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We establish both the uniqueness and the existence of the solutions to a hidden-memory variable-order fractional stochastic partial differential equation, which models, e.g., the stochastic motion of a Brownian particle within a viscous liquid medium varied with fractal dimensions. We also investigate the inverse problem concerning the observations of the solutions, which eliminates the analytic assumptions on the variable orders in the literature of this topic and theoretically guarantees the reliability of the determination and experimental inference.

Building extraction from high spatial resolution imagery (HSRI) plays an important role in the remotely sensed imagery application fields. However, automatically extracting buildings from HSRI is still a challenging task due to such factors as large size variations of buildings, background complexity, variations in appearance, etc. Especially, it is difficult to extract both crowded small buildings and large buildings with accurate boundaries. To address these challenges, this paper presents an end-to-end encoder–decoder model to automatically extract buildings from HSRI. The designed network, called AEUNet++, is based on UNet++, attention mechanism and multi-task learning. Specifically, the AEUNet++ introduces the UNet++ as the backbone to extract multiscale features. Then, the attention block is used to effectively fuse different-layer feature maps instead of direct concatenation in the output of traditional UNet++, which can assign adaptive weights to different-layer feature maps as their relative importance to enhance the sensitivity of the mode and suppress the background influence of irrelevant features. To further improve the boundary accuracy of the extracted buildings, the boundary geometric information of buildings is integrated into the proposed model by a multi-task loss using a proposed distance class map during training of the network, which simultaneously learns the extraction of buildings and boundaries and only outputs extracted buildings while testing. Two different data sets are utilized for evaluating the performance of AEUNet++. The experimental results indicate that AEUNet++ produces greater accuracy than U-Net and the original UNet++ architectures and, hence, provides an effective method for building extraction from HSRI.

A time-fractional substantial diffusion equation of variable order is investigated, in which the variable-order fractional substantial derivative accommodates the memory effects and the structure change of the surroundings of the physical processes with respect to time. The existence and uniqueness of the solutions to the proposed model are proved, based on which the weighted high-order regularity of the solutions, in which the weight function characterizes the singularity of the solutions, are analyzed.