I need good sources of gene expression data in the form of 3rd order tensors. Typically the commonly available datasets are in the form of a matrix, for instance, $sample \times gene$ or $gene \times time$. I need good sources for 3-dimensional data, for example $gene \times sample \times tissue$ or $gene \times sample \times time$ ... etc.
I familiar with the inflammatory bowel disease which is quite complicated: It affects multiple sites and depending on when it appears it is different. So you can look at the GEO for microarrays and RNA-seq data of this disease and I'm sure you'll find datasets from the same patients from several tissues or time.
In this disease there is a multi'omics project that fits your needs. You can find (and download) data of high dimensionality: there are samples for the same time, tissue (but different locations), gene, over a period of time and for several omics (not only expression data).
The only problem is that there are some mistakes on the metadata.
However, I don't think they are usually described as 3rd order tensors and I doubt you'll find enough data to train a machine learning method.