Question about SC 1.3.1 for caption in HTML 5 table

Hello, 
 
We have many tables on the website with no captions. 
WCAG success criterion 1.3.1 requires caption or summary for tables. Summary is obsolete in HTML 5. 
 
Does this mean that data tables require caption to pass sc 1.3.1 or having a heading with a paragraph would be acceptable? 
Sometimes the table is preceded by a heading or heading and a paragraph that is explaining the table content. 
 
We migrated to HTML 5 and looking for a data tables solution that wouldn't fail later on WCAG HTML 5. 
 
A feedback from WCAG Working Group would be appreciated. 
 
Thank you,
Rabab