Recently, learning-based partial differential equation (L-PDE) has achieved success in few-shot learning area, while its feature weighting mechanism and recognition stability require further improvement. To address these issues, we propose a novel model called “location-dependent PDE” (LD-PDE) based on Navier–Stokes equation and rotational invariants in this paper. To our best knowledge, LD-PDE is the first application of the Navier–Stokes equation to achieve image recognition as a high-level vision task. Specifically, we formulate the feature variation with respect to each time step as a linear combination of rotational invariants in LD-PDE. Meanwhile, we design location-dependent mechanism to adaptively weight each invariant in an attention-based approach, which provides hierarchical discrimination in the spatial domain. Once the ultimate feature is learned, we measure the model error with the cross-entropy loss and update the parameters by the coordinate descent algorithm. As a verification, experimental results on face recognition datasets show that LD-PDE method outperforms the state-of-the-art approaches with few training samples. Moreover, compared to L-PDE, LD-PDE achieves a much more stable recognition with low sensitivity to its hyper-parameters.

中文翻译：

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通过位置相关的偏微分方程的有效的一次性学习方法

近年来，基于学习的偏微分方程（L-PDE）在少拍学习领域取得了成功，而其特征加权机制和识别稳定性还需要进一步改进。为了解决这些问题，在本文中，我们基于Navier–Stokes方程和旋转不变量提出了一种称为“位置相关PDE”（LD-PDE）的新型模型。据我们所知，LD-PDE是Navier–Stokes方程的第一个应用，可实现将图像识别作为高级视觉任务。具体来说，我们将相对于每个时间步的特征变化公式化为LD-PDE中旋转不变量的线性组合。同时，我们设计了一种基于位置的机制，以一种基于注意力的方法自适应地对每个不变量进行加权，从而在空间域中提供了层次区分。一旦掌握了最终特征，我们就可以利用交叉熵损失来测量模型误差，并通过坐标下降算法来更新参数。作为验证，在面部识别数据集上的实验结果表明，LD-PDE方法在训练样本很少的情况下优于最新方法。此外，与L-PDE相比，LD-PDE对其超参数的敏感性较低，因此可实现更加稳定的识别。