpolation based on a Parabolic-Hyperbolic Chemotaxis System in 3D
For researchers studying tumor angiogenesis and chemotaxis, this work provides a more efficient numerical method for 3D simulations, though it is incremental as it combines existing techniques.
The authors developed a mesh-free particle-based neural network algorithm (NSIPF) for simulating a parabolic-hyperbolic Keller-Segel chemotaxis system in 3D. The method achieves faster computation than classical finite difference and SIPF methods while preserving mass and nonnegativity.
Tumor angiogenesis involves a collection of tumor cells moving towards blood vessels for nutrients to grow. Angiogenesis, and in general chemotaxis systems have been modeled using partial differential equations (PDEs) and as such require numerical methods to approximate their solutions in 3 space dimensions (3D). This is an expensive computation when solutions develop large gradients at unknown locations, and so efficient algorithms to capture the main dynamical behavior are valuable. Here as a case study, we consider a parabolic-hyperbolic Keller-Segel (PHKS) system in the angiogenesis literature, and develop a mesh-free particle-based neural network algorithm that scales better to 3D than traditional mesh based solvers. From a regularized approximation of PHKS, we derive a neural stochastic interacting particle-field (NSIPF) algorithm where the bacterial density is represented as empirical measures of particles and the field variable (concentration of chemo-attractant) by a convolutional neural network (CNN) trained on low cost synthetic data. As a new model, NSIPF preserves total mass and nonnegativity of the density, and captures the dynamics of 3D multi-bump solutions at much faster speeds compared with classical finite difference (FD) and SIPF methods.