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[Wataru Namiki](https://orcid.org/0000-0003-4053-7366), [Daiki Nishioka](https://orcid.org/0000-0002-3369-7700), [Takashi Tsuchiya](https://orcid.org/0000-0002-6950-6160), [Kazuya Terabe](https://orcid.org/0000-0003-3988-3456)

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[Fast physical reservoir computing, achieved with nonlinear interfered spin waves](https://mdr.nims.go.jp/datasets/f70bbe67-9654-4e0d-ac06-736dab9756aa)

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Fast physical reservoir computing, achieved with nonlinear interfered spin wavesNeuromorphic Computing andEngineering     PAPER • OPEN ACCESSFast physical reservoir computing, achieved withnonlinear interfered spin wavesTo cite this article: Wataru Namiki et al 2024 Neuromorph. Comput. Eng. 4 024015 View the article online for updates and enhancements.You may also likeReal-time economic and safe operationdiagnosis of hydropower unitsAizhen Wang and Shuxin Hu-Study on APD real time compensationmethods of laser Detection systemFENG Ying, ZHANG He, ZHANG Xiangjinet al.-Energy Dense Storage Using IntermediateTemperature Reversible Solid Oxide CellsArrigo Monti, Christopher Wendel,Massimo Santarelli et al.-This content was downloaded from IP address 144.213.253.16 on 02/08/2024 at 11:15https://doi.org/10.1088/2634-4386/ad561a/article/10.1088/1742-6596/2564/1/012059/article/10.1088/1742-6596/2564/1/012059/article/10.1088/1742-6596/276/1/012120/article/10.1088/1742-6596/276/1/012120/article/10.1149/MA2015-03/1/419/article/10.1149/MA2015-03/1/419Neuromorph. Comput. Eng. 4 (2024) 024015 https://doi.org/10.1088/2634-4386/ad561aOPEN ACCESSRECEIVED29 December 2023REVISED18 April 2024ACCEPTED FOR PUBLICATION10 June 2024PUBLISHED20 June 2024Original content fromthis work may be usedunder the terms of theCreative CommonsAttribution 4.0 licence.Any further distributionof this work mustmaintain attribution tothe author(s) and the titleof the work, journalcitation and DOI.PAPERFast physical reservoir computing, achieved with nonlinearinterfered spin wavesWataru Namiki1, Daiki Nishioka1,2, Takashi Tsuchiya1,∗ and Kazuya Terabe11 Research Center for Materials Nanoarchitectonics, National Institute for Materials Science, Ibaraki, Japan2 Department of Applied Physics, Tokyo University of Science, Tokyo, Japan∗ Author to whom any correspondence should be addressed.E-mail: TSUCHIYA.Takashi@nims.go.jpKeywords: Reservoir computing, spin wave interference, artificial intelligenceSupplementary material for this article is available onlineAbstractReservoir computing is a promising approach to implementing high-performance artificialintelligence that can process input data at lower computational costs than conventional artificialneural networks. Although reservoir computing enables real-time processing of input time-seriesdata on artificial intelligence mounted on terminal devices, few physical devices are capable ofhigh-speed operation for real-time processing. In this study, we introduce spin wave interferencewith a stepped input method to reduce the operating time of the physical reservoir, andsecond-order nonlinear equation task and second-order nonlinear autoregressive mean averaging,which are well-known benchmark tasks, were carried out to evaluate the operating speed andprediction accuracy of said physical reservoir. The demonstrated reservoir device operates at theshortest operating time of 13 ms/5000-time steps, compared to other compact reservoir devices,even though its performance is higher than or comparable to such physical reservoirs. This study isa stepping stone toward realizing an artificial intelligence device capable of real-time processing onterminal devices.1. IntroductionReservoir computing is a promising approach to implementing high-performance artificial intelligence thatcan process input data at lower computational costs than conventional artificial neural networks since thereservoir computing system trains only output weights as parameters and exhibits the sufficient ability toprocess time-series data [1–4]. This feature of reservoir computing arises from three key features: nonlinearresponse, short-term memory, and the ability to map input information in higher dimensional feature space.Said reservoir computing enables real-time processing of input information, such as time-series data onartificial intelligence mounted on the terminal device, namely ‘edge-computing.’ To implement such edgecomputing, it is needed physical reservoir computing that operates on a terminal device with limited electricpower can be carried out with a physical device satisfying the three features mentioned above. In recent years,various physical systems, such as electrical circuits, electrochemical cells, magnetics, optical elements,robotics, ion-gating devices, and so on, have been utilized to build reservoir computing systems based onphysical dynamics [4–49]. However, some issues still need to be addressed due to the slow operation speed,high electrical power consumption, insufficient accuracy rate, and enormous volumes associated with suchsystems. In particular, slow operation speed is a fatal issue when realizing real-time processing. While theoptical circuit shows the fastest operation speed of approximately 2 ms among physical reservoirs, the circuitsize is enormous due to the long feedback loop of 280 meters [26]. Thus, it is challenging to integrate suchcircuits on a terminal device consisting of compact elements. The spin wave, a collective excitation motion ofwave-shaped magnetization with phase, is attracting attention among related research fields due to itsadvantage of a charge-less transport method, which reduces electric power consumption [50–53]. In recentyears, reservoir computing systems with spin wave propagation in an active-ring resonator have been© 2024 The Author(s). Published by IOP Publishing Ltdhttps://doi.org/10.1088/2634-4386/ad561ahttps://crossmark.crossref.org/dialog/?doi=10.1088/2634-4386/ad561a&domain=pdf&date_stamp=2024-6-20https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://orcid.org/0000-0003-4053-7366https://orcid.org/0000-0002-3369-7700https://orcid.org/0000-0002-6950-6160mailto:TSUCHIYA.Takashi@nims.go.jphttp://doi.org/10.1088/2634-4386/ad561aNeuromorph. Comput. Eng. 4 (2024) 024015 W Namiki et alproposed. However, its capabilities could be better due to the large volume resulting from the circuit structureand its computational performance [17–19]. To overcome such problems, it has been shown that the physicaldevice with spin wave interference, proposed by micromagnetic simulation [40–45], satisfies the followingfeatures of the physical reservoir by an experimental demonstration in our previous study [46]. Although theexperimental demonstration revealed relatively fast operation comparable to the optical circuit using opticaldynamics [46], no experimental demonstration has explored realizing the fastest processing speed thus far.Herein, we improve the operation speed of compact physical reservoirs utilizing spin wave interference torealize real-time processing of physical reservoir computing. A spin wave is excited by resonance withmagnetic field switching induced by an electrical signal’s rising or falling edge [54]. In this study, theconversion method for input data is changed from pulsed voltage input used in the previous study to steppedvoltage input. This change can reduce the length of input time-series voltage dominating the operating timeof the physical device since the rising or falling edge of the pulsed signal, which consists of a rising edge and afalling edge, is no longer necessary. This is because a stepped signal has only a falling edge or rising edge,while a pulsed signal has falling and rising edges. This method can reduce operating time by up to 11.8%while maintaining high performance superior to other physical reservoirs.2. Experimental methods2.1. Fabrication of the physical device for reservoir computing with interfered spin wave multi-detectionThe physical device is the same one used in the previous study [46]. A Y3Fe5O12 single crystal with adiameter and thickness of 5.0 and 0.5 mm, respectively, was used in this study, and its surface is 111 plane.Coplanar waveguides (CPWs) were fabricated using 90 nm-thick Au thin films on ten nm-thick Ti adherentlayers. A signal line (S) and two ground lines (G) of a CPW are 10 µm wide and 20 µm wide, respectively.The edges of adjacent CPWs are separated by 30 µm [46].2.2. Measurement setup of the physical deviceAll experimental measurement was performed at a room temperature of 295± 1 K. A magnetic field H wasapplied to the 111 direction. The exciters and detectors shown in figure 1(a) were connected to an arbitrarywaveform generator and an oscilloscope through GSG probes, respectively. Stepped voltage with transienttimes (i.e. rise time or fall time) of 280 ps was input to the exciter antenna for spin wave excitation. Theamplitude of the stepped voltage was set to 280 mV. Then, the input and detected signals were amplifiedthrough amplifiers. Five hundred times accumulation was carried out to reduce the noise component in thedetected signal. On a nonlinear time-series data processing task, a random waveform was transformed into astepped voltage waveform with various durations ranging from 2 to 20 ns. The converted waveform has adiscrete time step k of 5000. To remove the history of the last waveform, an interval of 4 µs was inserted afterthe input waveform signal.This study used two exciters (Exciter A and Exciter B) and two detectors (Detector A and Detector B). Tomeasure spin wave, (1) In the case of ‘with interference’ and ‘with multi-detection’, both exciters and bothdetectors were used simultaneously. (2) In the case of ‘with interference’ and ‘without multi-detection’, bothexciters and one detector (i.e., Detector A) were used. (3) In the case of ‘without interference’ and ‘withoutmulti-detection’, one exciter (i.e. only Exciter A) and one of the detectors (i.e. Detector A or Detector B) wereused. The previous work also shows the details of the measurement setup [46].H ranges from 100–400 mT at a 10 mT step to evaluate the spin wave property and from 150–250 mT toevaluate the processing performance of the physical reservoir.2.3. Evaluation of spin wave propertyThe wave number kw of the electric current density j(x) reflects that of a spin wave excited by an oersted fieldinduced by the j(x) flowing in the CPW. The j(x) in a CPW is written as follows [55];j(x) =−1 −(lS2 + l1+ lG)⩽ x⩽−(lS2 + l1),(lS2 + l1)⩽ x⩽(lS2 + l1+ lG)1 − lS2 ⩽ x⩽ lS20 other(1)where lS, l1, and lG denote the width of the signal line, the distance between the signal and ground lines, andthe width of the ground line, respectively. The j(x) in wave number space can be expressed by Fouriertransform as follows [55];j(k) = lSsinc(klS2)− 2lG cos(kl1+ lG+ 2l12)sinc(klG2). (2)2Neuromorph. Comput. Eng. 4 (2024) 024015 W Namiki et alFigure 1. A schematic illustration of reservoir computing utilizing nonlinear interfered spin waves in Y3Fe5O12 (YIG).Exciter A and Detector A were connected to a vector network analyzer, and the transmission signal (S21parameter) from Exciter A and Detector A was measured to reveal the actual wave number of spin waves. Thespectrum at a magnetic field of 0 mT was subtracted from the spectra obtained at a magnetic field of100–400 mT to de-embed the transmitted spin wave. The spin wave frequency variation at various H isdescribed as follows [55];f = γ√(H−Ha){(H−Ha)+M(1− 1− e−kwdkwd)}(3)where f, γ, Ha,M, kw, and d represent spin wave frequency, gyromagnetic ratio, magnetic anisotropy field ofthe YIG, saturation magnetization of the YIG, wave number of spin wave, and YIG thickness, respectively.Here, γ andM were fixed at 2.8 MHz Oe −1 [53] and 1984 Gauss [46]. Note that the d used in the calculationis thinner than its thickness. The assumption is that a spin wave propagates at a region that does notcorrespond to the whole YIG but near the surface at a depth of several micrometers.2.4. Nonlinear time-series data processing taskThe subject reservoir computing system was trained and tested with a random waveform to predict theoutput from a second-order nonlinear dynamic system and a second-order nonlinear autoregressive movingaverage (NARMA2) system. These systems are described as follows [56],d(k) = 0.4d(k− 1)+ 0.4d(k− 1)d(k− 2)+ 0.6u3 (k)+ 0.1. (4)andd(k+ 1) = 0.4d(k)+ 0.4d(k)d(k− 1)+ 0.6u3 (k)+ 0.1. (5)Here, d(k) and u(k) are the output values from the system at k, and the input values at k, respectively. Toinput to a reservoir computing system, the original random waveform u(k), which ranges from 0 to 0.5, wasconverted to stepped waveforms. The waveform was separated into training phases with k of 3500 and testphases with k of 500 after the first k of 1000 was discarded. Stepped waveforms were input to the physicalreservoir through Exciter A and Exciter B. In the training phase, the weight parametersW , shown infigure 1(a), connecting the reservoir part and the readout layer, were learned to minimize the error betweenthe d(k) of equations (4) and (5), and the reservoir output y(k) is described as follows [30, 31, 36, 46]:y(k) =n∑i=1WiXi (k)+ b. (6)Here, Xi(k),Wi, n, and b are an i-th node state at k, a weight coefficient connecting Xi(k) and the readoutlayer, the total number of nodes, and a bias term of 0.1, respectively.Wi was optimized by ridge regression astraining for the system to correspond to d(k). On the nonlinear time-series data processing tasks, 50 virtualnodes per detector were from each induced voltage. Thus, 100 reservoir states (‘with multi-detection’) could3Neuromorph. Comput. Eng. 4 (2024) 024015 W Namiki et albe obtained (i.e. n= 100). The inputs were applied sequentially. In order to process this task accurately, usingthe physical reservoir satisfying nonlinearity, short-term memory, and high dimensionality required for thereservoir is necessary. The NARMA task described in equation (4) represents the required properties (i.e.nonlinearity, short-term memory, and high dimensionality). Because of these system features, predicting theoutput of the NARMA system is a benchmark task widely used in physical reservoirs, and in this study, whichaims to improve speed, it is also an indicator of how long it takes to perform the same task. The relationshipbetween the physical reservoir and time-series data, including short-term memory and nonlinearity, isdescribed in S1 of the supporting information.2.5. Evaluation of processing performance of the physical reservoir computingIn the time series data analysis tasks, the readout network of the nonlinear interfered spin wavemulti-detection reservoir was trained by ridge regression [30, 31, 36, 37, 46]. The reservoir output y(k)shown in equation (6) is transformed to;y(k) =W ·X(k) . (7)Here,W = (w0, w1, wi,…, wn) and X(k)= (X0(k), X1(k), Xi(k),…, Xn(k))T are the weight vector and thereservoir state vector with a reservoir size (i.e., the number of nodes) of n, respectively [4]. Note that w0 = band X0(k)= 1 to introduce the bias b shown in equation (6). The cost function J(W) in the ridge regressionis defined as follows;J(W) =12T∑k=1(d(k)− y(k))2+β2N∑i=0Wi2, (8)where T, β and d(k) are the data length in the training phase, the ridge parameter, and the target outputgenerated by equation (4) or equation (5), respectively. T = 3500 and β = 0 were fixed for all the tasksdemonstrated in this study. The weight matrix Ŵ, which minimizes the J(W), is written as follows;Ŵ= YXT(XXT +λI)−1. (9)Here, Y = (d(1), d(2), …, d(T)), X(k)= (X(1), X(2), …, X(T)), and I (⊆ R(N+1)×(N+1)) are the targetoutput vector, the reservoir state matrix and the identify matrix, respectively. Here, T is the length of thetraining phase (T = 3500) or test phase (T = 500).After optimizing theW , the computational performance of the physical reservoir was evaluated bycalculating ‘NMSE’ for the second-order nonlinear dynamics equation task and ‘NMSEvar.’ for the NARMA2task to compare its performance with that of other physical reservoirs as following equations [7, 10, 12, 13,28, 30, 31, 36, 37, 46],NMSE=∑Tk=1 (d(k)− y(k))2∑Tk=1 (d(k))2 (10)andNMSEvar. =∑Tk=1 (d(k)− y(k))2∑Tk=1 (d(k)− dave.)2 . (11)Here, dave is the time average of d(k). T is 3500 in the training and 500 in the testing phases.Operating times of various physical reservoirs were calculated as follows: The number of steps requiredfor training and testing by the physical reservoir in the NARMA task was multiplied by the input time perstep (i.e., duration). One waveform consists of 5000 total steps (= discarded 500 steps/training phase of 3500steps)/discarded 500 steps/testing phase of 500 steps) and a 4 µs interval, where one step corresponds to thetotal transition time and duration of the input voltage. Since this process is repeated 500 times due toaccumulation, all the time required was defined as operating time as follows,Operating time= (5000 step× duration + 4µs)× 500. (12)In order to compare the operating time for different physical reservoirs, we calculated it in the same wayas above, using the information found in the literature (total number of steps, which are various lengths inthe range 500–10 000 depending on studies, and input time per discrete time) [5, 7, 9, 11, 23, 25–27, 30–32,36, 38, 46, 57].4Neuromorph. Comput. Eng. 4 (2024) 024015 W Namiki et alFigure 2. (a) YIG single crystal with deposited coplanar waveguides (CPWs) and its measurement configuration. Magnetic fieldHis applied perpendicular to the YIG surface. (b) A cropped image of CPWs and their dimensions. G and S represent ground andsignal lines. (c) Wave number distribution of electric current density. (d) Magnetic field dependence of spin wave magneticfield.resonance frequency. A dashed line represents a fitting result. Ha denotes an anisotropic.3. Results3.1. Physical reservoir computing with nonlinear interfered spin wave and characteristics of the physicaldeviceFigure 1 shows a schematic illustration of a reservoir computing network with a physical device (i.e., physicalreservoir computing) in case of utilizing nonlinear interfered spin waves in magnetic material. A reservoircomputing network generally has three input, reservoir, and output layers. In particular, the reservoir layermust satisfy critical requirements: nonlinearity, the ability to map input time series data u(k) in highdimensional space, and short-term memory. Thus, such reservoir layer plays a role of nonlinear functionmapping u(k) to i-th node state Xi(k) based on its past state Xi(k−1) as follows [4],Xi (k) = f [Winu(k) , WrXi (k− 1)] . (13)Here,W in andWr denote the weight connecting u(k) to Xi and the weight connecting Xi to j-th node Xj(i ̸= j) in the reservoir layer, respectively, and f represents the nonlinear activation function. Increasing thenumber of i leads to improvement of the ability to map u(k) in high dimensional space. Physical reservoircomputing is an innovative scheme that replaces the reservoir layer with a physical device satisfying the abovethree requirements to implement ‘edge computing’. Here, a physical device with nonlinear interfered spinwave multi-detection is realized by utilizing magnetic material and some antennas and is a promisingmethod to realize edge computing with low electrical power consumption.Figure 2(a) shows an optical microscope image of the physical device with a YIG single crystal. Fourcoplanar antennas were deposited on the YIG to excite and detect spin waves under the H application. Acropped image of antennas is shown in figure 2(b). The details of fabrication and dimensions of the physicaldevice are described in the experimental section. Since the spin wave is excited by the slope of the electricalsignal flowing in the antenna [54], wave number kw of the excited spin wave corresponds to kw of j(x). Theelectric current density calculated from equation (2) is shown in figure 2(c), and a prominent peak and twosatellite peaks can be seen. kw1 is denoted at the prominent peak, with a full-width half maximum (FWHM)of 0.095 µm−1 located at 0.118 µm−1. The remaining two peaks, kw2 and kw3 with FWHM of 0.067 and0.095 µm−1, are located at 0.311 and 0.492 µm−1, respectively. Spin wave resonance frequency measurementwas carried out to investigate the actual kw of the excited spin wave, as shown in figure 2(d). It was revealedthat the wave number of the prominent peak is kw of 0.134 µm−1, which is in good agreement with kw1 of0.118 µm−1 calculated from the distribution of the j(x) shown in figure 2(c). Anisotropic magnetic field Hacan be determined from an intersection with the horizontal axis, and Ha was 1464 Oe.Figure 3(a) shows a schematic illustration of measurement configuration to detect interfered spin wavesat the time domain. In Detector A, spin waves excited at Exciter A and Exciter B interfere and are detected. InDetector B, spin waves excited at Exciter A interfere with spin waves excited at Exciter B and are also affectedby the residual precision of the preceding spin wave excited at Exciter B. Figure 3(b) shows spin wave signals5Neuromorph. Comput. Eng. 4 (2024) 024015 W Namiki et alFigure 3. (a) A schematic illustration of measurement configuration to detect interfered spin wave at time domain. Black andwhite solid arrows represent the magnetic fields induced by electric current and propagating spin waves. Exciter A and Exciter Bare CPWs connected to an arbitrary waveform generator, and Detector A and Detector B are CPWs connected to an oscilloscope.(b) (Upper panel) Stepped input voltage. (Middle panel) Spin wave signals, which are excited at Exciter A and Exciter B, detectedat Detector A. (Lower panel) Nonlinear interference of spin waves, which are excited at Exciter A and Exciter B, at Detector A. (c)(Upper panel) Stepped input voltage. (Middle panel) Spin wave signals, which are excited at Exciter A and Exciter B, detected atDetector B. (Lower panel) Nonlinear interference of spin waves, which are excited at Exciter A and Exciter B, at Detector B.Difference represents subtracting the linear sum of two waveforms from the Interfered waveform. (d) (Upper panel) Spin wavevariation at various base amplitudes of input voltage. (Lower panel) Normalized stepped input. Input [0.5, 1.0] and Input [0.0,0.5] denote a base level of 0.5 and amplitude of 1.0 and a base level of 0.0 and amplitude of 0.5, respectively.detected at Detector A and input signals at Exciter A and Exciter B. As shown in the upper panel, a spin wavepropagating between Excitor A and Detector A differs from one between Excitor B and Detector A. Thus, avariety of signals can be extracted by the multi-detection. As shown in the lower panel, a finite differencebetween an interference and a linear summation of spin waves, shown in the upper panel after a time domainof 5 ns, shows that the interfered wave exhibits nonlinearity. This nonlinear interference of spin waves resultsfrom magnetic dipole interaction between nonlinearly excited spin waves, as observed in theoretical studies[40, 44, 45]. As shown in figure 3(c), this nonlinear interference was observed at another detector, DetectorB. The intensity of the spin wave excited by the step input, defined by the difference between the two values,depends on the source and destination of the step transition, as shown in figure 3(d). Even though inputsignals have the same amplitude of 0.5, detected signals have different wave packets due to different baseamplitudes of 0.0 and 0.5 for input [0.0, 0.5] and input [0.5, 1.0]. This found feature leads that the physicalreservoir can distinguish between time series of the same difference (e.g. six different ‘0.5’ expressed by0.0–0.5, 0.1–0.6, 0.2–0.7, …, and 0.5–1.0) that the input data contains.3.2. Process flow of physical reservoir computingTime-series data processing task (e.g., second-order nonlinear equation task and NARMA2 task) is widelyperformed to evaluate the computational performance of a physical reservoir [7, 10, 12, 13, 28, 30, 31, 36–39,46]. The process flow for such tasks is shown in figure 4. In solving a second-order nonlinear equation task,u(k) is input to a second-order nonlinear dynamical system. Its d(k) is described as equation (4). The d(k)depends on the input u(k) and the two states d(k− 1) and d(k− 2) at past steps k− 1 and k− 2 in additionto a second-order nonlinear term resulting from the product of d(k− 1) and d(k− 2). On the other hand, anoutput d(k+ 1) is described as equation (5) in the case of the NARMA2 task; the output depends on the u(k)6Neuromorph. Comput. Eng. 4 (2024) 024015 W Namiki et alFigure 4. A process flow of time-series data processing task performed by the physical reservoir utilizing interfered spin wavemulti-detection. Original random input u(k) is converted to stepped voltage u’(k) with arbitrary duration (e.g. 5 ns). u(k) andu’(k) are input to the theoretical model and physical reservoir. One hundred reservoir states are generated by two physical nodesand 50 virtual nodes taken from nonlinear interfered spin wave multi-detection. Reservoir output y(k) is constructed by linearsummation of readout weightWi and reservoir states Xi(k). Theoretical output d(k) and y(k) are compared to cost function J(W)by optimizingW in y(k).and the two states d(k) and d(k− 1) in addition to a second-order nonlinear term resulting from the productof d(k) and d(k− 1). Thus, the physical reservoir is required to satisfy the property shown in equation (12) tosolve these tasks precisely. The error of these tasks plays the role of an indicator to evaluate nonlinearity andshort-term memory required as the reservoir to process more concrete application tasks, such asspoken-digit recognition task, blood glucose level prediction, abnormal electrocardiogram detection, andsunspot data prediction [4, 9–11, 30, 31, 36–39, 46, 57]. The u(k) is converted to a stepped voltage signal u′(k), and five different u′ (k) with five durations of 2, 5, 10, 15, and 20 ns were prepared to explore optimizedconditions. Each of these u′ (k) is input to the physical reservoir to which perpendicular magnetic fields of150, 169, 176, 186, 200, and 250 mT are applied. One duration of a stepped signal was equivalent to onediscrete time. Then, various spin waves were excited depending on input amplitude. These spin wavespropagate toward Detector A and Detector B, interfering with other spin waves, and are converted to voltagesignals at two detectors (two physical nodes). The virtual nodes are equally spaced from the whole periodwithin one k. Fifty virtual nodes were taken from the voltage signal, which region does not contain thecross-talk component from the slope of input and which are denoted as black dots in the figure showing thespin wave signals measured at Detector A and Detector B for each k. Then, 100 waveforms were extractedusing Detector A and Detector B at k, and a ‘reservoir state’ was generated. The details of training and testingprocedures are described in section 2.4.3.3. Time-series data processing taskFigure 5(a) shows a comparison with the d(k) of equation (4) and the y(k) in the training phase at thecondition under H of 169 mT at a duration of 5 ns. The NMSE defined by equation (10) at this phase is1.44× 10−3. The compared results in the testing phase are shown in figure 5(b). NMSE for this phaseexhibits a similar value of 1.39× 10−3. Figure 5(c) shows NMSE changes at various durations and H. Thereis a tendency for NMSE to be lower in weaker H, while NSME in more robust H is higher under allmeasurement conditions. Although there was no systematic change with different durations, NMSE droppedsignificantly at durations of 5 and 20 ns, and the lowest NMSE of 1.39× 10−3 was achieved at H of 169 mT7Neuromorph. Comput. Eng. 4 (2024) 024015 W Namiki et alFigure 5. Predicted results at the (a) training phase and (b) test phase of a second-order nonlinear equation task using a durationof 5 ns. The black dotted line, green, and red solid lines represent the target and prediction results at the training and test phases,respectively. (c) NMSE variation of the reservoir computing with interfered spin wave multi-detection for prediction of asecond-order nonlinear equation task at various magnetic fields and step durations.and a duration of 5 ns. This value is lower than or comparable to the values for other physical reservoirs thathave been reported, in which the NMSEs of experimental physical reservoirs utilizing 90 metal-oxidememristors and with magnetization vector rotation manipulation were 3.13× 10−3 and 1.69× 10−3 [7, 37],and a theoretical physical reservoir utilizing 24 spin torque oscillators was 1.31× 10−3 [12]. The NARMA isa more difficult task than the former task since, to predict the output of a NARMA model, a physicalreservoir is required not only for nonlinearity but also for short-term memory. Here, we introduceNARMA2, which requires short-term memory from the previous two steps, as defined in equation (5).Figures 6(a) and (b) show comparisons with the d(k) from equation (5) and the y(k) in the training andtesting phases at the condition under an H of 169 mT and a duration of 5 ns, for NARMA2. NMSEvarvariations at various H and durations are summarized in figure 6(c). The measurement conditiondependence of NMSEvar is similar to that of the NMSE. The lowest NMSEvar is 1.71× 10−1, which is lowerthan and comparable to experimental physical reservoirs previously reported [10, 13, 28, 31, 57]. Here,NMSEsvar of surface-functionalized carbon nanotubes [57], the spin torque oscillator (simulated) [13], softbody [28], diode electrical circuit [10], and redox ion-gating reservoir [31] are 0.368, 0.270, 0.192, 0.177, and0.163, respectively. Thus, it was found that the physical reservoir computing in this study satisfied threecritical features as reservoirs and achieved high computational performance comparable to other physicalreservoirs [10, 13, 28, 31, 57]. Accumulation number dependence of a signal-to-noise ratio of the detectedsignal and the computational performance and reproducibility of the time-series data processing aredescribed in S2.1, optimization of length of a training phase and test phase is described in S2.2, and activityof selected nodes in the physical reservoir is described in S3 of the supporting information.The pulse input corresponding to the input frequency and magnetic field closely relates to equation (3).The frequency of the input pulse is estimated from the rise and fall times of the voltage (280 ps) to be about1.25 GHz. The magnetic field that satisfies the resonance condition of the spin wave is about 169 mT, asshown in figure 2(d). The spin wave is strongly excited by satisfying the resonance condition, and thecomputational performance is improved through spin wave interference. In other words, it can bereproduced by applying a pulse input and a magnetic field with a frequency that satisfies the resonancecondition expressed in equation (3).8Neuromorph. Comput. Eng. 4 (2024) 024015 W Namiki et alFigure 6. Predicted results at the (a) training phase and (b) test phase of a NARMA2 task using a duration of 5 ns. The blackdotted line, green, and red solid lines represent the target and prediction results at the training and test phases, respectively. (c)NMSEvar variation of the reservoir computing with interfered spin wave multi-detection for predicting a NARMA2 task at variousmagnetic fields and step durations.4. Discussion4.1. Processing time of physical reservoirsThe advantage of the physical reservoir with stepped input is not only its high performance but also itshigh-speed processing. Its operating time for a series of input data with 5000-time steps was 13 ms/500integrations at the duration of 5 ns, corresponding to an operating time of 11.8% faster than the previousstudy using a pulsed input method with an interval of 5 ns [46]. This dramatic reduction of operating timeresults from discarding the rising or falling edge and pulse-on time of the pulsed signal due to using astepped input signal instead of a pulsed one. As shown in figure 7(a), the time for input one pulse signal at k(i.e., tp) consists of rising and falling edges, pulse-on times, and pulse interval. In contrast, the time for inputof one stepped signal at k (i.e., ts) consists of only the rising edge or falling edge and step duration. Thus, thedifference between tp and ts is the total time t for the edge and pulse-on, and the absence of t makes thestepped signal shorter (i.e., ts = tp—t).The operating times of various physical reservoirs are summarized in figure 7(b). The operating timeachieved with the stepped input method is the fastest among other physical reservoir computing, which hasbeen carried out in a second-order nonlinear equation task. It corresponds to approximately a discrete timeof those physical reservoirs.Furthermore, the operating time of the physical reservoir utilizing spin wave interference is comparableto some physical reservoirs utilizing optical elements [23, 25–27]. The physical reservoir with spin waveinterference is approaching within one order delays, compared with the fastest operating time of 2 ms,achieved with optical element [26], among physical reservoir computing.The operating time can be further improved by reducing the input duration. This reduction can berealized by adopting magnetic material with higher spin wave frequency. Spin waves with higher frequenciesattenuate faster than those with lower frequencies since the duration time of stepped voltage must beadjusted to the decay time, and the duration time must be shortened for fast decay times. Furthermore, sincethe spin wave is excited by a magnetic field with a specific frequency corresponding to the transition time(i.e. slope) of stepped input voltage, this scheme leads to a reduced slope time per k. When the duration timeper k can be set to 500 ps, which is one order faster than this study, and the slope can be shortened, theoperating time can be reduced to approximately 1.3 ms, corresponding to 1 order faster than the fastestoperating speed in any physical reservoir device.9Neuromorph. Comput. Eng. 4 (2024) 024015 W Namiki et alFigure 7. (a) Difference between waveform length of pulsed and stepped input. tp and ts denote time corresponding to one k ofpulsed and stepped input, respectively. t is the total time of rise and pulse-on. (b) Benchmark of operating time of variousphysical reservoir devices reported as an experimental demonstration. These operating speeds were calculated from the total timesteps used to train and examine the reservoir network, including discarded time steps and time per discrete time [5, 7, 9, 11, 23,25–27, 30–32, 36, 38, 46, 57].4.2. Comparing the volume of the physical reservoirTo compare volumes of various physical reservoirs other than the active ring resonator, calculated volumes ofvarious physical reservoirs are summarized in figure 8. The reservoir system using YIG based active ringresonator has volume of 7.8× 10−3 ∼ 4.4× 10−2 cm3, which are estimated values from dimension (i.e.length× width× thickness) of two YIG devices (7.7 mm× 2.0 mm× 5.06× 10−1 mm and 22 mm×4.0 mm× 5.04× 10−1 mm) [17–19]. Here, the smaller one (i.e. 7.8× 10−3 cm3) is the smallest estimationsince the length of the YIG is not shown in the reference, and the length of 7.7 mm used in the estimation isthe antenna separation, which should be shorter than the YIG length, described in the [17]. The estimatedvolumes are comparable to or larger than the volume of 9.8× 10−3 cm3 (diameter of 5 mm× 0.5 mm) inthe present study. In the spin-wave reservoir using the active ring resonator demonstrated by Watt et al, it isnecessary to delay the spin wave transmission by lengthening the antenna separation in order to feed backthe voltage signal based on the spin wave to the input terminal with a long time difference ranging fromapproximately 50–215 ns [17–19]. In the studies, antennas were placed on the YIG with a very wide spacing10Neuromorph. Comput. Eng. 4 (2024) 024015 W Namiki et alFigure 8. (a) Comparison of volume of physical reservoirs. In addition to this study, physical reservoirs utilizing soft body [28],optical element [24–27], electrical element [5, 6], electrochemical cell [9], spin wave active ring resonator [17–19], carbonnanotube [11], MEMS [32], EDL-IGR [20], redox-IGR [30], AMR array [20], spin torque nano-oscillators [12], and memristorarray [7] are shown. Blue and light green bars denote volumes of smaller and larger values.of 7.7–12 mm to obtain enough delay times [17–19], so the volume of the YIG is large. (The feedback loop inthe resonator can be ignored because it does not pose a significant problem for the magnetic volume due tominiaturization techniques.) On the other hand, although we use the same YIG as Watt et al in the presentstudy, we reduced the antenna separation of 30 µm by using interfered spin waves and a multi-detectiontechnique. The use of antennas with this short separation allows the device to operate as a smaller reservoirdevice than the active ring resonator [17–19]. In the future, further miniaturization in thin-film form isexpected to reduce the volume through dramatical reducing chip area, and the achieved small volume will becomparable to the volume of the fine physical reservoir with a volume of 10−6 cm−3 such as spin torqueoscillator arrays [12] and memristor arrays [7]. A description of the removal of the electromagnet to simplifythe reservoir system is shown in S4 of the supporting information.4.3. Power consumption of the physical reservoirsIn this study, input information is transferred as a spin wave without Joule loss since a spin wave does notinvolve a flow of electric charge. Figures 9(a) and (b) show the input voltage signal and the square voltagedivided by terminated electrical resistance R of 50 Ω (i.e. V 2/R), respectively, at the region within 1 µs. Here,the total time length per accumulation of input voltage is approximately 26 µs. The electric power [J] isobtained as the time integration of V 2/R. The calculated electrical power is 628 µJ at the condition of inputsusing two exciters and 500 times accumulations. The electric power is reduced to 1.26 µJ when theaccumulation is no longer necessary.Echo state networks and liquid state machines, the recurrent neural networks implemented incomplementary metal oxide semiconductor (CMOS) circuits, utilize memristor arrays [58, 59]. Theelectrical power consumption of the echo state network and the liquid state machine are 34.4 and 936 µJ,respectively [58, 59]. Thus, the electric power consumption estimated under the condition without theaccumulation is much lower than those networks (i.e. 1.26 µJ (this study)< 34.4 µJ [58]< 936 µJ [59]). Thereservoir network and peripheral circuit implemented on the terminal device can be designed based on finedevices such as magnon transistors, operating with a meager energy consumption of 10−18 J (≪ CMOS of10−16 J) [50] and readout layer consisting of magnetic tunnel junction array (approximately 50× 10−15 J)[60]. In this case, the total power consumption of the physical reservoir will be approximately 50× 10−15 J,which is comparable to the skyrmion-based spintronic physical reservoir (approximately 50× 10−15 J) [48]in the future.4.4. Implementation of the output layerThe output of the physical reservoir in this study is an analog circuit. To obtain the reservoir output y(k), thereadout requires the following elements: (1) an adder circuit based on a general inverting amplifier circuit forintegration, (2) an analog-to-digital converter (ADC), (3) a demultiplexer (DEMUX) to extract the nodes,11Neuromorph. Comput. Eng. 4 (2024) 024015 W Namiki et alFigure 9. (a) Input voltage signal and (b) the square voltage divided by terminated electrical resistance R of 50 Ω (i.e. V 2/R),respectively, at the region within 1 µs. A total time of input per accumulation is approximately 26 µs.(4) a memristor crossbar array for readout weights, (5) a multiply-accumulate to compute the sum ofproducts (i.e. y(k)) of weights and reservoir states. All elements can be fabricated using conventionallithographic techniques [47]. It has been demonstrated that a memristor crossbar array implemented in aCMOS circuit can represent 2048 electrical resistances and can be used sufficiently as a weight matrix [61]. Itis known that an ADC can be omitted by restricting the input to the reservoir and the output of the readoutmemristor crossbar array to binary [47], but in this study, the ADC is necessary because the task is to takeanalog random waveforms as input.We assumed the peripheral circuits required for the physical reservoir in this study and estimated theirarea, power, and delay. The peripheral circuits of the physical reservoir were based on previous studies ofphysical reservoirs implemented up to the readout layer [47]. It is assumed that an adder will be used toaccumulate the analog voltage signal induced by the spin wave, and a demultiplexer will be used to extractthe 100 nodes multiplexed into that voltage signal. The retrieved reservoir states are then weighted in amemristor crossbar array and summed in a MAC circuit to produce a reservoir output. An ADC is insertedbetween the adder and DEMUX to convert the voltage signal to a digital signal. It is assumed that the targetdata for training the reservoir output is converted to digital signals by the ADC to adjust the electricalresistance (i.e. output layer weights) of the memristor crossbar array.The number of nodes and the length of the output vector is represented by N and L. The feature size (S)of the peripheral circuit is assumed to be 65 nm (i.e. S= 65 nm) [47]. The area of the NMOS and PMOStransistors are assumed to be 4S2 and 8S2, respectively [47]. For a 2-input adder, 11 NMOS and 11 PMOStransistors (i.e., 22 in total) are used each, and the area of the adder for 500 inputs corresponding to 500integrations is 250× 11× (4S2 + 8S2)= 13.94 µm2 [58]. An analog-to-digital converter (ADC) consists of2 n–1 comparators depending on the number of bits n. The area of an n-bit ADC is represented by(2 n–1)× 52S2, which is approximately 56–224 µm2 (n= 8–10 bits) [59]. A 4ch DEMUX consists of 14NMOS and 14 PMOS, and the area of a DEMUX with 100 channels is 14× (4S2 + 8S2)× 25= 17.8 µm2.The area of a memristor is 10S2 [47], and the area of the memristor crossbar array connected between the Nof 100 and the output layer is 10S2 × N × L= 423 µm2. Finally, assuming that each multiplication-and-accumulation consists of 3 NMOS [60], its area is (4S2 × 3)× 100 nodes= 5.07 µm2.The power consumption of the memristor is 5.05 fJ, and from 100 nodes and the length of outputvectors, the power consumption of the memristor crossbar array is 5.05 fJ× 100× 100= 50.5 pJ. For otherCMOS circuits, the power consumption of the adder is 0.176 fJ [58], the power consumption of the ADC is151 nJ [59]. Since the power consumption of the physical reservoir is 628 µJ (500 times integration) and12Neuromorph. Comput. Eng. 4 (2024) 024015 W Namiki et al1.26 µJ without integration, the power consumption of the memristor crossbar array and typical peripheralCMOS circuitry is sufficiently small compared to that of the physical reservoir.Assuming implementation, the delay due to the readout layer is 0.1–35 ns in CMOS circuits [58, 59].Since the operating time of the physical reservoir is 26 µs× 500 accumulation, the readout is completed in avery short time compared to the input time of the time series data.4.5. Reduction of applied magnetic fieldThe magnetic field ranging from 150 to 250 mT was used to investigate the measurement conditiondependence of the computational performance, and it was found that the magnetic field for the highestcomputational performance was 169 mT. The applied magnetic field must satisfy the resonance conditionunder which spin waves are strongly excited. For a given input frequency, the magnetic field that excites spinwaves is determined by equation (3), and the computational performance becomes high near this magneticfield. In this study, the magnetic field of 169 mT satisfies the resonance condition, and the magnetic field wasvaried around this field from 150 to 250 mT to verify the magnetic field dependence. The resonancecondition is described by anisotropic magnetic field Ha and saturation magnetizationM, as shown inequation (3). It thus varies depending on the magnetic material and shape (bulk or thin film) with differentmagnetic properties. If the use of a large magnetic field will affect high-speed processing, the use offerromagnetic materials with perpendicular magnetic anisotropy will reduce the external magnetic field, thuspreserving the high speed of reservoir computing using spin-wave interference demonstrated in this study.To reduce the amplitude of the magnetic field, the selection of magnetic material with strong magneticanisotropy as the physical reservoir is needed. The YIG single crystal used in this study has in-planespontaneous magnetization, meaning that a large magnetic field is required to align the magnetization withthe out-of-plane direction. Here, when any magnetic material with out-of-plane spontaneous magnetizationis used, a large magnetic field along the out-of-plane is not required. Thus, the required magnetic field willbe successfully reduced. It is known that some magnetic material in the thin film form shows perpendicularmagnetic anisotropy [62, 63]. Demonstrating physical reservoir computing, which can operate under orwithout a small magnetic field application, is a future challenge to improve its practicality.In conclusion, we demonstrated physical reservoir computing utilizing spin wave interference with astepped input method. The physical device used in this study consists of a typical ferrimagnetic YIG andmultiple CPWs. The time-series data processing task, second-order nonlinear equation task, and NARMA2prediction task were carried out to evaluate the performance of said physical reservoir. NMSE and NMSEvarof the former and the latter were 1.39× 10−3 and 1.71× 10−1, respectively. These errors are lower than orcomparable to other compact physical reservoirs. Operating time at 5 ns step duration was successfullyreduced by 11.8%, compared to the physical reservoir utilizing spin wave interference with a pulsed inputmethod at a pulse interval of 5 ns [46]. This physical reservoir achieved the fastest operating time amongcompact physical reservoirs. A much shorter operating time can be realized by utilizing spin waves withhigher frequency, and it can transcend the fastest operating speed of optical circuits and physical devices withhuge volumes in the future.Data availability statementThe data cannot be made publicly available upon publication because no suitable repository exists forhosting data in this field of study. The data that support the findings of this study are available uponreasonable request from the authors.AcknowledgmentsInnovative Science and Technology Initiative for Security Grant Number JPJ004596, ATLA, Japan supportedthis work. This work was partly supported by the Japan Society for the Promotion of Science (JSPS),KAKENHI Grant number, JP21J21982 (Grant-in-Aid for JSPS Fellows). A part of this work was supported by‘Advanced Research Infrastructure for Materials and Nanotechnology in Japan (ARIM)’ of the Ministry ofEducation, Culture, Sports, Science and Technology (MEXT). Proposal Number JPMXP1223NM5072. Partof this work was supported by the Electron microcopy unit of the National Institute for Materials Science.ORCID iDsWataru Namiki https://orcid.org/0000-0003-4053-7366Daiki Nishioka https://orcid.org/0000-0002-3369-7700Takashi Tsuchiya https://orcid.org/0000-0002-6950-616013https://orcid.org/0000-0003-4053-7366https://orcid.org/0000-0003-4053-7366https://orcid.org/0000-0002-3369-7700https://orcid.org/0000-0002-3369-7700https://orcid.org/0000-0002-6950-6160https://orcid.org/0000-0002-6950-6160Neuromorph. Comput. 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Fast physical reservoir computing, achieved with nonlinear interfered spin waves 1. Introduction 2. Experimental methods 2.1. Fabrication of the physical device for reservoir computing with interfered spin wave multi-detection 2.2. Measurement setup of the physical device 2.3. Evaluation of spin wave property 2.4. Nonlinear time-series data processing task 2.5. Evaluation of processing performance of the physical reservoir computing 3. Results 3.1. Physical reservoir computing with nonlinear interfered spin wave and characteristics of the physical device 3.2. Process flow of physical reservoir computing 3.3. Time-series data processing task 4. Discussion 4.1. Processing time of physical reservoirs 4.2. Comparing the volume of the physical reservoir 4.3. Power consumption of the physical reservoirs 4.4. Implementation of the output layer 4.5. Reduction of applied magnetic field References