Background and Objective: The topology of the circle of Willis (CoW) is crucial in cerebral circulation and significantly impacts patient management. Incomplete CoW structures increase stroke risk and post-stroke damage. Current detection methods using computed tomography and magnetic resonance scans are often invasive, time-consuming, and costly. This study investigated the use of machine learning (ML) to classify CoW topology through arterial blood flow velocity pulse waves (PWs), which can be noninvasively measured with Doppler ultrasound. Methods: A database of in silico PWs from 30,618 virtual subjects, aged 25 to 75 years, with complete and incomplete CoW topologies was created and validated against in vivo data. Seven ML architectures were trained and tested using 45 combinations of carotid, vertebral and brachial artery PWs, with varying levels of artificial noise to mimic real-world measurement errors. SHapley Additive exPlanations (SHAP) were used to interpret the predictions made by the artificial neural network (ANN) models. Results: A convolutional neural network achieved the highest accuracy (98%) for CoW topology classification using a combination of one vertebral and one common carotid velocity PW without noise. Under a 20% noise-tosignal ratio, a multi-layer perceptron model had the highest prediction rate (79%). All ML models performed best for topologies lacking posterior communication arteries. Mean and peak systolic velocities were identified as key features influencing ANN predictions. Conclusions: ML-based PW analysis shows significant potential for efficient, noninvasive CoW topology detection via Doppler ultrasound. The dataset, post-processing tools, and ML code, are freely available to support further research.