It's generally well-known that Kant was responding, amongst other influences, to Hume's critique of the empirical method on purely logical grounds. One could consider him as a modern-day Pyrrhonniste.
The problem of induction is sourced from a brief argument in Hume's Treatise, but the SEP shows a discussion in thoroughly analytic terms - Popper and Carnap are mentioned, for example.
What was Kant's response to the same question, if he in fact did respond to it? It seems likely that a response could be fashioned out of his philosophy on the basis of his categories as pure concepts of his understanding.
Question: How does Kant, or Kantians or neo-Kantians solve or understand the problem of induction?