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Due to controllable conductance and non-volatility, flexible memristors are regarded as a key enabler for building artificial neural network (ANN)-based learning algorithms in flexible and wearable systems. However, the existing flexible memristors are suffering from limited number of conductance values, issues limiting large-scale integration, and insufficient accuracy that cannot support accurate computation of ANN. In this paper, solutions are proposed for the three major challenges of the flexible memristor; the feasibility of a three-layer fully connected neural network on MNIST and a 13-layer convolutional neural network (CNN) on CIFAR-10 using the flexible memristor based on single-walled carbon nanotubes network/polymer composite and hydrophilic Al2O3 dielectric are studied. The evaluation result shows that in the fully connected neural network system, it is able to recognize MNIST with an accuracy above 90% after 4-bit quantization, 52.05% decrease in interconnection numbers in the circuit and up to 40% random error introduced, and in the CNN on CIFAR-10, the system can retain an accuracy above 86% with less than 4% accuracy loss after 5-bit quantization, 59.34% decrease in interconnection numbers in the circuit and up to 40% random error injected., QC 20180904