Solution

I notice that you are not mapping the mysql container port out. If you did, you would see this from the docker ps command:

...   0.0.0.0:3306->3306/tcp       network_mysql

The container network_mysql is attached to a bridge type network called tmp_wex_net. This means that the container is not accesible from the host, by its container name.

I appears that you are using a docker-compose.yml definition for the stack. In order to be able to access the container from the host, you need to use the ports section of your compose definition for this container:

serivces:
  mysql:
    ...
    ports:
      - "3306:3306"
    ...

If you are starting it with docker run, then you can acomplish the same thing with:

docker run -p 3306:3306 --name network_mysql --network="tmp_wex_net" -d mysql

And then use localhost in the hostname of your connection settings in PHPStorm. Like this:

Host: localhost

Port: 3306

Database: network

The problem

The reason that you are not able to connect, is that the host name network_mysql that you specify in the connection settings, does not resolve to any host that your machines knows of.

The container name of a docker container, is not a DNS name that the docker host can resolve.

If you have not specified any network for your mysql container, then it is connected to the default bridge network. And if you have created a new network, without specifying the type - it will also default to the bridge driver.

In order to access the container from the host, you need to either:

1. Connect the container to the host network
2. Or from a container on a bridge network, map the port to the host like suggested in the solution above. You can then address the specifically mapped port on that container with localhost:<portnum> from the host machine.