Title: Solving sparse finite element problems on neuromorphic hardware

Abstract: Neuromorphic computing’s relation to applied math has been relatively unexplored. This talk will present a brain-inspired, spiking, neuromorphic algorithm for solving sparse linear systems (Ax = b) such as those arising in Finite Element methods for solving partial differential equations (PDEs), one of the most important techniques in modern numerical science and engineering. The algorithm embeds the sparse matrix into the synaptic connections between subpopulations of neurons directly without training or learning. Neural dynamics are defined such that the collective spiking activity of the whole network flows to an efficient spiking representation of the solution vector x, with comparable numerical accuracy to traditional algorithms. We demonstrate this algorithm on real neuromorphic hardware (Intel’s Loihi 2) and show close to ideal strong and weak scaling. We demonstrate the generality of the algorithm through several examples of PDEs in 2 and 3 dimensions, with nontrivial mesh topologies, and different boundary conditions. Our work establishes a direct connection between established numerical methods for PDEs and brain-like spiking neural networks, demonstrating the value of brain inspiration, and expanding the neuromorphic footprint in scientific computing. Further details are available in our preprint: https://arxiv.org/abs/2501.10526