用深度算子网络实现微分方程解的超分辨率重建，效果优于传统插值方法。
Super Resolution Based on Deep Operator Networks
- 基于DeepONet学习算子映射，从低分辨率输入预测高分辨率解。
- 在1D和2D问题中，对高频振荡和小尺度结构重建误差显著低于三次样条插值。
- 引入卷积层降低计算成本，且训练集大小变化时模型表现稳定。
我们使用深度算子网络（DeepONets）对两类偏微分方程的解进行超分辨率重建，并与传统插值方法的结果对比，验证DeepONets的优势。采用两种下采样方法生成不同分辨率的输入图像，在三种分辨率下完成重建。结果表明，DeepONet能准确预测低分辨率输入中的高频振荡与小尺度结构。对于二维问题，引入卷积层以更低成本提取输入图像特征。通过调整训练集规模，观察预测误差变化。在1D与2D场景中，DeepONet的重建精度均显著优于三次样条插值，凸显算子学习方法在该类问题上的优越性。
原文摘要 · Abstract (English)
We use Deep Operator Networks (DeepONets) to perform super-resolution reconstruction of the solutions of two types of partial differential equations and compare the model predictions with the results obtained using conventional interpolation methods to verify the advantages of DeepONets. We employ two pooling methods to downsample the origin data and conduct super-resolution reconstruction under three different resolutions of input images. The results show that the DeepONet model can predict high-frequency oscillations and small-scale structures from low-resolution inputs very well. For the two-dimensional problem, we introduce convolutional layers to extract information from input images at a lower cost than purer MLPs. We adjust the size of the training set and observe the variation of prediction errors. In both one-dimensional and two-dimensional cases, the super-resolution reconstruction using the DeepONet model demonstrates much more accurate prediction results than cubic spline interpolation, highlighting the superiority of operator learning methods in handling such problems compared to traditional interpolation techniques.
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