Musk is said to have made a beeline for Heard after first meeting her on the set of 2013 film Machete Kills, in which he made a cameo.
It's the first time the couple have publicly posted about their relationship on social media.
The pair is shown in paparazzi photos zip-lining in Australia, where Heard is filming "Aquaman".
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"Over time I think we will probably see a closer merger of biological intelligence and digital intelligence", Musk told an audience at the World Government Summit in Dubai.
Musk, the high-profile billionaire who founded Hawthorne-based SpaceX and owns Tesla electric vehicle company, broke with his tradition of mostly just sharing work-related photos on his Instagram account, showing a photo of Heard's arm draped over his shoulder. Now, the actor has turned all the rumours into truth with a picture that she had posted on her Instagram account recently. In fact, the first rumors on a possible romance emerged a year ago, soon after Amber and her former husband, actor Johnny Depp, filed for divorce. As she said, their relationship was intensely passionate and at times volatile, but always bound by love.
"One of the things they want to do is settle down and have a family". According to the insider, they also worked together to plan the previous year's gala.
Heard and Musk reportedly struck up their romance a few months after Musk filed for divorce from ex-wife Talulah Riley, whom he was married to and divorced from twice. In the image, Heard has just planted a kiss on Musk's cheek, leaving some lipstick residue behind. Also Amber was smitten with the billionaire as well, Daily Mail previously reported.