1. INTRODUCTION

The ability to decipher non-trivial patterns in data using computational techniques has led to the development of sophisticated machine intelligence approaches with a plethora of scientifically and technologically important applications [1–4]. Such approaches have predominantly been performed on general-purpose digital electronic processors (i.e., GPUs and CPUs), but this can introduce unwanted and deleterious computational latency and limitations to data throughput. Thus, special-purpose hardware accelerators designed intentionally for use in machine learning applications are an essential development [5–9]. Harnessing the wavelength-multiplexing capabilities of photonics to carry out parallel operations simultaneously in special-purpose accelerators can greatly increase the capacity of intelligent information processing [10–13].

Practical associative learning hardware accelerators require a hardware device structure that can associate inputs to a device, but current implementations using electronic [14–26], optoelectronic [27], and synthetic biological [28] approaches are limited at the device level. Specifically, the ability to monadically associate at least two inputs together is distinctly absent at a device level. In this paper, we experimentally demonstrate such a single associative learning element, one that exploits the ultrahigh bandwidth capabilities of photonics in a readily scalable architecture with the potential to deliver future artificial neural networks with very significantly faster and lower energy cost training as compared to conventional approaches.

2. ASSOCIATIVE MONADIC LEARNING ELEMENT CONCEPT

In biological systems, a fundamental associative learning process—classical conditioning—can be described using the neural circuit in Fig. 1(a) [29,30]. For a motor neuron to generate an action potential, it must receive a sensory signal. Pavlov in his experiment showed that salivation in a dog can be “stimulated” by associating the ringing of a bell with food [31], i.e., the two sensory stimuli were associated so as to generate an identical response. The process of associating stimulus ${s_2}$ from a sensory neuron with natural stimulus ${s_1}$ is the process of association, and this is a learning mechanism. Once the association between the two signals is established, the response is triggered when either ${s_1}$ or ${s_2}$ is sent to the motor neuron through a synaptic weight [${w_1}$ or ${w_2}$ in Fig. 1(a)]—the network has now learned this response. Thus, this simplified neural circuitry has two main roles: to converge and associate the two inputs, and to store memories of these associations in situ.

 

Fig. 1. Optical associative monadic learning element (AMLE). (a) Simplified illustration of the neural circuitry for associative learning. After stimulus ${s_2}$ is associated with stimulus ${s_1}$, both elicit the same response. (b) SEM image (false colored) of a fabricated AMLE, consisting of $\text{Si}_3{\text N}_4$ directional couplers (cyan), ${\text{SiO}_2}$ undercladding (gray), and GST cell (marigold). Inset shows detail of coupling area with GST. (c), (d) Corresponding electric field profiles of AMLE (c) before and (d) after learning, with ${s_1}$ and ${s_2}$ inputs. Inset: output cross-section optical field at locations denoted by black vertical bars. The color bar is normalized.

Download Full Size | PDF

In our optical device, we embed the above functionality in an associative monadic learning element (AMLE) [Fig. 1(b)]. The device has two coupled waveguides and a thin film of phase-change material ${\text{Ge}_2}{{\text{Sb}}_2}{\text{Te}_5}$ (GST) on the lower waveguide that effectively modulates the coupling between the waveguides. The GST exists in two states (amorphous or crystalline). These two states (and the fractional volumes between the two states) govern the amount of coupling between the waveguides. As shown in Fig. 1(c), when the material is crystalline, there is no association between inputs ${{s}_1}$ and ${s_2}$. However, when the two inputs (learning pulses) arrive at the same time, the material has sufficient absorption to amorphize the GST, changing the coupling between the waveguides. As the material amorphizes, inputs ${s_1}$ and ${s_2}$ begin to “associate” as shown in Fig. 1(d). As the number of learning pulses increases, a larger volume of material switches from the crystalline to the amorphous state, until reaching a point when the two inputs result in an output that is nearly indistinguishable—we set this level and term this the learning threshold. Unlike previous non-volatile phase-change material photonic memories that relied primarily on optical amplitudes for their operation [11], we here employ the optical phase difference between inputs ${s_1}$ and ${s_2}$ to precisely control the phase state of the GST cell. This enables us to establish the precise extent of association between the inputs (details in Supplement 1). Our phase-change material GST is known to have an ultrafast structural phase transition time (sub-ns amorphization and few-ns crystallization time [32]), high cycling endurance (${\sim}{{10}^{12}}$ cycles [33,34]), and long retention time (${\gt}\!{10}$ years at room temperature [34]). A thin capping layer of indium tin oxide (ITO) is deposited on the GST to prevent oxidation, and to help localize optically induced heat to enable low-power phase switching [11,35].

 

Fig. 2. Photonic Pavlovian learning process. (a) Input–output relation of AMLE. ${s_1}$ and ${s_2}$ inputs denote “food” and “bell” inputs; the corresponding output transmission $r$ (bottom chart) represents transmission responses ${r_1}$ and ${r_2}$. Blue, red, and purple bars of the top and middle charts denote ${s_1}$, ${s_2}$, and both ${s_1} - {s_2}$ input incidences, respectively (${\unicode[Times]{x1D706}}_{1}$, ${\unicode[Times]{x1D706}}_{2}$, and ${\unicode[Times]{x1D706}}_{0}$ are the wavelengths used to selectively address each one). Bottom chart inset: simplified diagram of AMLE. (b) Corresponding real-time measurement of output probe transmission $r$ of a single cycle learning and forgetting processes in (a). (c) Repeatability of the processes on AMLE over 80 cycles. The levels are denoted by filled circles of different colors that correspond to (b).

Download Full Size | PDF

The association between inputs ${s_1}$ and ${s_2}$ during the learning process occurs only when the two inputs are paired at a specific optical phase delay $\Delta \varphi$. This results in a change in the synaptic weight $\Delta w$ between stimulus and response signals. In our AMLE, the optical delay $\Delta \varphi$ is introduced by the optical phase difference between ${s_1}$ and ${s_2}$ inputs. We found that the slightest vibration and/or temperature change in the measurement environment can cause the optical phases to vary erratically. We mitigated this by using an on-chip layout (details in Supplement 1 Fig. S2.1) that greatly reduced effects of environmental disturbances on phase control.

The use of directional couplers ensures that the design is applicable over a broad optical wavelength range. Our simulations of the design show the natural response, as outlined in Fig. 1(c) (prior to associative learning taking place) and Fig. 1(d) (after the association). Before learning, only ${s_1}$ leads to a high transmission response, whereas ${s_2}$ does not. After learning, both ${s_1}$ and ${s_2}$ produce high transmission responses, which indicates that the two inputs are now associated, i.e., the system has “learned” to associate the two inputs such that both trigger the same response.

3. OBSERVATION OF PHOTONIC ASSOCIATIVE LEARNING

We now experimentally characterize the dynamic response of the AMLE. To achieve this, it is necessary to control the input signal combinations (${s_1}$ only, ${s_2}$ only, and both ${s_1}$ and ${s_2}$) to the AMLE. We achieve this through the use of wavelength-selective, critically coupled ring resonators to each of the inputs leading to the AMLE (details in Supplement 1). This allows us to characterize the real-time dynamics of the associative learning process.

The starting point of the AMLE for our experiments is its crystalline state. We probe output transmission ${r_1}$ and ${r_2}$ of the AMLE in real-time using wavelengths ${\unicode[Times]{x1D706}}_{1}$ and ${\unicode[Times]{x1D706}}_{2}$. As shown in Fig. 2(a), transmission readouts ${r_1}$ and ${r_2}$ remained the same for single input pulses at 1.45 nJ (pulse widths $\tau = {100}\;\text{ns}$) as expected (for events 1 to 4). However, when inputs ${s_1}$ and ${s_2}$ were sent simultaneously at pump wavelength ${\unicode[Times]{x1D706}}_{0}$ with a fixed phase delay of $\pi /{2}$ at 0.66 nJ ($\tau = {100}\;\text{ns}$) each in event 5, transmission changes $\Delta {r_1}$ and $\Delta {r_2}$ for ${s_1}$ and ${s_2}$ probe readouts are ${\sim}- {4}\%$ and ${\sim}+ {4}\%$ respectively. As the input pump pulse power was increased from 0.87 nJ ($\tau = {100}\;\text{ns}$) to 1.45 nJ ($\tau = {100}\;\text{ns}$) in events 6 to 8, the probe readouts changed by approximately ${-}{7}\%$ and ${+}{7}\%$, respectively, both of which are well above our output transmission threshold of ${r_{\text{th}}}\;\sim{5}\%$. Details of optical pulses are provided in Appendix A. Effectively, these experiments show that the two inputs can be “taught” to associate with each other such that either triggers the response, i.e., the associations are learned.

We then show in Fig. 2(a) (bottom chart) that these learned associations can also be reversed. A set of pulses at 0.43 nJ ($\tau = {100}\;\text{ns}$) in event 9, followed by 0.19 nJ pulses ($\tau = {100}\;\text{ns}$) in event 10 resulted in the “forgetting” process, where the readouts $r$ reverted to the baselines (${r_{1\:}}\sim{0.14}$ for ${s_1}$ input probe and ${r_{2\:}}\sim{0}$ for ${s_2}$ input probe). For our measurements, readout above a threshold ${r_{\textit{th}}}\;\sim{0.05}$ is designated as the learned state.

 

Fig. 3. Supervised Pavlovian associative learning. (a) Pavlovian learning involves pairing the inputs (IN) with the correct outputs (Teacher) to supervise the learning process. (b) Current conventional supervised learning networks use backpropagation. The network diagram depicts the network for supervised learning. (c) Optical on-chip hardware diagram of supervised learning with two inputs using AMLE. Input signals ${{IN}_1}$ to ${{IN}_4}$ are fed into the system during the learning process to be supervised by the Teacher input signal. ${{IN}_1}$ to ${{IN}_4}$ and Teacher inputs lead to the AMLE being controlled by thermo-optic (with NiCr heaters) MZMs. (d) Network representation of the hardware diagram in (c). (e) Optical micrograph of supervised learning network in (c), which consists of four AMLEs (boxed, purple). The arrows show the optical input–output connections coupled to the on-chip network via grating couplers. (f), (g) Optical micrographs of a single AMLE and heaters respectively correspond to those in (e).

Download Full Size | PDF

Figure 2(b) shows a single cycle of the real-time output readout of associative learning in events 5 to 8 and the forgetting process in events 9 to 11 of Fig. 2(a). To test the repeatability of our associative learning and forgetting processes, we subjected the AMLE through 80 learning cycles, examined over a period of 50 minutes. After 80 cycles [Fig. 2(c)], the individual learning weights were clearly identifiable with a standard deviation of $\pm {0.69}\%$ in readout transmission.

4. ASSOCIATIVE NETWORK FOR SUPERVISED LEARNING

Up to this point, we have observed associative learning and characterized the workings of our AMLE via single device photonic measurements. The general concepts of Pavlovian associative learning [Fig. 3(a)] and supervised learning [a class of machine learning; Fig. 3(b)] are in essence comparable, both involving the pairing of input (IN) with the correct output (Teacher) to supervise the learning process. However, the mapping of the input to the desired output in conventional supervised learning architectures is relatively complicated as well as time and energy consuming—the Teacher signal propagates backward layer by layer to collectively adjust the network weights such that the actual output $R$ better resembles the desired output (Teacher) after each learning iteration. The learning process for our AMLEs is by comparison far more straightforward (and faster and more energy efficient), and to elucidate this, we consider a scaled-up network of AMLE devices, which we illustrate schematically in Fig. 3(c).

To implement the system of Fig. 3(c), we use cascaded Mach–Zehnder modulators (MZMs) to provide a reliable means to split both the Input (IN) and Teacher signals equally with stable optical phases (obtained via the use of integrated NiCr thermo-optic heaters) and feed them to the inputs of the AMLEs. The MZMs also allow for the use of wavelength multiplexing to feed multiple signals to the inputs of multiple AMLEs, before these signals are paired with Teacher signals at the associative layer to cumulatively produce and output a transmission response. A more detailed illustration of the resulting integrated AMLE network chip is thus as shown in Fig. 3(e). The system consists of the associative layer (purple bordered boxes) between an input layer (Input data and Teacher denoted using red and blue arrows, respectively) and an output layer (response $R$; green arrow). Optical micrographs with an enlarged view of the AMLE and thermo-optic heaters are shown in Fig. 3(f) and Fig. 3(g), respectively (see Supplement 1 and Figs. S3.1 and S3.2).

We use the AMLE network chip of Fig. 3(e) to carry out a rapid pattern recognition task to verify that once the associations are formed by the AMLEs, simultaneous parallel recognition can be achieved. The GST cells are initially set to their crystalline states (details in Appendix A). We then program the bit pattern “0110” to the AMLEs by respectively sending the pump Teacher signal $T = {1.47}\;\text{nJ}$ ($\tau = {100}\;\text{ns}$) and pump Input signals ${{\mathit{IN}}_1} = {0}$, ${\textit{IN}_2} = {1.47}\;\text{nJ}$ ($\tau = {100}\;\text{ns}$), ${\textit{IN}_3} = {1.47}\;\text{nJ}$ ($\tau = {100}\;\text{ns}$) and ${\textit{IN}_4} = {0}$ simultaneously, essentially associating these patterns within the AMLE network [measurement setup is detailed in Supplement 1 and shown in Figs. S4(a) and (b)]. Pattern recognition is then carried out by sending a series of randomized optical binary signals modulated at 1 GHz speed to the AMLEs (with 0 mW for “0” and 0.7 mW for “1”), using different wavelengths (in the range of 1548.51 nm to 1550.92 nm, C33 to C36 in the telecom band) for each of the individual AMLE inputs [see Fig. 4(a)] (further details in Supplement 1). The output from each AMLE device in the network passes to four photodiodes to generate the system outputs (or responses), $R$. The output transmission response of a single AMLE ${r_{\textit{ij}}}$ measured at wavelength ${\unicode[Times]{x1D706}}_{\!j}$ from the photodetectors (PDs) is then summed to obtain the cumulative transmission response ${R_j} = {\Sigma _i}$ ${R_{\textit{ij}}}$ for the specific wavelength, where ${R_{\textit{ij}}}$ is the measured output transmission of the $i$th AMLE and $j$th wavelength after normalizing (between zero and one range) using ${R_{\textit{ij}}} = ({r_{ij\:}} \!-\! \;{r_{\text{base}}})/{r_{\text{base}}}$, i.e., the average value of two consecutive output transmission values using a trigger pulse (with ${r_{\text{base}}}$ being the baseline of the optical transmission. The average value is taken to minimize the effects of electrical or optical noise. After recording the output pulses from the experiments at four optical wavelengths, we systematically retrieve the cycle(s) at which the cumulative response ${R_c} = {\Sigma _{ij\:}}{R_{\textit{ij}}}$ is above a pre-determined learning threshold. Figure 4(c) shows the inputs sent at the four different wavelengths C33 to C36, while the corresponding output response of the AMLEs is shown in Fig. 4(d). Figure 4(e) shows the cumulative ${R_c}$ for all four wavelengths, which exceeds the learning threshold only at cycle 30; this enables us to pinpoint the output bit combinations at which input signals from all four wavelengths match the bit pattern “0110” to be at cycle 30. Figure 4(f) shows the performance of the AMLEs for detection predictions. We measured the prediction error $\textit{Error} = R\!-\!{R_{\text{calculated}}}$ from the difference between output transmission $R = {\Sigma _{i\:}}{R_{\textit{ij}}}$ and the calculated transmission ${R_{\text{calculated}}}$. The error is within the $\pm {0.5}$ threshold bands.

 

Fig. 4. Pattern recognition with AMLE. (a) Measurement setup for pattern recognition. Bit patterns to be detected “0110” are associated into the AMLE via a learning step. 1 GHz optical detection pulses defined by computer-generated pseudo-random bits at wavelengths from C33 to C36 (1548.51 nm to 1550.92 nm) are sent through AMLE devices and detected by photodiodes as the signals are temporally multiplexed. Measured average output transmission $R$ cumulatively summed to determine if pattern is recognized. (b) Eye-diagram obtained from non-return-to-zero (NRZ) pseudo-random binary sequence (PRBS) pulses modulated at 1 GHz shows clear distinction between states “0” and “1.” (c) Inputs ${\textit{IN}_1}$ to ${\textit{IN}_4}$ sent at four different wavelengths i.e., C33 to C36 (input = “1” represented in red, yellow, green, and blue respectively). Purple shades represent “0110” pattern sent to the devices. (d) $R$ for the different wavelengths (C33 to C36). Purple shades represent “0110” pattern detected. (e) Cumulative $R$ obtained for the respective wavelengths and detection cycles. Values above cumulative transmission threshold $R = {7}$ (dashed black line) denote all-matching response (i.e., in cycle 30). (f) Calculated error $R\!-\!{R_{\text{calculated}}}$ (difference between measured average transmission response and expected response) shows mean error is negligibly low with standard deviation well within the $\pm {0.5}$ threshold band.

Download Full Size | PDF

It is important to point out that the detection speed in the experiments shown in Fig. 4 is limited only by the number of pulses to average (in our case, two) and the signal modulation and detection speed of the modulators and PDs. In principle, this detection speed can be significantly improved by increasing the signal to noise ratio (by using on-chip PDs instead of external PDs connected via grating couplers in our case) and increasing the input signal modulation speed while maintaining detected signal integrity. Although we demonstrated detection specifically for the pattern “0110,” this hardware system can be used to detect all other possible (and more) patterns (as shown in Supplement 1 Fig. S6) by additionally representing on other AMLEs another separate set of bits toggled to the bit pattern to be detected, and then additionally sending toggled sets of input bits to these additional AMLEs.

Now, we demonstrate how AMLE-based hardware can achieve generalization on an image recognition task using associative learning, based on the network architecture shown in Fig. 5(a). The network is similar to those in Fig. 3(d), which consists of three main network layers (i.e., input layer, associative layer, and output layer) [36]. During the training process, images to be trained are first pixelated to ${15} \times {13}$ input pixel data (195 pixels altogether). These data ${\textit{IN}_1}\!-\!{\textit{IN}_{195}}$ are then sent to the associative layer, while simultaneously, the Teacher signals are likewise preprocessed and sent to the associative layer. Notably, images fed into ${\textit{IN}_1}\!-\!{\textit{IN}_{195}}$ and Teacher signals are exchangeable in the training process. The change in the learning states of AMLEs is determined from their transmission readouts, when preprocessed Input data of maximum amplitude (blank white image) and the same dimensions as the training pixel data are fed through the layers. The transmission of these individual AMLEs is summed up at the output layer and rearranged to form a ${15} \times {13}$ pixel model representation. The complete computation is obtained by cumulatively adding the model representation from each training pair. A pixel-by-pixel comparison between the model (which generalizes the training images) and measurement is then made to determine whether the testing images are of the image to be detected.

 

Fig. 5. Scaling architecture for image recognition using associative learning network. (a) General associative neural network composed of an input layer (${\textit{IN}_1}\!-\!{\textit{IN}_{195}}$, ${T_1}\!-\!{T_{195}}$), associative layer (stimuli ${s_1}\!-\!{s_2}$ association), and output layer (transmission $R$). The input signal is the pattern (pixels from image) to be classified, and the external teacher provides the desired set of output values chosen to represent the class of patterns being learned. During training, the input layer (${\textit{IN}_1}\!-\!{\textit{IN}_{195}}$, ${T_1}\!-\!{T_{195}}$) is fed into the associative layer. Associative layer consists of AMLEs with states that are modified when both the input and external teacher signals are paired together. A model representation of the trained images is then obtained by sending a preprocessed input signal of a blank white image to propagate through the layers. (b) Optical on-chip implementation of the input layer, associative layer, and output layer. The associative layer, which consists of thermo-optic NiCr heaters, distributes the combination of the input and external teacher signals (from the input layer) as ${s_1}$ and ${s_2}$ inputs to the respective AMLEs. The output layer consists of a summation unit to sum up the output response from the AMLEs to form the net output. Conventional ${2} \times {1}$ combiner has 3 dB loss each. (c) Training the associative learning network to identify cat images, with bounding box corners to indicate region of interest. (d) After five training iterations, the network learns the model representation of the cat from the output response $R$. (e) Images used to test whether the network can correctly classify pictures as cat and non-cat. (f) The network successfully recognizes cat and non-cat images based on the squared Euclidean distance $\Sigma \;{(\textit{rep}_{j}\!-\!\textit{im}_{j})^2}$ measured over the pixelated testing images for each of the ${15} \times {13}$ pixels.

Download Full Size | PDF

The photonic implementation of the associative learning network architecture for image recognition is shown in Fig. 5(b). Here, the GST elements on the AMLEs are first initialized to the crystalline state before the signals are sent to the on-chip photonic structure. During training, depending on the optical input signal and input pump power sent to the AMLEs, the state of the GST cells on AMLEs either remains in crystalline or structurally switches to an amorphous state. This results in a change in the optical probe output response of the AMLEs. In our experiment, because we have only four AMLEs, we raster the 195 pairs of input pixels across them four.

We examine the cat image classification capabilities of our associative learning network using the “Dogs versus Cats” dataset from fast.ai [37] (single dataset collected from CIFAR10, CIFAR 100 [38], Caltech 101 [39], Oxford-IIIT Pet [40], and Imagewoof [41]). The cat images that we used for the training process are shown in Fig. 5(c). After the training process using these images, we obtain the model representation of a cat shown in Fig. 5(d) by feeding in a ${15} \times {13}$ pixel blank white test input of maximum probe input power magnitude (1.3 mW; further details of the experiment in Supplement 1).

Thus far, we have demonstrated searching for patterns (in the form of pixel amplitudes) from the ${15} \times {13}$ pixel image sent to the network of monadic AMLEs in a single step. After each training iteration with a model image, the network’s ability to distinguish the appearance of a cat improves. The feature subtleties can then be captured from the model representation, giving us a valuable means to distinguish a cat from other objects. To test the model representation, we use the testing images shown in Fig. 5(e) to compare them (im) with the representation of the cat rep and measure the error function with respect to pixel $j$, given by $\tanh\;[{({\textit{rep}_j}\!-\!{{im}_j})^2}\;e]$ for every pixel $j$. We set the threshold $\textit{min}(\textit{Error}) + (\textit{max}(\textit{Error})\!-\!\textit{min}(\textit{Error}))/{2}$, which is 22.625 for our case, to determine the Testing Images that resemble the model representation in Fig. 5(d). Our results, summarized in Fig. 5(f), reveal that Testing Images 1, 3, and 5 resemble the cat representation. With the error [Error = squared Euclidean distance $\Sigma \;{(\textit{rep}_{j}\!-\!\textit{im}_{j})^2}]$ of Images 2, 4, and 6 above the threshold, the network predicts that these images are not of a cat. The associative learning network thus accurately classifies images of cats from the model representation obtained from the training iterations. In contrast to conventional numeric-based artificial intelligence (AI) approaches, our symbolic associative learning system interprets from the “cat representation” results to logically conjure up the proposition that an image must have the shown clear, distinctive, and comprehensible physical attributes to constitute a cat image. Our image recognition example adopts symbolic AI, while conventional image recognition uses connectionist AI. Our approach is typically simpler and faster, but comes at the expense of not having the ability to acquire deep features—an ability that may not be needed for many less complex machine learning tasks.

5. EVALUATION METRICS FOR ASSOCIATIVE LEARNING DEVICES

Our application-specific system offers convergence in one training step due to its straightforward approach of detecting similarities without having to randomize network weights and perform backpropagation as in conventional neural networks.

We identify relevant device-level evaluation metrics by contextualizing the AMLE with a typical machine learning data load, which requires data to be transferred back and forth from the data source to be run using cloud computing and/or supercomputers. For a more energy-efficient locally run neural network, it is important to shrink the network for greater portability and reduce the energy consumption of the learning process. Table 1 summarizes the minimum active device volume (in our case, volume of GST and the waveguide below) and learning energy of other associative learning devices [14–27]. The electronic and optoelectronic associative learning devices range from ${\sim}{0.{1\!-\!10}^{10}}\;\unicode{x00B5}\text{m}^3$ in active volume and consume ${\sim}{2.{63\!-\!10}^5}\;\text{nJ}$ of energy per learning event [14–27]. In comparison, the all-optical AMLE in our work fares favorably relative to these devices in terms of dimensions and very favorably in terms of energy usage, with a low active volume at ${0.12}\;{{\unicode{x00B5}\text{m}}^3}$ and minimum learning energy at only 1.8 nJ. The overall size of the single-element device is of dimensions ${2}\;{\unicode{x00B5}\text{m}} \times {17}\;{\unicode{x00B5}\text{m}} \times {0.33}\;\unicode{x00B5}\text{m}$. Despite the relatively low active volume (of GST) in our case, compared to electronic hardware, our devices are still larger; however, the speed and multiplexing of photonic devices can lead to higher computational density. Experimentally, we have demonstrated $118\,\,\text{TOPS/mm}^{2}$, which was limited by the available setup in the laboratory (see Supplement 1). Scaling the device to more wavelength channels and modulating at a higher speed potentially leads to significant gains in the compute density. Considering, for example, modulation at 50 GHz [42] and using 16 wavelengths, the device would deliver approximately $2.5 \times 10^{4}\,\,\text{TOPS/mm}^{2}$. The overall net energy usage of the learning system is directly related to the number of iterations required to train a learning system (details in Supplement 1). The energy–speed trade-off—particularly evident in CMOS devices [43]—is another important evaluation metric that requires further investigation and research. Our device learns in ${\sim}{100}\;\text{ns}$, compared to ${\sim}\text{ms}$ in a previous associative learning device [15].

Table 1. Comparison of Active Volume and Learning Energy in Associative Learning Devices

View Table

6. CONCLUSION

Our results show the first demonstration of an AMLE implemented on a photonic platform. We provide a supervised learning framework that facilitates the transition from a monadic Pavlovian single Input–Teacher association on an AMLE to any arbitrary $n$ Input–Teacher associations, thus enabling backpropagation-free, single-layer weight artificial neural network architectures.

We have elucidated the inner workings of the network building block, which can spatiotemporally correlate two initially distinct inputs (${s_1}$ and ${s_2}$) to a same output when both inputs are simultaneously applied at a predetermined $\Delta \varphi$ optical delay. Given that light signals inherently do not interfere at different wavelengths in linear media (including the AMLE), such input–input association can handle associations of multiple data streams consisting of different wavelengths over a single element, as we have experimentally demonstrated. Our photonic platform allows for wavelength multiplexing, which is inherently suited to the highly parallel nature of machine learning. We anticipate further improvements in other relevant metrics (e.g., overall device volume and learning energy) on different material platforms and with other optimization methods.

More generally and interestingly, our work can extend the one-way learning (${s_2}$ becomes associated to ${s_1}$) to a customizable form of learning, for example, mutual/two-way learning (${s_2}$ becomes associated to ${s_1}$ while ${s_1}$ is associated to ${s_2}$; details in Supplement 1. This customizable feature when combined with demonstrations of deterministic weights using identical, fixed-energy, and fixed-duration pulses [35] will provide unprecedented design flexibility for a wide range of machine learning applications. As to whether the nonlinear scaling with the number of inputs for nonlinear classification problems is an inherent attribute in associative learning is still an open question (discussed in Supplement 1). However, as shown in Figs. 4 and 5, practical applications such as pattern recognition and image recognition can be readily demonstrated with an associative learning network that scales linearly with the number of inputs. The compact single-element implementation in our work will allow the use of the AMLE as a building block in machine learning/statistical inference in general, thus potentially opening up new avenues of research in machine learning algorithms and architectures.