Abstract
Memristors with tunable resistance states are emerging building blocks of artificial neural networks. However, in situ learning on a largescale multiplelayer memristor network has yet to be demonstrated because of challenges in device property engineering and circuit integration. Here we monolithically integrate hafnium oxidebased memristors with a foundrymade transistor array into a multiplelayer neural network. We experimentally demonstrate in situ learning capability and achieve competitive classification accuracy on a standard machine learning dataset, which further confirms that the training algorithm allows the network to adapt to hardware imperfections. Our simulation using the experimental parameters suggests that a larger network would further increase the classification accuracy. The memristor neural network is a promising hardware platform for artificial intelligence with high speedenergy efficiency.
Introduction
With the introduction of hardware accelerators^{1,2,3,4} for inference in deep neural networks (DNNs)^{5,6,7,8,9}, the focus on improving overall energy and time performance for artificial intelligence applications is now on training. One promising approach is inmemory analog computation based on memristor crossbars^{10,11,12,13,14,15,16,17,18}, for which simulations have indicated potentially significant speed and power advantages over digital complementary metaloxidesemiconductor (CMOS)^{19,20,21,22,23}. However, experimental demonstrations to date have been limited to discrete devices^{24,25} or small arrays and simplified problems^{26,27,28,29,30,31}. Here we report an experimental demonstration of highly efficient in situ learning in a multilayer neural network implemented in a 128 × 64 memristor array. The network is trained on 80 000 samples from the Modified National Institute of Standards and Technology (MNIST)^{32} handwritten digit database with an online algorithm, after which it correctly classifies 91.71% of 10 000 separate test images. This level of performance is obtained with 11% devices in the crossbar unresponsive to programming pulses and the training algorithm blind to the defectivity, demonstrating the selfadapting capability of the in situ learning to hardware imperfections. Our simulation based on the memristor parameters suggested that the accuracy could be higher than 97% with a larger (e.g., 1024 × 512) memristor array. Our results indicate that analog memristor neural networks can achieve accuracy approaching that of stateoftheart digital CMOS systems with potentially significant improvements in speedenergy efficiency.
Memristors offer excellent size scalability (down to 2 nm)^{33}, fast switching (faster than 100 ps)^{34}, and low energy per conductance update (lower than 3 fJ)^{34}. Their tunable resistance states can be used both to store information and to perform computation, allowing computing and memory to be integrated in a highly parallel architecture. However, given the level of technology maturity, attempts to implement memristive neural networks have struggled with device nonuniformity, resistance level instability, sneak path currents, and wire resistance, which have limited array sizes and system performance. In particular, learning in memristor neural networks has been hampered by significant statistical variations and fluctuations in programmed conductance states and the lack of linear and symmetric responses to electric pulses^{35}.
Here we develop a reliable twopulse conductance programming scheme utilizing onchip series transistors to address the challenges in memristor conductance programming. This in situ training scheme enables the network to continuously adapt and update its knowledge as more training data become available, which significantly improves accuracy and defect tolerance.
Results
Linear and symmetric conductance tuning
We used recently developed Ta/HfO_{2}/Pt memristors to achieve stable tunable multilevel behavior with a linear current–voltage (IV) relationship^{36,37}. The memristors were monolithically integrated with foundrymade transistor arrays on a 6inch wafer (see Methods). Each memristor was connected to a series transistor in a “1T1R” configuration (Fig. 1a–e shows the integrated memristor array from wafer scale to nanometer scale). To increase the conductance of a given cross point, we applied synchronized positive voltage pulses from a driving circuit board to the memristor top electrode and the gate of the series transistor. The gate voltage, which specifies a compliance current, determines the resulting memristor conductance. We decreased the conductance by first applying a sufficient positive pulse to the memristor bottom electrode to initialize the state, and then used the conductance increase scheme to set the memristor to the desired level (illustrated in Supplementary Fig. 1). With this scheme, we achieved linear and symmetric conductance increase and decrease with minimal cycletocycle (Fig. 1f, g) and devicetodevice (Fig. 1f, h) variations. We were able to set the conductance values across the entire 128 × 64 array, except for the stuck devices, with reasonably high accuracy using only two electrical pulses to each memristor (Fig. 1i and Supplementary Fig. 2). The speed and reliability of the conductance update scheme make it possible to train the network in situ with almost any standard algorithm. Here the network was trained using stochastic gradient descent (SGD)^{32} to classify handwritten digits in MNIST dataset. For each new sample of training data, the network first performs inference to get the logprobability of the label for each output by the softmax function, and then the weights in each layer are updated accordingly (see Methods).
In situ training in memristor crossbar
To implement the SGD algorithm in the memristor crossbar, each synaptic weight was encoded as the difference of the conductance between two memristors. Inference was performed by biasing the top electrodes of memristors in the first layer with a set of voltages whose amplitudes encode an image, then reading the currents from the bottom electrodes of devices in the final layer (Fig. 2a, b) using custombuilt circuit boards that can address up to 64 channels in parallel (Supplementary Fig. 3 and Supplementary Fig. 4). During inference, all transistors operate in the deep triode region, and the memristor array becomes a pseudocrossbar capable of performing matrix multiplication following Ohm’s law and Kirchhoff’s current law^{37,38} (see Supplementary Fig. 5a). The hidden neurons after each layer apply a nonlinear activation (in this work, a rectified linear function in software) to the weighted sums computed in the crossbar. The desired weight update (ΔW) for each layer was calculated in software using Eq. 1, then applied to the crossbar by the measurement system (see Fig. 2c for the algorithm flow chart, Supplementary Fig. 6 for a unified modeling language class diagram, and Supplementary Fig. 5b, c for a parallel weight update scheme).
where η is the learning rate, v is the input voltage column vector for the lth layer, δ_{ l } the output error column vector for the layer, n indexes over the sample, and B is the batch size. For a network with L layers, the error row vectors are computed using
where y_{ j } is the Bayesian probability computed by the network and t_{ j } is 1 if this sample belongs to class j and 0 otherwise. This calculation ensures that the network maximizes the loglikelihood of the correct classification for each example.
The error backpropagation^{39,40} in this work is calculated in software from the values of the readout weights (see Methods). In the future, backpropagation can be implemented within the memristor crossbar by applying a voltage vector representing the currentlayer error to the bottom electrodes of the crossbar and reading out the current vector from the top electrodes for the previouslayer error. An onchip integrated peripheral for full hardware implemented functionality is under development, which has been discussed and simulated in the literature as well^{41,42,43,44,45}.
Classification of MNIST handwritten digits
We partitioned a single 128 × 64 array and constructed a twolayer perceptron with 64 input neurons, 54 hidden neurons, and 10 output neurons to be trained on the MNIST dataset of handwritten digits “0” through “9”, which has become a standard benchmark by which to gauge new machine learning algorithms. Each input image was rescaled to 8 pixels by 8 pixels (see Supplementary Fig. 7, and sample images in Fig. 3a) to match our network size. The intensities of each pixel of the grayscale images were unrolled into 64dimensional input feature vectors, which were duplicated to produce 128 analogue voltages to enable negative effective weights (Fig. 2b). The twolayer network used 7992 memristors (see Fig. 3b for the partition on a 128 × 64 array), each of which was initialized with a single pulse with a 1.0 V gate voltage from a lowconductance state. The network was then trained on 80 000 images drawn from the training database (some images were drawn more than once), with a minibatch size B = 50 for a total of 1600 training cycles. The smoothed minibatch experimental accuracy (compared with a defectfree simulation) during online training is shown in Fig. 3c. Figure 3d shows the linear relationship between the conductance and the applied gate voltage during each update cycle, which was critical for this demonstration. More analyses are shown in Supplementary Figs. 8, 9. After utilizing the entire training database, the network correctly classified 91.71% of the 10 000 images in the separate test set (Figs. 3e–j, Supplementary Table 1). Many of the misclassified images are in fact difficult for humans to identify at the available resolution (Fig. 3h, and more in Supplementary Fig. 10).
To understand the potential of the memristor array, we developed a simulation model (see Supplementary Fig. 6 for detailed architecture) based on measured parameters such as the unresponsive rate, conductance update error, and limited memristor conductance dynamic range. We found that the simulated accuracy agrees well with the experimentally achieved one (Fig. 4a), validating the simulation model. A further simulation on a defectfree network shows that the MNIST classification accuracy is similar to that of the same network architecture trained in TensorFlow^{46} that uses 32bit floating point numbers. This result suggests that even though the analogue memristor network has limited precision due to conductance variation, they do not have a significant impact on MNIST classification accuracy (more analysis on conductance update variation is shown in Supplementary Fig. 11). Our finding is consistent with previous theoretical and simulation studies on more sophisticated problems^{20,24,47,48,49,50,51}. The analog precision could potentially be improved, if needed for other applications, by device engineering for better IV linearity^{37,38}, or using pulse width instead of amplitude to represent analog input (with increased time overhead)^{27,28,37}, or employing multiple memristors to represent one synaptic weight (at the expense of chip area)^{52}, etc.
A multilayer neural network trained with the online algorithm is more tolerant to hardware imperfections. The experimental accuracy of 91.71% in this work was achieved with 11% devices unresponsive to conductance updates. Our simulation showed that even with 50% of the memristors stuck in a lowconductance state, a >60% classification accuracy is still possible through online training (Fig. 4b), although the accuracy is much more sensitive to shorted devices (Supplementary Fig. 12). On the other hand, if pretrained weights are loaded to the memristor crossbar (i.e., ex situ training), the classification accuracy decreases quickly with the defect rate (Fig. 4b)). There are approaches to improve the robustness of ex situ training^{19,38}, but most of them require that the parameters be tuned based on specific knowledge of the hardware (e.g., peripheral circuitry) and memristor array (e.g., device defects, wire resistance, etc.), while the in situ training adapts the weights and compensates them automatically. The selfadaption is more powerful in a deeper network in which the hidden neurons are able to minimize the impact of defects, as suggested by the higher classification accuracy from a twolayer network than that from a singlelayer one on the same images (Fig. 4c). The online training is also able to update the weights to compensate for possible hardware and memristor conductance drift over time (see Supplementary Fig. 13).
We expect that the classification accuracy can be improved substantially with a larger network that has more hidden units and/or more inputs to support images with higher resolution. We performed a simulation with our model on a 1024 × 512 memristor array, which is likely to be available in the near future, to recognize images of 22 × 22 pixels cropped from the MNIST dataset. The network consists of 484 input neurons, 502 hidden neurons, 10 output neurons, and a total of 495 976 memristors in the two layers to represent the synaptic weights (see Supplementary Fig. 14). After training on 1,200,000 images (20 epochs), a 97.3 ± 0.4% classification accuracy is achieved on the test set even with 11% stuck devices, approaching that demonstrated with traditional hardware. It will be straightforward to build deeper fully connected neural networks on an integrated chip with multiple large arrays in the near future, for even better accuracy and application to more complicated tasks. It is also noteworthy that most stateofart DNNs involve sophisticated microstructures, e.g., convolutional neural networks (CNNs) or long shortterm memory units (LSTMs). It may be worth investigating how to implement CNNs^{37} or LSTMs efficiently on memristor crossbars in the future. But on the other hand, such microstructurebased algorithms have been developed for use on conventional hardware, on which it is more efficient to process sparse matrices. Since the advantages of using sparse matrices in a memristor crossbar are minimal, the optimal architectures for sophisticated tasks may look different.
Discussion
A further potential benefit of utilizing analog computation in a memristorbased neural network is a substantial improvement in speedenergy efficiency. The advantages mainly come from the fact that the computation is performed in the same location used to store the network data, which minimizes the time and energy cost of accessing the network parameters required by the conventional vonNeumann architecture. The analog memristor network is also capable of handling analog data acquired directly from sensors, which further reduces the energy overhead from analogtodigital conversion. The memristors we used maintain a highly linear IV relationship, allowing for the use of voltageamplitude as the analog input for each layer. This also minimizes circuit complexity and hence energy consumption for future hardware hidden neurons and output current readout. While the external control electronics we use in this work is not optimized for fast speed and low power consumption yet, previous literature on circuit design^{45,51} and architecture^{21,53} suggest an onchip integrated system would yield significant advantages in speedenergy efficiency.
In summary, we have demonstrated the in situ and selfadaptive learning capability of a multilayer neural network built by monolithically integrating memristor arrays onto a foundrymade CMOS substrate. The transistors enable reliable, linear, and symmetric synaptic weight updates, allowing the network to be trained with standard machine learning algorithms. After training with a SGD algorithm on 80 000 images drawn from the MNIST training set, we achieved 91.71% accuracy on the complete 10,000image test set. This accuracy is 2.4% lower than an idealized simulation despite an 11% defect rate for the memristors used. The demonstrated performance with in situ online training and inference suggests that memristor crossbars are a promising high speed and energy efficiency technology for artificial intelligence applications. The software neurons used in this demonstration indicate that a hybrid digital processor and neuromorphic analogue approach for DNNs can be effective, but all the software functions used in the present demonstration can be integrated as hardware onto a fullfunction chip in the near future.
Methods
Device fabrication and array integration
The transistor array and interconnection between cells are taped out from a commercial foundry with 2 μm technology node to achieve low wire resistance. We monolithically integrate memristors on top of asreceived chip in house. Pd/Ag contact metals are first deposited on both vias after argon plasma treatment to remove the native oxide. The chip is then annealed at 300 °C for 30 min in 20 s.c.c.m nitrogen flow to achieve good electrical contact. The memristor bottom electrode is deposited by evaporating 20 nm Pt on top of a 2 nmthick Ta adhesive layer and patterned by photolithography and liftoff in acetone. A switching layer of 5 nmthick HfO_{2} is deposited by atomic layer deposition using water and tetrakis(dimethylamido)hafnium as precursors at 250 °C, and then patterned by photolithography and reactive ion etch. Finally, the top electrode of 50 nmthick Ta is deposited by sputtering and liftoff, followed by sputtering of a 10 nmthick Pd protection layer.
Dataset
The dataset is composed of the input feature vector (x(n) for sample n) and the target output vector (t(n)). For the classification problem, t_{ c } (n) = 1 if sample n belongs to class c, and is 0 otherwise. For the MNIST dataset, feature vectors are the unrolled grayscale pixel values of the handwritten digital twodimensional images. The original images are 28 pixels by 28 pixels. They were cropped to 20 × 20 and then further down sampled to 8 × 8 (using bicubic interpolation) to match the input size of the memristor neural network (Supplementary Fig. 7). The 8 × 8 grayscale images were then unrolled to 64dimensional input feature vectors. The input feature vectors were converted to voltage values v_{1}(n)by a scaling factor, which was the same for all images. The output vectors have 10 dimensions, each corresponding to one digit.
Inference
The in situ online training was composed of two stages: feedforward inference and feedback weight update. The multilayer inference was performed layer by layer sequentially. The input voltage vector to the first layer was a feature vector from the dataset, while the input vector for the subsequent layer was the output vector of the previous layer. The analog weighted sum step was performed in the memristor crossbar array as indicated by Eq. 3, or equivalently by Eq. 4:
where v_{ l } is the lth layer input voltage vector that is applied to the top electrodes of the memristor crossbar, i_{ l } is the readout current vector from the bottom electrodes of the crossbar, and W_{ l } is the weight matrix of layer l. The total current is the sum of the currents through each device in the same column (Kirchhoff’s current law), while each current is the product of the conductance and the voltage across the memristor (Ohm’s law). Each weight value is represented by the difference in conductance between two memristors: \(W_{ij} = G_{ij}^{\, + } G_{ij}^ \), so that weights can be negative. This is accomplished by duplicating the voltage vector v_{ l }, with +v_{ l } applied to half of the array and −v_{ l } applied to the other half, as shown in Fig. 2c.
We chose a rectified linear unit activation function for the hidden layer, which is defined in
where c is a scaling factor to match the voltage range. For the MNIST network in this work, c was set to 200 V/A, and elements of the resulting voltage vector that exceed 0.2 V were clipped to avoid altering the memristor states. This particular step was implemented by software in this investigation, and can be easily implemented in the future with a rectifying diode and amplifier on an integrated chip. The most active (highest amplitude) output current was interpreted as the classification result.
Softmax and crossentropy loss function
The inference result can also be calculated as a Bayesian probability, using the conversion defined by Eq. 6.
where \(y_c\left( n \right)\) is the probability that sample n belongs to class c, and C is the total number of classes. k was set to 5 × 10^{5}/A for the MNIST network in this work.
The goal of the training process was to adjust the weight values to maximize the loglikelihood of the true class. We used a crossentropy loss function, which is defined in Eq. 7.
where N is the total number of samples.
SGD with backpropagation
In order to estimate the gradient of the loss function for training the weights, we withdrew a subset of B samples (termed a minibatch) from the training set without replacement at each round of training. The SGD algorithm was used to update the weights along the direction of steepest descent for \({\Bbb E}\left[ \xi \right]\). The desired weight update was given by Eq. 1 in the main text. For a network with L layers, the error vector is computed by
where σ is the nonlinear activation function for the hidden layer and ξ the loss function of the output layer. For the loss function utilized in this study and rectified linear activation, Eq. 8 reduces to Eq. 2 in the main text and was evaluated in software. With some improvements to the measurement system, we will be able to implement this step in the crossbar, as described in the main text.
The weight update is then applied to the crossbar. We first adjust the gate voltages of the transistors following Eq. 9. The changes in the gate voltage for the memristors in differential pairs are of the same magnitude but in opposite directions. We enforced a maximum and a minimum gate voltage of 1.7 and 0.6 V, respectively, for the current chip to make sure the transistorgatevoltage and memristor conductance relation remains in the linear region.
For memristors for which \({\mathrm{\Delta }}V_{{\mathrm{gate}},l,ij} \, < \, 0\), we first apply a voltage pulse (1.6 V) on the bottom electrodes of the array to initialize the memristor to a very lowconductance state. We then apply a voltage pulse (2.5 V) to the top electrodes with an updated voltage matrix applied to the gates of the series transistors, which raises the memristor conductance state up to match the gate voltage. As shown in the main text, the resulting memristor conductance depends linearly on the transistor’s gate voltage during this update process (Fig. 1f).
Reading the conductance matrix
To read the conductance of cross point (i, j) directly, we turn off all transistors except for column j, which is left fully on, then apply a voltage to row i and ground all other rows. The current out of column j is read, and we use the relation
where \(R_w(i,j)\) is the total wire resistance along the unique conductive path. For a known wire resistance per segment R_{s} in an N × M array with voltages applied from the left (j = 0) edge and the lower (i = N) edge grounded, we calculate
Together, these equations give
as the exact conductance of the memristor itself. However, because of wire resistance and the sneak path problem, this conductance cannot be used directly to predict the behavior of the array during vectormatrix multiplication operations. Also, each memristor must be read sequentially, so the time complexity of this approach is proportional to NM.
For any linear physical system with N inputs and M outputs, there is an equivalent linear transformation implemented by the system that can be represented as an N × M matrix. We can use this fact to read the array more efficiently. In particular, for any possible network of Ohmic resistors with N voltage inputs and M current outputs, there is a matrix G_{ eq } such that for any matrix V in\({\Bbb R}^{N \times P}\), I = G_{ eq }V. Because the IV relationship in the Ta/HfO_{2}/Pt memristor is highly linear, such a matrix exists for our array, which is the equivalent conductance matrix. This matrix is electrically indistinguishable from our physical array (to the extent that the components used are linear). We can determine G_{ eq } empirically by running matrixmatrix multiplication with some large matrix of inputs V and using the equation G_{ eq } = IV^{−1}. In practice, we usually choose V to be the N × N identity matrix. The runtime of this calculation is proportional to N.
Data availability
The data that support the findings of this study are available from the corresponding author upon request.
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Acknowledgements
This work was supported in part by the Air Force Research Laboratory (AFRL) (Grant No. FA87501520044) and the Intelligence Advanced Research Projects Activity (IARPA) (contract 201414080800008). D.B. is supported by a Research Experience for Undergraduates (REU) supplement grant from NSF (ECCS1253073). P.Y. acknowledges the support from the Chinese Scholarship Council (CSC) under grant 201606160074. This work was performed in part at the Center for Hierarchical Manufacturing (CHM), an NSF sponsored Nanoscale Science and Engineering Center (NSEC) at University of Massachusetts, Amherst, and in part at the Center for Nanoscale Systems (CNS), a member of the NSF National Nanotechnology Infrastructure Network (NNIN) at Harvard University (ECS0335765).
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C.L. and D.B. did the programming, measurements, data analysis, and simulation. C.L., P.Y., N.G., H.J., and P.L., built the integrated chips. Y.L., C.L., W.S., M.H., E.M., Z.W., and J.P.S. built the measurement system and firmware. C.L. and H.J. took the SEM and TEM images. Q.X. and J.J.Y. designed and supervised the project. Q.X., C.L., D.B., R.S.W., and J.J.Y. wrote the manuscript. M.B. and Q.W. and all other authors contributed to the result analysis and commented on the manuscript.
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Li, C., Belkin, D., Li, Y. et al. Efficient and selfadaptive insitu learning in multilayer memristor neural networks. Nat Commun 9, 2385 (2018). https://doi.org/10.1038/s41467018044842
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