Best architecture solution for heavy data mining system, that can run on cloud
The question is quite open to discussion with such few details. What kind of calculations are performed with what density? What kind of an application server are you using? What amount and what kind of data comes in and goes out? What's the acceptable latency between the application server and the database / clients and application server? Thus, are there any requirements to keep the application server in-house / will it need to be clustered? Will there be any business logic implemented in the database as stored procedures and stored functions? If so, will it be in the native SQL dialect, or will it be in any external language like C/C++/Python/Java etc?

I won't be able to single out and suggest a provider or strategy, due to the lack of details and due to the fact that (for the last few years) I'm generally managing DB servers deployed in-house or on dedicated servers on the cloud.
But first of all, I strongly suggest you phase out MySQL.

Having the power of Oracle or the ease of use of MSSQL would be nice, of course, but I won't push such a proprietary system with (possibly) high cost of licencing.

Instead I strongly suggest moving towards PostgreSQL. Especially for stability and scalability issues.
But the strategy to pick (will it be rented service, a platform or rented dedicated servers? Will there be a cluster, and if yes, which kind? What should be the population of the cluster, and what kind of network topology should be implemented) is all up to the details.
It's kind of hard to suggest more than saying "build a cluster of PostgreSQL DB servers, consisting of a couple of dedicated hardware, within the compounds of a trusted infrastructure provider" with so few details. And maybe: "Keep an acceptable amount of business logic in stored functions and procedures if the application server will be residing in-house and the network latency between it and the DB cluster will be an issue."
Maybe you need more, or maybe you don't even need all that hassle.

And if by an unlucky decision, you end up sticking with MySQL, do not even think about using MyISAM.