  Digital twins, the cornerstone of Industry 4.0, replicate real-world entities
through computer models, revolutionising fields such as manufacturing
management and industrial automation. Recent advances in machine learning
provide data-driven methods for developing digital twins using discrete-time
data and finite-depth models on digital computers. However, this approach fails
to capture the underlying continuous dynamics and struggles with modelling
complex system behaviour. Additionally, the architecture of digital computers,
with separate storage and processing units, necessitates frequent data
transfers and Analogue-Digital (A/D) conversion, thereby significantly
increasing both time and energy costs. Here, we introduce a memristive neural
ordinary differential equation (ODE) solver for digital twins, which is capable
of capturing continuous-time dynamics and facilitates the modelling of complex
systems using an infinite-depth model. By integrating storage and computation
within analogue memristor arrays, we circumvent the von Neumann bottleneck,
thus enhancing both speed and energy efficiency. We experimentally validate our
approach by developing a digital twin of the HP memristor, which accurately
extrapolates its nonlinear dynamics, achieving a 4.2-fold projected speedup and
a 41.4-fold projected decrease in energy consumption compared to
state-of-the-art digital hardware, while maintaining an acceptable error
margin. Additionally, we demonstrate scalability through experimentally
grounded simulations of Lorenz96 dynamics, exhibiting projected performance
improvements of 12.6-fold in speed and 189.7-fold in energy efficiency relative
to traditional digital approaches. By harnessing the capabilities of fully
analogue computing, our breakthrough accelerates the development of digital
twins, offering an efficient and rapid solution to meet the demands of Industry
4.0.
