Depends on the independent and dependent variables, e.g. if the dependent variable is a factor, then a logistic regression link function is used (generalization of the step function to make it smooth/differentiable for binary classification). Sometimes, the prior knowledge can help form the prior distribution term (hence residual term, such as the  L1 absolute norm  for laplacian distributed terms, resulting in sparsity, which is a viable model for certain types of noise in images).

You can always use cross validation to select between different linear and non linear models, which is a data driven approach.

Any statistics book will have sections dedicated to GLMs (generalized linear models) and non linear models.

I suggest Tibshirani and Hastie's book.

Dear Professor Thomas E. Becker,

we use non-linear transformations (NLTs) for solving the following problems:

 - anomaly detection in non-Gaussian data

https://www.researchgate.net/publication/282329599_Statistical_anomaly_detection_techniques_based_on_normalizing_transformations_for_non-Gaussian_data

- non-linear regression analysys (building non-linear regressions, their confidence and prediction intervals)

https://www.researchgate.net/publication/303962243_DEVELOPING_THE_SOFTWARE_DEFECT_PREDICTION_MODELS_USING_REGRESSION_ANALYSIS_BASED_ON_NORMALIZING_TRANSFORMATIONS

- modeling a dependent variable of nonlinear regression

https://www.researchgate.net/publication/308510035_MODELING_A_DEPENDENT_VARIABLE_OF_NON-LINEAR_REGRESSION_ON_THE_BASIS_OF_NORMALIZING_TRANSFORMATIONS

- modelling the Gaussian random variables

https://www.researchgate.net/publication/307212325_SIMULATION_OF_GAUSSIAN_RANDOM_VARIABLES_USING_JOHNSON_SU_TRANSFORMATION_in_Ukrainian_Modeluvanna_gausivskih_vipadkovih_velicin_iz_vikoristannam_peretvorenna_Dzonsona_iz_sim%27i_SU_S_B_Prihodko_Informati

- parameter estimation of nonlinear stochastic differential equations

https://www.researchgate.net/publication/305281544_Parameter_estimation_of_nonlinear_stochastic_differential_equations_based_on_the_normalizing_transformations_in_Russian_Ocenka_parametrov_nelinejnyh_stohasticeskih_differencialnyh_uravnenij_na_osnove_

- accurate linearization of some classes of nonlinear stochastic differential equations

https://www.researchgate.net/publication/309677910_The_method_of_accurate_linearization_of_some_classes_of_nonlinear_stochastic_differential_equations_in_Ukrainian_Metod_tocnoi_linearizacii_deakih_klasiv_nelinijnih_stohasticnih_diferencialnih_rivnan_S

- constructing mathematical models of non-linear stochastic differential systems

https://www.researchgate.net/publication/303803493_Application_of_normalizing_transformations_for_constructing_mathematical_models_of_non-linear_stochastic_differential_systems_Electronic_Modeling_2011_V_33_No2_-_P13-23_-_ISSN_0204-3572_in_Russian_Pri

Conference Paper Statistical anomaly detection techniques based on normalizin...

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Most of the time some non-linear transformation is used to linearize or normalize a dependent variable before analysis. 

One contrarian view is expressed in the following article.  Perhaps you will find the proof and argument useful. 

Wainer, Howard (1976)  Estimating coefficients in linear models:  It don't make no nevermind.  Psychological Bulletin, 83(2), 213-217.

Best wishes,

David Drehmer 

.

you might find the paper attached interesting

THE LOG TRANSFORMATION IS SPECIAL

OLIVER N. KEENE

STATISTICS IN MEDICINE, VOL. 14,811-819 (1995)

.

T.P. Ryan (1997, p.71,  Modern regression methods, Wiley, 529 p. ISBN 0471529125) underlined that transformations are made for essentially four purposes:

1. To transform a nonlinear model to a linear model

2. To transform X and/or Y in such a way that the strength of the linear relationship between the new variables is better than the linear relationship between X and Y

3. To obtain a relatively constant error variance

4. To obtain approximate normality for the distribution of the error term

These uses are covered in the book.