We introduce VPNet, a novel model-driven neural network architecture based on variable projection (VP). Applying VP operators to neural networks results in learnable features, interpretable parameters, and compact network structures. This paper discusses the motivation and mathematical background of VPNet and presents experiments. The VPNet approach was evaluated in the context of signal processing, where we classified a synthetic dataset and real electrocardiogram (ECG) signals. Compared to fully connected and one-dimensional convolutional networks, VPNet offers fast learning ability and good accuracy at a low computational cost of both training and inference. Based on these advantages and the promising results obtained, we anticipate a profound impact on the broader field of signal processing, in particular on classification, regression and clustering problems.