Transfer learning for a regression problem
The question is if you can provide the trained model with the data it expects to provide a good output.

In your example, for the model to provide an output, you will have to provide 15 inputs. If you have only 14, can you put a value for input 15 that will make sense to the model? If feature 15 is a person's age, can you make an estimate? If you trained a model on 16x16 images (256 features) but now you have images that are 4x4 (16 features), you may be able to just provide 256 - 16 = 240 black pixels and the model may work fine. Or you could reformat your data to multiply each pixel into 4x4 patch, and get a 16x16 image that way.

What you cannot do, is simply reshape the input so that it's 15 columns wide. If you do that, the first example you provide to your model will contain 14 features from the first real example, plus feature number 1 from the next case. That way, the model is using data from two different examples to make a prediction and it's getting a value for feature 1 on the input node where it expects a value feature 15, and feature 1 and feature 15 can have very different meaning.