High speed Supercomputers enable advanced computational modeling and data analytics applicable to all areas of science and engineering. They are being widely used in applications like Astrophysics, to understand stellar structure, planetary formation, galactic evolution and other interactions; Material Sciences to understand the structure and properties of materials and creation of new high-performance materials; Sophisticated climate models, which capture the effects of greenhouse gases, deforestation and other planetary changes, that have been key to understanding the effects of human behavior on the weather and climate change.

However, such enormous processing power comes at a cost. Sunway TaihuLight at the National Supercomputing Center in Wuxi, Chinavers a whopping 93 petaflops (one petaflop equals a quadrillion floating-point operations per second). But it requires 10.649.600 processing units, so-called cores, that consume 15.371 megawatts – an amount of electricity that could power a small city of about 16.000 inhabitants based on an average energy consumption equal to that of San Francisco.

In contrast human brain’s processing power is estimated at about 38 petaflops, about two-fifths of that of TaihuLight. But all it needs to operate is about 20 watts of energy. Watts, not megawatts! And yet it performs tasks that no machine has ever been able to execute. It is simply “programmed” by the interconnections between its active components, mostly so-called neurons.

Electronic computers are extremely powerful at performing a high number of operations sequentially at very high speeds. However, they struggle with combinatorial tasks that can be solved faster if many operations are performed in parallel for example in cryptography and mathematical optimisation, which require the computer to test a large number of different solutions.

Tomorrow’s applications demand stronger computing powers at much lower energy consumption levels. But digital computers simply can’t provide this out of the box. Therefore many alternative approaches are being pursued by the researchers.

There have been significant efforts in conceiving parallel-computation approaches in the past, for example: Quantum computation and microfluidics-based computation. However, these approaches have not proven, so far, to be scalable and practical from a fabrication and operational perspective.

Analog computer can be described as a model for a certain problem that can then be used to solve that very problem by means of simulating it. Typically such analogs are based on analog electronic circuits such as summers, integrators and multipliers. But they can also be implemented using digital components in which case they are called digital differential analyzers.

There is no stored program that controls the operation of such a computer. Instead, you program it by changing the interconnection between its many computing elements – kind of like a brain. All of the machine’s computing elements work in complete parallelism with no central or distributed memory to access and to wait for.

Such analog computers reach extremely high computational power for certain problem classes. Among others, they are unsurpassed for tackling problems based in differential equations and systems thereof – which applies to many if not most of today’s most relevant problems in science and technology.

For instance, in a 2005 paper, Glenn E. R. Cowan described a Very-Large-Scale-Integrated Analog Computer (VLSI), i.e. an analog computer on a chip, so to speak. This chip delivered whopping 21 gigaflops per watt for a certain class of differential equations, which is better than today’s most power-efficient system in the Green500-list.

Another proposed approach is Hybrid Approach. That is instead of full analog computer developing modern analog co-processors that take off the load of solving complex differential equations from traditional computers. The result would be so-called hybrid computers.

### Metamaterials based Analog Computers

One of the technologies that is being used to create analog computers is metamaterials.

Metamaterials are synthetic, compound materials that are structured in ways that give them specific properties — such as a negative refractive index – that are rare or absent in natural materials. To design optical metamaterials, researchers often rely on a branch of mathematics called transformation optics, which transforms the coordinates of space to control the path of light through a material. A famous example is the invisibility cloak whereby transformation optics is used to control the refraction of light in the cloak so that incident light travels smoothly around the cloaked object rather than scattering off it. The result is that an observer will conclude that the cloaked object is not present.

In 2014, researchers led by Nader Engheta of the University of Pennsylvania proposed another possible use for transformation optics. They pointed out that electromagnetic waves encode mathematical functions in their amplitudes and phases – both of which can be transformed by metamaterials. This led the team to suggest that metamaterials could perform mathematical operations on these functions.

Now Metamaterials have been used by researchers in the US to solve mathematical problems by transforming data that are encoded into electromagnetic waves. The researchers believe their new analogue computing paradigm offers several advantages over conventional digital computers and are now working to make it compatible with traditional silicon photonics devices.

Compact analogue computer based on an acoustic metamaterial has been proposed by Farzad Zangeneh-Nejad and Romain Fleury at the Federal Institute of Technology (EPFL) in Lausanne, Switzerland. They have shown that the system should be capable of rapid differentiation, integration, and instantaneous image processing, and the duo believe it could achieve yet more impressive feats in the future.

### Metamaterial computer solves integral equations encoded in electromagnetic waves

Now, Engheta and colleagues have designed a metamaterial that not only performs mathematical operations but can also find solutions to an important class of equations called integral equations.

“In almost any field of science and engineering you can describe the numerical values of the phenomena that you are after using integral equations,” explains Engheta. Solving these equations is therefore vital to modelling a wide range of phenomena. Algebraic solutions are often impossible, however, so researchers often must rely on computational analysis. This involves rearranging the equation so that the unknown solution appears on both sides. Starting from an arbitrary point, the calculation is then run repeatedly in a feedback loop until the correct solution is reached. At this point, performing the mathematical operation described by the equation does not change the value, so the solution remains stable.

“That takes time,” explains Engheta, which is why finding numerical simulations can often require significant computational resources.

#### Speed of light

The researchers believed metamaterials could offer several important advantages over this conventional digital process. One benefit is that the computational process could be extremely fast because electromagnetic waves pass through metamaterials at the speed of light. Also, the same metamaterial can process multiple waves simultaneously: “Waves can pass through each other, giving you a parallel system,” explains Engheta.

To test their ideas, the researchers designed metamaterials from carefully-patterned dielectrics to perform mathematical transformations related to three different integral equations. Computational modelling of how electromagnetic waves interact with the metamaterials suggests that the solutions provided by the hypothetical systems should agree very well the solutions obtained from traditional numerical methods. Furthermore, the computational modelling suggests that the metamaterial systems can reach the correct solutions very quickly.

The team also created a metamaterial in the lab for one of the integral equations (see figure). It was made from patterned low-loss polystyrene and is designed for use with microwaves. The team found that its performance was in very good agreement with computational predictions.

In future, the researchers aim to build their metamaterials from a silica dielectric, which would make integration with standard silicon photonics devices easier. A silica dielectric metamaterial would also allow infrared light at telecom wavelengths to be used to perform calculations. This means that future devices could be much smaller than the microwave prototype.

The team also hopes that in the future reconfigurable metameterials could be developed, effectively creating a kind of reprogrammable analogue computer. Nevertheless, stresses Engheta, the present platform does not offer the prospect of an alternative to the conditional logic of a true computer, in which one computation depends on the outcome of another: “We don’t have any optical logic here,” he says.

Andrea Alù at the City University of New York was involved in the 2014 research and continues to work independently on computing based electromagnetic waves. He praises Engheta and colleagues for turning the original idea into reality. “I find it interesting because it’s not at all trivial that this can be worked out, especially given all the tolerances present.”

### Analogue computer could use sound to make rapid calculations

Analogue computers use interactions involving physical entities such as light, electrical current or a mechanical system to perform specific calculations. Some of the most sophisticated analogue computers were developed in the early to mid-20th century to help guide artillery and aerial bombing strikes.

While the advent of digital computers made these computers obsolete, they are now enjoying a resurgence thanks to ongoing research into artificial materials called metamaterials. These materials can be engineered to manipulate the light or sound waves passing through them in new ways – opening the door to new types of analogue computer.

#### Subtle engineering

“Metamaterials are artificial structures composed of periodic subwavelength inclusions, which can be subtly engineered to provide the desired macroscopic characteristics of the overall material,” explains Zangeneh-Nejad.

Metamaterials have already been used to create analogue computers that manipulate electromagnetic waves to perform mathematical operations. Zangeneh-Nejad and Fleury set out to design a device comparable to these optical computers, but using sound waves. However, the distinctive properties of sound waves meant that the researchers first needed to carefully consider how to design their metamaterial.

“Usually, when sound is incident on a hard wall, it reflects without being subject to any particular transformation, and the only thing that happens is the direction of propagation changes,” says Fleury. “Our metamaterial is capable of performing complex signal processing tasks on sound waves when they are reflected, directly on the fly and without delay. It can achieve this instantaneously without converting [sound] into electrical signals.” Through their calculations, the physicists uncovered the physical properties required of their metamaterial. “It requires a very special acoustic property that does not exist in nature: an acoustic refractive index larger than that of air,” explains Fleury.

#### No transform required

An important feature of the proposed device is that it performs operations directly in the spatial domain. Previous metamaterial-based computers have worked in the frequency, or Fourier domain, requiring bulky Fourier transform sub-blocks to convert signals into the spatial domain. The new metamaterial has no need for these additional elements. “In our computing system, the mathematical operator of choice is directly performed in the spatial domain using a metamaterial known as a high-index acoustic slab waveguide,” Zangeneh-Nejad explains.

The duo have shown how their device could perform differentiation and integration, as well as instantaneous image detection. Writing in a preprint on arXiv, they explain how future generations of their design could be used to solve more complex differential equations, such as the Schrödinger equation. “We showed how more complex operators such as second order differentiator can be constructed simply by cascading more and more slab waveguides,” says Zangeneh-Nejad. Importantly, the researchers have worked-out that computing devices made from acoustic metamaterials could be entirely compatible with current computing infrastructure. “Our system is free of any Fourier bulk lens, highly miniaturized and potentially integratable in compact architectures, and can be implemented easily in practice.”

The physicists will now further explore the capability of their waveguide to perform calculations at faster rates than conventional computers. “We are investigating applications of our metamaterial in compressive sensing, ultrafast equation solving, neural networks, and a large variety of other applications necessitating real-time and continuous signal processing,” Fleury explains. Their device also has the potential for exploring the dynamics of complex biological systems, allowing for new advances in medicine. As Zangeneh-Nejad adds, “our system could explore the computation processes in human brains, and many other natural systems like DNA, membranes and protein-protein interactions”.

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