Fully analog iteration for solving matrix equations with in-memory computing

Memristive in-memory computing has demonstrated potential for solving matrix equations in scientific computing. However, the inherent inaccuracies of analog mechanisms create challenges in achieving high-precision solutions while maintaining low-energy consumption. This study introduces a memristive matrix equation solver that considerably accelerates solutions by performing mathematical iterations directly within an analog domain. Our approach facilitates rapid approximate solutions with a scalable circuit topology and expedites the high-precision refinement process by substantially reducing the digital-to-analog conversion overhead. We experimentally validated this methodology using a heterogeneous computing system. We performed simulations of multiple scientific problems on these circuits, including solving the diffusion equation and modeling equilibration in silicon P-N junctions. Notably, our memristive solver, combined with digital refinement, achieved software-equivalent precision (with an error of 10^-12). Compared to conventional digital processing units, this approach offered a 128-fold improvement in solution speed and a 160-fold reduction in energy consumption. This work establishes a foundation for future scientific computing using imprecise analog devices.