Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Article
  • Published:

A general memristor-based partial differential equation solver


Memristive devices have been extensively studied for data-intensive tasks such as artificial neural networks. These types of computing tasks are considered to be ‘soft’ as they can tolerate low computing precision without suffering from performance degradation. However, ‘hard’ computing tasks, which require high precision and accurate solutions, dominate many applications and are difficult to implement with memristors because the devices normally offer low native precision and suffer from high device variability. Here we report a complete memristor-based hardware and software system that can perform high-precision computing tasks, making memristor-based in-memory computing approaches attractive for general high-performance computing environments. We experimentally implement a numerical partial differential equation solver using a tantalum oxide memristor crossbar system, which we use to solve static and time-evolving problems. We also illustrate the practical capabilities of our memristive hardware by using it to simulate an argon plasma reactor.

This is a preview of subscription content, access via your institution

Access options

Buy this article

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: High-precision PDE solver based on memristor crossbar.
Fig. 2: Hardware set-up and device measurement.
Fig. 3: Experimental demonstration of solving a Poisson’s equation.
Fig. 4: Experimental demonstration of solving a damped 2D wave equation.
Fig. 5: Argon plasma reactor simulation.

Similar content being viewed by others


  1. Simon, H., Zacharia, T. & Stevens, R. Modeling and Simulation at the Exascale for Energy and the Environment (Department of Energy Technical Report, 2007).

  2. Palmer, T. Build imprecise supercomputers. Nature 526, 32–33 (2015).

    Article  Google Scholar 

  3. Aage, N., Andreassen, E., Lazarov, B. S. & Sigmund, O. Giga-voxel computational morphogenesis for structural design. Nature 550, 84–86 (2017).

    Article  Google Scholar 

  4. Altrock, P. M., Liu, L. L. & Michor, F. The mathematics of cancer: integrating quantitative models. Nat. Rev. Cancer 15, 730–745 (2015).

    Article  Google Scholar 

  5. Bauer, P., Thorpe, A. & Brunet, G. The quiet revolution of numerical weather prediction. Nature 525, 47–55 (2015).

    Article  Google Scholar 

  6. Achdou, Y., Buera, F. J., Lasry, J.-M., Lions, P.-L. & Moll, B. Partial differential equation models in macroeconomics. Philos. Trans. R. Soc. A 372, 20130397 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  7. Dongarra, J. J. et al. The International Exascale Software Project roadmap. Int. J. High. Perform. Comput. 25, 3–60 (2011).

    Article  Google Scholar 

  8. Nair, R. Evolution of memory architecture. Proc. IEEE 103, 1331–1345 (2015).

    Article  Google Scholar 

  9. Kogge, P. et al. Exascale Computing Study: Technology Challenges in Achieving Exascale Systems (DARPA, 2008).

  10. Nair, R. et al. Active memory cube: a processing-in-memory architecture for exascale systems. IBM J. Res Dev. 59, 1–7 (2015).

  11. Jeddeloh, J. & Keeth, B. Hybrid memory cube new DRAM architecture increases density and performance. In Proc. IEEE Symposium on VLSI Technology (VLSIT) 87–88 (IEEE, 2012).

  12. Strukov, D. B., Snider, G. S., Stewart, D. R. & Williams, R. S. The missing memristor found. Nature 453, 80–83 (2008).

    Article  Google Scholar 

  13. Yang, J. J., Strukov, D. B. & Stewart, D. R. Memristive devices for computing. Nat. Nanotech. 8, 13–24 (2013).

  14. Wong, H.-S. P. et al. Metal–oxide RRAM. Proc. IEEE 100, 1951–1970 (2012).

    Article  Google Scholar 

  15. Prezioso, M. et al. Training and operation of an integrated neuromorphic network based on metal-oxide memristors. Nature 521, 61–64 (2015).

    Article  Google Scholar 

  16. Sheridan, P. et al. Sparse coding with memristor networks. Nat. Nanotech. 12, 784–789 (2017).

    Article  Google Scholar 

  17. Zidan, M. A., Strachan, J. P. & Lu, W. D. The future of electronics based on memristive systems. Nat. Electron. 1, 22–29 (2018).

    Article  Google Scholar 

  18. Ielmini, D. Modeling the universal set/reset characteristics of bipolar RRAM by field- and temperature-driven filament growth. IEEE Trans. Electron Devices 58, 4309–4317 (2011).

    Article  Google Scholar 

  19. Kim, K.-H. et al. A functional hybrid memristor crossbar-array/CMOS system for data storage and neuromorphic applications. Nano Lett. 12, 389–395 (2011).

    Article  Google Scholar 

  20. Waser, R. & Aono, M. Nanoionics-based resistive switching memories. Nat. Mater. 6, 833–840 (2007).

    Article  Google Scholar 

  21. Li, C. et al. Analogue signal and image processing with large memristor crossbars. Nat. Electron. 1, 52–59 (2018).

    Article  Google Scholar 

  22. Feinberg, B., Vengalam, U., Whitehair, N., Wang, S. & Ipek, E. Enabling scientific computing on memristive accelerators. In ACM/IEEE Int. Symp. on Computer Architecture (ACM/IEEE, 2018).

  23. Hu, M. et al. Dot-product engine for neuromorphic computing: programming 1T1M crossbar to accelerate matrix-vector multiplication. In ACM/EDAC/IEEE Design Automation Conf. 1–6 (ACM/EDAC/IEEE, 2016).

  24. Shafiee, A. et al. ISAAC: a convolutional neural network accelerator with in-situ analog arithmetic in crossbars. In ACM/IEEE Ann. Int. Symp. on Computer Architecture 14–26 (ACM/IEEE, 2016).

  25. Chi, P. et al. PRIME: a novel processing-in-memory architecture for neural network computation in ReRAM-based main memory. In ACM/IEEE Ann. Int. Symp. on Computer Architecture 27–39 (ACM/IEEE, 2016).

  26. Zidan, M. A. et al. Field-programmable crossbar array (FPCA) for reconfigurable computing. IEEE Trans. Multi-Scale Comput. Syst. (2017).

  27. Song, L., Qian, X., Li, H. & Chen, Y. PipeLayer: a pipelined ReRAM-based accelerator for deep learning. IEEE Int. Symp. on High Performance Computer Architecture 541–552 (IEEE, 2017).

  28. Bojnordi, M. N. & Ipek, E. Memristive Boltzmann machine: a hardware accelerator for combinatorial optimization and deep learning. IEEE Int. Symp. on High Performance Computer Architecture 1–13 (IEEE, 2016).

  29. Zidan, M. A., Chen, A., Indiveri, G. & Lu, W. D. Memristive computing devices and applications. J. Electroceram. 39, 4–20 (2017).

    Article  Google Scholar 

  30. Neftci, E., Pedroni, B. U., Joshi, S., Al-Shedivat, M. & Cauwenberghs, G. Stochastic synapses enable efficient brain-inspired learning machines. Front. Neurosci. 10, 241 (2016).

    Article  Google Scholar 

  31. Yu, S. et al. Scaling-up resistive synaptic arrays for neuro-inspired architecture: challenges and prospect. In IEEE Int. Electron Devices Meeting 17.3.1–17.3.4 (IEEE, 2015).

  32. Alibart, F., Gao, L., Hoskins, B. D. & Strukov, D. B. High precision tuning of state for memristive devices by adaptable variation-tolerant algorithm. Nanotechnology 23, 075201 (2012).

    Article  Google Scholar 

  33. Richter, I. et al. Memristive accelerator for extreme scale linear solvers. In Government Microcircuit Applications & Critical Technology Conf. (GOMACTech) (2015).

  34. Gallo, M. L. et al. Mixed-precision in-memory computing. Nat. Electron. 1, 246–253 (2018).

    Article  Google Scholar 

  35. Jeong, Y., Zidan, M. A. & Lu, W. D. Parasitic effect analysis in memristor array-based neuromorphic systems. IEEE Trans. Nanotechnol. 17, 184–193 (2018).

    Article  Google Scholar 

  36. Choi, S., Shin, J. H., Lee, J., Sheridan, P. & Lu, W. D. Experimental demonstration of feature extraction and dimensionality reduction using memristor networks. Nano Lett. 17, 3113–3118 (2017).

    Article  Google Scholar 

  37. Guan, X., Yu, S. & Wong, H.-S. P. On the switching parameter variation of metal-oxide RRAM—Part I: Physical modeling and simulation methodology. IEEE Trans. Electron Devices 59, 1172–1182 (2012).

    Article  Google Scholar 

  38. Jo, S. H., Kim, K.-H. & Lu, W. Programmable resistance switching in nanoscale two-terminal devices. Nano Lett. 9, 496–500 (2008).

    Article  Google Scholar 

  39. Alibart, F., Gao, L., Hoskins, B. D. & Strukov, D. B. High precision tuning of state for memristive devices by adaptable variation-tolerant algorithm. Nanotechnology 23, 075201 (2012).

    Article  Google Scholar 

  40. Kim, K. M. et al. Voltage divider effect for the improvement of variability and endurance of TaOx memristor. Sci. Rep. 6, 20085 (2016).

  41. Gilbarg, D. & Trudinger, N. S. Elliptic Partial Differential Equations of Second Order (Springer, Berlin, 2015).

  42. Ames, W. F. Numerical Methods for Partial Differential Equations (Academic, New York, 2014).

  43. Nishidate, Y. & Nikishkov, G. P. Fast water animation using the wave equation with damping. Int. Conf. on Computational Science 232–239 (Springer, 2005).

  44. Kushner, M. J. Hybrid modelling of low temperature plasmas for fundamental investigations and equipment design. J. Phys. D 42, 194013 (2009).

    Article  Google Scholar 

  45. SLAP Sparse Matrix Library (accessed 6 Jan 2017);

  46. Eymard, R., Gallouët, T. & Herbin, R. in Handbook of Numerical Analysis (eds Ciarlet, P. G. & Lions, J. L.) 713–1018 (Elsevier, 2000).

Download references


We acknowledge inspiring discussions with Z. Zhang, J. Moon and T. Chen. This work was support by the Defense Advanced Research Projects Agency (DARPA) through award HR0011-17-2-0018 and by the National Science Foundation (NSF) through grant CCF-1617315.

Author information

Authors and Affiliations



M.A.Z. and W.D.L. conceived the project and constructed the research frame. M.A.Z., Y.J., J.L. and B.C. prepared the memristor arrays and built the hardware and software package. M.A.Z. and Y.J. performed the hardware measurements. M.A.Z, Y.J., S.H., M.J.K. and W.D.L. analysed the experimental data and simulation results. W.D.L. directed the project. All authors discussed the results and implications and commented on the manuscript at all stages.

Corresponding author

Correspondence to Wei D. Lu.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary information

Supplementary Information

Supplementary Figures 1–12 and Supplementary Notes 1–2

Supplementary Video 1

Solution obtained from the memristor hardware system showing the wave propagation in a shallow water system at different times.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zidan, M.A., Jeong, Y., Lee, J. et al. A general memristor-based partial differential equation solver. Nat Electron 1, 411–420 (2018).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:

This article is cited by


Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing