Neural Networks (NNs) are steering a new generation of artificial intelligence (AI) applications at the micro-edge. Examples include wireless sensors, wearables and cybernetic systems that collect data and process them to support real-world decisions and controls. For energy autonomy, these applications are typically powered by energy harvesters. As harvesters and other power sources which provide energy autonomy inevitably have power variations, the circuits need to robustly operate over a dynamic power envelope. In other words, the NN hardware needs to be able to function correctly under unpredictable and variable supply voltages. In this paper, we propose a novel NN design approach using the principle of pulse width modulation (PWM). PWM signals represent information with their duty cycle values which may be made independent of the voltages and frequencies of the carrier signals. We design a PWM-based perceptron which can serve as the fundamental building block for NNs, by using an entirely new method of realising arithmetic in the PWM domain. We analyse the proposed approach building from a 3x3 perceptron circuit to a complex multi-layer NN. Using handwritten character recognition as an exemplar of AI applications, we demonstrate the power elasticity, resilience and efficiency of the proposed NN design in the presence of functional and parametric variations including large voltage variations in the power supply.