You can connect HDInsight to your on-premises network by using Azure Virtual Networks and a VPN gateway.
Note: To allow HDInsight and resources in the joined network to communicate by name, you must perform the following actions:
Create Azure Virtual Network.
Create a custom DNS server in the Azure Virtual Network.
Configure the virtual network to use the custom DNS server instead of the default Azure Recursive Resolver.
Configure forwarding between the custom DNS server and your on-premises DNS server.
References:
https://docs.microsoft.com/en-us/azure/hdinsight/connect-on-premises-network
https://docs.microsoft.com/en-us/azure/vpn-gateway/vpn-gateway-howto-site-to-site-resource-manager-portal