Uploading file to HTTP via Powershell
A year late, so you probably don't need the answer anymore, but what worked for me was:
Invoke-RestMethod -Uri $uri -Method Post -InFile $uploadPath -UseDefaultCredentialsMine needed to use windows auth as the currently logged in user... From this other answer, looks like you can use -cred in some cases and have to build out an Authorization header in others.
Even later post but these Invoke-RestMethod solutions (and a few others) didn't work for me. My service endpoint was failing with missing boundaries and other "content" issues, depending on my trial and error.
I tried using WebClient's Uploadfile() functionality and it worked for me.Thankfully, it's concise.
$wc = New-Object System.Net.WebClient$resp = $wc.UploadFile($uri,$uploadPath)