  Mixed-signal artificial neural networks (ANNs) that employ analog
matrix-multiplication accelerators can achieve higher speed and improved power
efficiency. Though analog computing is known to be susceptible to noise and
device imperfections, various analog computing paradigms have been considered
as promising solutions to address the growing computing demand in machine
learning applications, thanks to the robustness of ANNs. This robustness has
been explored in low-precision, fixed-point ANN models, which have proven
successful on compressing ANN model size on digital computers. However, these
promising results and network training algorithms cannot be easily migrated to
analog accelerators. The reason is that digital computers typically carry
intermediate results with higher bit width, though the inputs and weights of
each ANN layers are of low bit width; while the analog intermediate results
have low precision, analogous to digital signals with a reduced quantization
level. Here we report a training method for mixed-signal ANN with two types of
errors in its analog signals, random noise, and deterministic errors
(distortions). The results showed that mixed-signal ANNs trained with our
proposed method can achieve an equivalent classification accuracy with noise
level up to 50% of the ideal quantization step size. We have demonstrated this
training method on a mixed-signal optical convolutional neural network based on
diffractive optics.
