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Yu Yamaguchi, [Daiki Nishioka](https://orcid.org/0000-0002-3369-7700), [Wataru Namiki](https://orcid.org/0000-0003-4053-7366), [Takashi Tsuchiya](https://orcid.org/0000-0002-6950-6160), [Masataka Imura](https://orcid.org/0000-0002-4236-9549), [Yasuo Koide](https://orcid.org/0000-0001-8321-9822), Tohru Higuchi, [Kazuya Terabe](https://orcid.org/0000-0003-3988-3456)

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[Inverted input method for computing performance enhancement of the ion-gating reservoir](https://mdr.nims.go.jp/datasets/9a40799c-ef9c-4c90-ab85-1b2d5f4a7ec2)

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Inverted input method for computing performance enhancement of the ion-gating reservoirApplied Physics Express     LETTER • OPEN ACCESSInverted input method for computing performanceenhancement of the ion-gating reservoirTo cite this article: Yu Yamaguchi et al 2024 Appl. Phys. Express 17 024501 View the article online for updates and enhancements.You may also likeSynthesis and Electrochemical Studies ofIonic Liquid Electrolytes for Lithium-AirBatteriesYuji Ono, Md Mijanur Rahman,Byambasuren Delgertsetseg et al.-Infrastructure performance of irrigationcanal to irrigation efficiency of irrigationarea of Candi Limo in Mojokerto DistrictS Kisnanto, R R R Hadiani and C Ikhsan-A method to develop performanceindicators based on performance criteria ofIndonesian National OccupationalCompetency Standards (SKKNI) forconstruction safety technician competencyA Okviana and Y Latief-This content was downloaded from IP address 220.144.117.252 on 27/02/2024 at 22:17https://doi.org/10.35848/1882-0786/ad2906/article/10.1149/MA2020-024840mtgabs/article/10.1149/MA2020-024840mtgabs/article/10.1149/MA2020-024840mtgabs/article/10.1088/1757-899X/333/1/012096/article/10.1088/1757-899X/333/1/012096/article/10.1088/1757-899X/333/1/012096/article/10.1088/1757-899X/930/1/012010/article/10.1088/1757-899X/930/1/012010/article/10.1088/1757-899X/930/1/012010/article/10.1088/1757-899X/930/1/012010/article/10.1088/1757-899X/930/1/012010https://googleads.g.doubleclick.net/pcs/click?xai=AKAOjstAhGji0EDNA-jdh4vcGOvX5G9rhuVOajAAxLo7zlAwMe7n04o6pwsRlPBSnSvM5Bva47CPI1KG8PElG0z4Bevit-u64Ahrhn07cw-vAxo9cIt-TiB74IFq_3uVrex_Xp9L_VyF-9SOORjMtvfCt2AZCMEcH3oYrUjR910v-X0FPIT0_Ni8hPjlH9CIng79vdzm0FXDyQJiIMNl5f0kDpXR6YFiXBTl9ZlncVJccCF60VOOAAV7O_HR3hd_A6ImekKIiPcYezufRu0NMQM3LVQZ_vEeYR8Ki-fseyAdhcK-WlzP0bOXX3QBN7tyh1hBnssBiEOlRg&sai=AMfl-YR0cAxTlbhne83YLs5C5O0SR4a1qNRy3cAuRYxzelTyP_M7TJsGJAdqbEkrUtIXWtm3WQej3OKtfdXBg7M&sig=Cg0ArKJSzE1UR3ZPvw7l&fbs_aeid=%5Bgw_fbsaeid%5D&adurl=https://ecs.confex.com/ecs/prime2024/cfp.cgi%3Futm_source%3DIOP%26utm_medium%3Dbanner%26utm_campaign%3Dprime_abstract_submissionInverted input method for computing performance enhancement of the ion-gatingreservoirYu Yamaguchi1,2, Daiki Nishioka1,2, Wataru Namiki1 , Takashi Tsuchiya1* , Masataka Imura3, Yasuo Koide4,Tohru Higuchi2, and Kazuya Terabe11Research Center for Materials Nanoarchitectonics (MANA), National Institute for Materials Science (NIMS), 1-1 Namiki, Tsukuba, Ibaraki, 305-0044,Japan2Department of Applied Physics, Faculty of Science, Tokyo University of Science, 6-3-1 Niijuku, Katsushika, Tokyo, 125-8585, Japan3Research Center for Functional Materials, NIMS, 1-1 Namiki, Tsukuba, Ibaraki, 305-0044, Japan4Research Center and Facility Services Division, NIMS, 1-2-1 Sengen, Tsukuba, Ibaraki, 305-0047, Japan*E-mail: TSUCHIYA.Takashi@nims.go.jpReceived January 21, 2024; accepted February 12, 2024; published online February 27, 2024Physical reservoir computing (PRC) is useful for edge computing, although the challenge is to improve computational performance. In this study,we developed an inverted input method, the inverted input is additionally applied to a physical reservoir together with the original input, to improvethe performance of the ion-gating reservoir. The error in the second-order nonlinear equation task was 7.3 × 10−5, the lowest error in reported PRCto date. Improvement of high dimensionality by the method was confirmed to be the origin of the performance enhancement. This inverted inputmethod is versatile enough to enhance the performance of any other PRC. © 2024 The Author(s). Published on behalf of The Japan Society ofApplied Physics by IOP Publishing LtdSupplementary material for this article is available onlineAmong various state-of-art neuromorphic computing ap-proaches,1–6) physical reservoir computing (PRC) is a parti-cularly promising computation scheme to reduce energyconsumption in AI-based information processing.1) In thisscheme, materials and devices are utilized as physicalreservoirs to efficiently process information, by mappinginput to a higher-dimensional feature space using their ownintrinsic nonlinearity.1) Whereas PRC using various materialsand devices has been reported so far, the performance is notsufficient.1,3–23) Although nonlinearity, high dimensionality,and short-term memory (STM) are known to be required tobe improved for high-performance PRC,1) a universal meth-odology to improve them has not yet been established. Inparticular, improving high dimensionality is difficult in PRCbecause of limitations in the structure of PRC devices. Amasking process improves high dimensionality by adding acertain amount of modulation to the desired signal andinputting it into the physical reservoir to maintain its transientstate.20,24) However, this has its drawbacks, such as requiringcomplex processing with adjustment of hyperparameters andmaking the data length of input very long. This causes anurge for the development of different approaches for theimprovement of high dimensionality. Herein, we demonstratean inverted input method as a new approach for highdimensionality improvement and evaluate its effect oncomputational performance. An inverted input method is asimple method in which inverted input is generated by aninversion (linear transformation) of the original input andadditionally applied to PRC with original input, leading to farlower processing costs for maintaining transient states of thereservoir than general masking methods. An “ion-gatingreservoir (IGR)” which uses the nonlinear response oftransistors with ions,20–23) was employed as a model caseto evaluate the effect of using inverted inputs on theprediction error of a second-order nonlinear equation task.Furthermore, to investigate the cause of the error reduction bythe inverting input, we quantitatively evaluated high dimen-sionality and STM, which are requirements for PRC.1,25)A schematic diagram of the IGR is shown in Fig. 1(a).This IGR is based on an electric double-layer (EDL)transistor22,23,26,27) fabricated with a Li-Si-Zr-O (LSZO)lithium ion-conducting amorphous electrolyte thin film,which is deposited by pulsed laser deposition, and ahydrogen-terminated diamond homoepitaxial (100) substrate,which is deposited by microwave-plasma chemical vapordeposition. Electronic carrier density on the diamond chan-nel’s surface is tuned by EDL formation due to Li+ transportdriven by applied gate voltage (Vg).22,23,26–30) As shown inFig. 1(a), the drain current (Id) is largely modulated by the Vgapplication. Please refer to the Supplementary Material fordetails of the device’s fabrication and electrical measure-ments.A general scheme of PRC is shown in Fig. 1(b). In theinput layer, time series data u(k) are inputted, where k is thetime step. Then, in the reservoir layer, input time series datais transformed nonlinearly to high dimensional space asreservoir states Xi(k) at given node i (i = 1, 2, …, N).1,31)For the output layer, the readout weight wi connecting Xi(k)and output yi(k) is trained by linear regression to obtain thedesired output. The reservoir output y(k) is described as alinear combination of Xi(k) and the wi, as follows,y k w X k b 1iNi i1å= +=( ) ( ) ( )where N and b are the number of the reservoir state and bias,respectively.As shown in Fig. 1(c), we prepared the input u(k) as arandom value (0.0 ≦ u(k) ≦ 0.5) and inverted input u′(k) (u′(k) = 0.5-u(k)). By combining both input u(k) and u′(k), theinverted input method enhances high dimensionality in PRC.The principle is explained below. The function of thereservoir in reservoir computing is to map the input to aContent from this work may be used under the terms of the Creative Commons Attribution 4.0 license. Any further distribution of thiswork must maintain attribution to the author(s) and the title of the work, journal citation and DOI.024501-1© 2024 The Author(s). Published on behalf ofThe Japan Society of Applied Physics by IOP Publishing LtdApplied Physics Express 17, 024501 (2024) LETTERhttps://doi.org/10.35848/1882-0786/ad2906https://crossmark.crossref.org/dialog/?doi=10.35848/1882-0786/ad2906&domain=pdf&date_stamp=2024-02-27https://orcid.org/0000-0003-4053-7366https://orcid.org/0000-0003-4053-7366https://orcid.org/0000-0002-6950-6160https://orcid.org/0000-0002-6950-6160https://orcid.org/0000-0003-3988-3456https://orcid.org/0000-0003-3988-3456mailto:TSUCHIYA.Takashi@nims.go.jphttps://doi.org/10.35848/1882-0786/ad2906https://creativecommons.org/licenses/by/4.0/https://doi.org/10.35848/1882-0786/ad2906high-dimensional feature space.1,31) The reservoir’s mappingfunction fi for the input ui and reservoir state Xi is shown inFig. 1(a). In a simulation reservoir such as an echo statenetwork, nonlinear functions such as tanh are employedidentically for all nodes (i.e., fi = fj≠i).25,31) On the otherhand, in PRC, a variety of nonlinear functions that originatefrom nonlinear dynamics that are inherent in the physicalsystem are employed as mapping functions for each node(i.e., for each measurement terminal or sampling time/virtualnode).1,3–23) These mapping functions change dynamicallyaccording to the driving conditions (e.g., input intensity andtime scale) of the physical system (i.e., fi ≠ fj≠i). In this IGR,the Id-Vg and gate current (Ig)-Vg characteristic were utilizedas mapping functions for the nonlinear transformation to highdimensional space. The relationship between the reservoirstate and the mapping function is defined by the followingequation.X k f u k X k1 1 , 2i i i+ = +( ) [ ( ) ( )] ( )As shown in Fig. 1(c), the relation between u(k) and u′(k)is linear. Yet, the nonlinearity of the fi causes them to bedifferent outputs from each other (i.e., two outputs are not ina linear relationship), allowing them to map to a higherdimensional feature space than is possible with a single u(k).This is the mechanism of higher performance with invertedinputs, which is a particularly effective scheme for physicalsystems with diverse and dynamic mapping functions,compared to simulation reservoirs with uniform mappingfunctions. In addition, the effective number of nodes in thereservoir increases with the improvement in high dimension-ality; therefore, STM can also be improved.1,25)In this present study, the effect of the inverted input hasbeen investigated with a second-order nonlinear equationtask, which is a typical benchmark task of PRCs.16–22) Thetarget output yt(k) for the task is determined from the(a)(b) (c)Fig. 1. (a) Schematic illustration of electric double layer effect-based IGR. (b) General scheme of PRC. (c) Basic role of inverted input in PRC.024501-2© 2024 The Author(s). Published on behalf ofThe Japan Society of Applied Physics by IOP Publishing LtdAppl. Phys. Express 17, 024501 (2024) Y. Yamaguchi et al.equation below.y k y k y k y ku k0.4 1 0.4 1 20.6 0.1 3t t t t3= - + - -+ +( ) ( ) ( ) ( )( ) ( )For the input data, input u(k) and inverted input u′(k) wereprepared. As referenced in Fig. S2, u(k) and u′(k) wereconverted to a voltage pulse stream with a pulse period T(10–80 ms) and duty rate D (40%–80%). The constant drainvoltage Vd was set to 0.1 V. Voltage pulse stream was appliedto the common gate and the output current was measuredfrom 9 terminals of drains (i.e., 9 drain current) which havedifferent channel lengths and 1 terminal of the gate (i.e.,1 gate current), in a total of 10 physical nodes. Underconditions without u‵(k), 20 virtual nodes were extracted witheven spacing from each current response from 10 terminals.Therefore, 200 virtual nodes (10 physical nodes × 20 virtualnodes) were utilized in total for the task. Under conditionwith u′(k), 10 virtual nodes were extracted with even spacingfrom each current response from 10 terminals. Thus,200 virtual nodes (2 ways of input × 10 physical nodes× 10 virtual nodes) were utilized similarly in total for thetask. By utilizing both physical and virtual nodes, uniquediverse reservoir states can be extracted, which enables thehigher dimensionality necessary for high performance.To evaluate the performance of the task, we calculated thenormalized mean squared error (NMSE) of the yt(k) fromEq. (3) and the prediction output y(k) trained by Eq. (1), asfollows.y k y ky kNMSE 4kLtkLt1212=å -å=={ ( ) ( )}( )( )where L(=500) is a data length.Figures 2(a) and 2(b) show yt(k) and y(k) of the second ordernonlinear equation task without u′(k) and with u′(k) when theNMSE was at a minimum under the best conditions of T and D(i.e., T = 70ms and D = 70%), which suggest the long T andlarge D enhance at least one of the three properties required forPRC, leading to precise prediction performance in the task.Please refer to the Supplementary Material for details of T andD dependence on the performance. By combining u(k) andu′(k), it was possible to predict more precisely which achievedthe NMSE of 7.3 × 10−5, whereas it was 1.4 × 10−4under conditions without u′(k). Moreover, the NMSEunder condition with u′(k) was the smallest NMSE in thesecond-order nonlinear equation task among reportedPRC.16–22)In order to clarify the origin of such performanceenhancement of the IGR due to the introduction of theinverted input method, we quantitatively analyzed the highdimensionality and STM of the system under the twoconditions (i.e., input without u′(k) and with u′(k)). Asdiscussed above, introducing an inverted input providesdiverse reservoir states due to the nonlinearity and asym-metry of the mapping function. Figures 3(a) and 3(b) showall reservoir states used for the task under conditions withoutu′(k) and with u′(k) respectively. The system diversity isclearly enhanced by adding the reservoir state X′(k) for theinverted input u′(k) as shown in Fig. 3(b) to the reservoir stateX(k) for the original input u(k) as shown in Fig. 3(a). Inparticular, X(k) and X′(k) are not inversely symmetric due tothe nonlinearities of the mapping functions, despite u(k) andu′(k) being inversely symmetric. Therefore, the inverted inputsurely improved the high dimensionality of this system. Thehigh dimensionality in PRC is achieved by the presence ofmultiple nodes that behave independently, which can becharacterized by a correlation coefficient r between eachnode Xi and Xj≠i defined by following equation20,21)5r X XX k X X k XX k X X k X,i jkLi i j jkLi i kLj j11501150 21150 2=å - -å - å -======( )( )( )( )( ( ) ̅ ) ( ) ̅( ( ) ̅ ) ( ) ̅where Xi̅ is the average value of Xi. Please refer toSupplementary Material for the details of the correlationcoefficient analysis.Figure 3(c) shows the correlation coefficients betweeneach node under the conditions with u(k) and u′(k) combined.X1∼X100 are nodes under the condition with u(k) andX′1∼X′100 are nodes under the condition with u′(k). Inaddition, X90∼X100 and X′90∼X′100 were extracted from theIg response, and all other reservoir states were extracted fromthe Id response, resulting in lower correlation coefficientsbetween reservoir states extracted from Ig and Id. Overall, thecorrelation coefficients between X1∼X100 and X′1∼X′100 werelow compared to pairs of both X1∼X100 and both X1∼X100,which suggests that X(k) from u(k) and X′(k) from u′(k) arenot linear relationships due to the nonlinear transformation bymapping function of the IGR. Therefore, the inverted inputmethod effectively boosts PRC’s high-dimensionality, whichmaps the input data to high-dimensional feature space basedon the nonlinear dynamics inherent in the physicalsystem.20,21)(a)(b)Fig. 2. The target and prediction waveforms under conditions without u′(k)(a) and with u′(k) (b) in pulse period of 70 ms and duty rate of 70%.024501-3© 2024 The Author(s). Published on behalf ofThe Japan Society of Applied Physics by IOP Publishing LtdAppl. Phys. Express 17, 024501 (2024) Y. Yamaguchi et al.High dimensionality is also dependent on the diversity ofeach node and the number of nodes.1) To evaluate the overallhigh dimensionality of the system, the sum of 1-|r| is rathersuitable for quantification.20,21) Figure 3(d) shows the rela-tionship between NMSE of the second-order nonlinearequation task under all T and D conditions and the sum of 1-|r| under both conditions without u′(k) (red dot) and with u′(k)(blue dot). The sums of 1-|r| tended to be larger with u′(k)than without u′(k), meaning that the overall high dimension-ality is enhanced with u′(k). More importantly, there appearsto be a clear trend where NMSE is low when the sum of 1-|r|is high. Therefore, it is evidenced that utilizing the invertedinput method improves the high dimensionality, leading tohigh performance.Moreover, improvement of the high dimensionality canfurther increase STM, which can be evaluated with a delaytask in which the reservoir reconstructs historical time seriesdata.1) The input of this task is the same random input as usedin the second-order nonlinear equation task, and the u(k-τ)before the delay time τ was reconstructed by a linearcombination of X(k) and w obtained from the currentresponse. The difference between the target waveformu(k-τ) and the reconstructed waveform y(k) was evaluatedby the following equation of determination coefficient r2ru k y ku k y kCov ,Var Var622tt=-´( ) ( ( ) ( ))( ( ) ( ( )))( )where Cov() and Var() are covariance and variance respec-tively. Figure 3(e) shows the forgetting curve, the τdependence of r2. The ability to reconstruct past data,represented by r2, decreased with increasing τ. Memorycapacities (MCs) were calculated by integration of theforgetting curves described in the following equationrMC 712å t=t=¥( ) ( )MCs were calculated as 4.34 without u′(k) and 4.71 withu′(k). The theoretical limit of the MC is the same as thereservoir size (the number of nodes), but generally, inphysical systems, the MC is much lower than the number ofnodes.25) That is because, in a physical system, there aremany nodes that behave similarly, which causes a lowereffective reservoir size than a number of nodes.1) Thus, it isessential to improve high dimensionality, which correspondsto an effective reservoir size to increase the MC. To evaluateoverall STM, we analyzed the relationship between the MCand the sum of 1-|r|, as shown in Fig. 3(f). Under thecondition with u′(k), both the sum of 1-|r| and the MC tend tobe large. Conversely, under the condition without u′(k), thesum of 1-|r| and the MC are small. Therefore, it is suggestedthat the MC increased due to the improvement in highdimensionality despite using the same number of nodes.Introducing the inverted input significantly enhances theperformance of this IGR due to the improvement in highdimensionality and STM.In conclusion, the inverted input method was applied toEDL-based IGR for PRC performance improvement. TheNMSE in a second-order nonlinear equation task was(a) (b) (c)(d) (e) (f)Fig. 3. (a) All reservoir states X under conditions without u′(k) (a) and with u′(k) (b), (c) The heatmap of r for 200 nodes consists of u(k) and u′(k) atT = 70 ms and D = 70%. X1 ∼ X100 corresponds to condition without u′(k). X′1 ∼ X′100 corresponds to condition with u′(k) (d) The relation between NMSE ofsecond order nonlinear equation task and SUM of 1-|r| (e) The forgetting curve under both conditions without u′(k) and with u′(k) at T = 70 ms and D = 70%(f) The relation between MC and SUM of 1-|r|.024501-4© 2024 The Author(s). Published on behalf ofThe Japan Society of Applied Physics by IOP Publishing LtdAppl. Phys. Express 17, 024501 (2024) Y. Yamaguchi et al.significantly reduced from 1.4 × 10−4 to 7.3 × 10−5,confirming the effectiveness of the additional inverted inputand achieving the best performance of any physical reservoirreported to date. High dimensionality was evaluated bycalculating the correlation coefficient r, r between eachnode notably decreased by applying the additional invertedinput. Higher dimensionality due to additional inverted inputwas the main origin of performance improvement of thereservoir by analyzing the relationship between NMSE andthe sum of 1-|r|. Also, the MC increased from 4.34 to 4.71 byapplying the inverted input method. The MC increase isaccompanied by a decrease in r. This inverted input methodis versatile enough to be applied to any physical reservoir andmay improve performance. Compared to masking, thismethod is easy to use and improves computational perfor-mance at a low computational cost without hyperparameters,because it can generate diverse reservoir states with a simpleprocess and avoid increasing the input data length.Acknowledgments This work was supported in part by the Japan Societyfor the Promotion of Science (JSPS) KAKENHI Grant No., JP22H04625 (Grant-in-Aid for Scientific Research on Innovative Areas “Interface Ionics”) andJP22KJ2799 (Grant-in-Aid for JSPS Fellows), and JST PRESTO (Grant No.,JPMJPR23H4). A part of this work was supported by the Iketani Science andTechnology Foundation and JFE 21st century foundation. A part of this work wassupported by “Advanced Research Infrastructure for Materials andNanotechnology in Japan (ARIM)” of the Ministry of Education, Culture, Sports,Science and Technology (MEXT). 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