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This paper aims to demonstrate that, beyond the small world of Riemann–Liouville and Caputo derivatives, there is a vast and rich world with many derivatives suitable for specific problems and various theoretical frameworks to develop, corresponding to different paths taken. The notions of time and scale sequences are introduced, and general associated basic derivatives, namely, right/stretching and left/shrinking, are defined. A general framework for fractional derivative definitions is reviewed and applied to obtain both known and new fractional-order derivatives. Several fractional derivatives are considered, mainly Liouville, Hadamard, Euler, bilinear, tempered, q-derivative, and Hahn.

Making machines learn like humans is the ultimate goal of artificial intelligence. Few-shot learning attempts to simulate the learning mechanism of humans, which is a task that can learn novel concepts from very few labeled samples. Due to the lack of sufficient labeled training data, existing methods often increase the risk of over-fitting or cause a considerable gap between clean and augmented data. To solve these problems, we present a novel compare network to perform robust few-shot learning in a meta-learned end-to-end manner. Specifically, we argue that it is desirable to learn a robust encoder that can draw inferences about other cases from one example. To this end, we improve the accuracy of few-shot learning by mining the internal mechanism of deep networks, which can leverage label information more effectively. By introducing shift-invariant blocks and a self-attention block in our architecture, all these components are seamlessly integrated into our framework, which can give feedback to each other without data augmentation. Furthermore, we provide ablative analyses of different blocks to help understand how each term contributes to performance. Extensive experiments demonstrate that our method can provide more robust results and outperform state-of-the-art few-shot learning methods.