Analogue resistive memory circuit solves linear algebra problems in one step
In March 2019, Daniele Ielmini and colleagues at Politecnico di Milano published results in PNAS demonstrating a crosspoint resistive-memory circuit that solves linear systems, matrix eigenvector problems, and differential equations by physical analogue relaxation, substantially reducing the energy and latency costs of conventional iterative digital solvers.
Background
Most computation happens digitally: a processor fetches numbers from memory, performs arithmetic on them, then writes the result back. That back-and-forth is fine for many tasks, but it becomes a serious bottleneck when the numbers involved are large matrices. Solving a linear system, finding the eigenvectors of a matrix, or integrating a set of differential equations all require many repeated arithmetic steps, and each step costs time and energy. The bigger the matrix, the more iterations a conventional solver needs before the answer converges.
Researchers had been looking at analogue circuits as a way around this. The idea was not new: analogue computers predate digital ones. But analogue hardware had largely been set aside because it is noisy and hard to control precisely. What changed the calculation, so to speak, was the arrival of resistive random-access memory, or RRAM, sometimes called a memristor. An RRAM device is a component whose electrical resistance can be set and later changed by applying a voltage. Arrange a grid of these devices into a crossbar array and you have a structure where currents flow according to Ohm’s law and sum at each wire junction according to Kirchhoff’s current law. Those two physical laws together perform a matrix-vector multiplication automatically, just by letting current flow.
The question was whether a circuit built from these devices could do something more than multiply. Solving a linear system or finding an eigenvector requires the circuit to settle on an answer, not just pass signals through once. Nobody had shown, with a real fabricated circuit, that RRAM crossbars could handle that class of problem reliably.
What happened
In March 2019, Daniele Ielmini and colleagues published their results in the Proceedings of the National Academy of Sciences. The team included Sebastian Ambrogio, Pritish Narayanan, Charles Mackin, Matthew J. Sherwood, and An Chen from IBM Research Almaden, alongside Geoffrey W. Burr, also at IBM. Ielmini’s group at Politecnico di Milano led the work, with support from the National Science Foundation.
They built a crosspoint circuit using RRAM devices and connected it in a feedback loop. When the circuit was presented with a problem encoded as conductance values in the array, current began to flow, voltages adjusted, and the system settled into a stable state. That equilibrium state was the answer. The process is physical relaxation: the circuit does not execute a sequence of arithmetic instructions but instead reaches its solution the way a mechanical system finds its lowest-energy configuration. For linear systems, matrix eigenvector problems, and differential equations, the team showed this approach worked on real hardware, not just in simulation.
The energy and time costs compared favourably with conventional iterative digital solvers, which must perform many sequential operations to reach the same answer. The analogue circuit arrived at its solution through the physics of the crossbar rather than through arithmetic, which is where the efficiency gain comes from. Ielmini and his colleagues were careful to describe the process as physical equilibration rather than instant single-step computation: the circuit still takes time to settle, but that time is set by the RC time constants of the hardware, not by a loop running in software.
The result was a demonstration on fabricated hardware of a general class of mathematical problems being solved by an analogue resistive crossbar. At that point it remained a laboratory result, not a deployed system, but it was the kind of concrete experimental evidence that moves a field from theoretical proposal to something engineers can build on.
Why it mattered
The work showed that analogue resistive crossbar arrays could exploit Ohm's law and Kirchhoff's current law to perform matrix-vector operations directly in hardware, reaching a solution through physical equilibration rather than sequential arithmetic. This offered a potential route to energy-efficient acceleration of the linear-algebra kernels that underpin neural-network inference and scientific computing. It provided early experimental evidence that memristive devices could serve as general-purpose analogue co-processors rather than merely as storage elements.
People
Daniele Ielmini, Sebastian Ambrogio, Pritish Narayanan, Charles Mackin, Matthew J Sherwood, An Chen, Geoffrey W Burr
Organisations
Polytechnic University of Milan, IBM Research Almaden, National Science Foundation
Sources
- Equivalent-accuracy accelerated neural-network training using analogue memory.Proceedings of the National Academy of Sciences.Primary source
- PNAS vol. 116 no. 10 – Ielmini et al. analogue resistive memory circuit.Proceedings of the National Academy of Sciences.Primary source
- In-memory computing with resistive switching devices.Nature Electronics.Secondary
Cite this page
AI Achievements. (2019). Analogue resistive memory circuit solves linear algebra problems in one step. Retrieved 2026-08-22, from https://achievements.ai/milestone/politecnico-di-milano-algebraic-solutions
@misc{achievements_politecnico_di_milano_algebraic_solutions,
title = {Analogue resistive memory circuit solves linear algebra problems in one step},
author = {{AI Achievements}},
year = {2019},
url = {https://achievements.ai/milestone/politecnico-di-milano-algebraic-solutions}
}