Getting the field of form angular

I have a form with some fields.
I'm validating the fields with css classes:(if the field is invalid and the user touched it, then input's border-color = red.)

select.ng-invalid.ng-touched,
input.ng-invalid.ng-touched,textarea.ng-invalid.ng-touched {
    border-color: red;
}

If the user submits the form without filling one or more field, there would be a danger alert.

HTML:

  <div ng-show="formInvalid>
       error!
  </div>

JS:

 if ($scope.pniyaForm.$valid) {
    $scope.formInvalid = false;
      .....
 } else {
    $scope.formInvalid = true;
 }

But, If the user submits the form and has not touched any of the field, the css classes don't influence.(because user didn't touch...)
I want to add the class in the code.
Does anyone have an idea for an elegant way to do this without writing it on each field separately?