Refer to attached photo1.png we understand in order to calculate the mean of squares errors for training sample, we need to divide "Sum of Squares Error" in model summary by the number of sample size assigned for training. The same procedure is applicable for the mean of squares errors for testing.

Now, for calculating Root Mean Squares of Error (RMSE)  we need calculate square root of Mean Squares of Error.

It is very good to look at this video to get what RMSE is really is:

https://www.youtube.com/watch?v=U4BTzF3Wzt0&list=PLUUr_zGTc4ETe2aZC1RUR4tmDBrOfB3Y_&index=2

I recommend to watch these following playlists following one after another to understand well ANN:

https://www.youtube.com/playlist?list=PL6DA09D4349A1EB86

https://www.youtube.com/playlist?list=PLC112AD1C69432FDB

https://www.youtube.com/watch?v=CVJOseIJnww

https://www.youtube.com/watch?v=U4BTzF3Wzt0

https://www.youtube.com/watch?v=p1-FiWjThs8

https://www.youtube.com/watch?v=IruMm7mPDdM

[Special thanks to Mr. Jeff Heaton for this remarkable training]http://www.heatonresearch.com/about/

I was thinking that RMSE for ANN is as same as Std. Error for the estimate in model summary in multiple regression model in SPSS. I mean we can compare them to conclude whether ANN predict better or MRA. Any idea?

"The standard error of the estimate is the square root of the residual mean square, which is an estimate of the average squared error in prediction and is printed in the Model Summary table of the Regression output. Some textbooks on regression analysis use the term "standard error of estimate" for the square root of the mean square error, such as Pedhazur (1997) and Cohen et al. (2003), while other texts use the term "standard error of regression" (Montgomery et al., 2001 ; Weisberg, 1985).

Cohen, J., Cohen, P., West, S.G., & Aiken, L.S. (2003). Applied Multiple Regression/Correlation Analysis for the Behavioral Sciences (3rd Ed.). Mahwah NJ: Erlbaum. (pp. 37-40).

Montgomery, D.C., Peck, E.A., & Vining, G.G. (2001). Introduction to Linear Regression Analysis (3rd Ed.). New York: Wiley. (pp. 22-23).

Pedhazur, E. J. (1997). Multiple Regression in Behavioral Research: Explanation and Prediction (3rd Ed.). (pp. 28-29)

Weisberg, S. (1985). Applied Linear Regression (2nd Ed.). New York. (pp 12-13).

..

To replicate the standard error of the estimate as printed by Regression, you would square the errors in prediction and then sum these squares across cases, then divide that sum by (N-P), where N is the sample size and P is the number of parameters in the model, including the intercept. The square root of this result is the standard error of estimate. .."

Source: http://www-01.ibm.com/support/docview.wss?uid=swg21481473

Base above argument, I think that we can square  "standard error of estimate from Regression (model)" and then multiply by (N-P). The result is Sum of Squares Error which is comparable with Sum of Squares Error of ANNs.

* where N is the sample size and P is the number of parameters in the model, including the intercept  

The MRA technique give us a better option to test causality in the model since it gives us a line which can be interpreted to know whether there is a significant relationship between variables or not. Whereas, the output of ANNs can be a not linear graph such as carve which can not be tested to say that there is a significant relationship between variables or not.

I think SPSS does not calculate RMSE. You need to calculate it yourself.

I applied a combination of multiple regression analysis and ANN in this paper:

Preprint Intention to Use SMS Vaccination Reminder and Management Sys...

Thank you.

An excel file is attached to calculate RMSEs and draw a diagram. You need to copy the relevant outputs of ANNs (SPSS) into the sheet. Please let me know if any errors are noticed. I hope you find this attachment helpful. Best.