Target classification is of great value when the number of transceivers in a sensing system is relatively small. This article proposes a high-accuracy and efficient classification method with machine learning techniques on frequency-modulated-continuous-wave (FMCW) radar. We first establish the novel mapping relationship from physical space to range-Doppler (PS-RD) image based on wave propagation theory, by which four physical features that effectively capture the kinematic and geometrical characteristics of targets, including speed, total reflectivity (ToRe), area, and incidence angle, are extracted from range-Doppler (R-D) image. Then, a multilayer perceptron (MLP) with a single hidden layer is employed to realize the classification. Since the above-mentioned four physical features, derived from wave physics, are chosen as the input of the neural network, our classifier does not work in a black-box way. The computational complexity of the whole classifier is the same as that of a 2-D fast Fourier transform (FFT), which guarantees a real-time operation. As an example, the proposed classifier is applied to the automotive radar system, where road targets are to be classified into five categories, including pedestrian, bike, sedan, truck/bus, and other static objects. Real-world data obtained from 77 GHz FMCW radars are provided for validation, where the proposed physics-assisted classifier turns out to outperform the state of the art in automotive radar application. The overall accuracy of the real data is about 99% even with complex multiple-target cases.