Spatially Embedded Neural Networks: Scalar-Field Updating via a Dual PDE-Optimised Point Process

The purpose of this research note is to describe a signal-dependent point process, optimised by a two-stage scalar-field update. The proposed model utilises a stochastic partial differential equation (SPDE) to ’collapse’ the scalar field on RD into a discrete non-i.i.d. set of points. Then, an updating PDE transforms the initial scalar field based on the system’s task-reward signal. The update equation utilises a reward-based reaction-diffusion equation. This dual PDE approach constitutes a complete learning loop that can run in continuous learning settings.