I have a list of factors (and their returns) as well as a set of mutual fund returns. What are some techniques I could use to select relevant factors for the funds. For example, fixed income factors to be selected for fixed income funds. I have tried stepwise regressions and then filtering on the $p$-value, but I was wondering if there are other methodologies. Would neural nets be a candidate for this?
You can formulate this as a machine learning problem of predicting mutual funds return based on factor returns.
Any machine learning model can be used, such as neural nets, although tree based models such as Random Forest would be more suitable as they provide feature importance.
Then your problem of selecting relevant factors is known as "feature selection" for which you can start by using univariate methods such as described in https://scikit-learn.org/stable/modules/feature_selection.html (i.e. calculating correlation and rank features by correlation) , or more time-consuming solutions such as RFECV or backward feature elimination such as http://rasbt.github.io/mlxtend/user_guide/feature_selection/SequentialFeatureSelector/ .
Also you can look into eli5 and SHAP libraries.
I'm assuming you want to check factors for a group of funds (from what you've written in the question) and not a single fund. Simple thing to do would be PCA on the set of different mutual fund returns and get the factors explaining most variance. Again PCA factors are completely black-box so now you'll have to check correlation of the highest variance explaining factor (PC1) with the group of pre-defined factors you want to check for.
Better approach is to decompose the returns matrix based on the set of pre-defined factors. There is a paper by Meucci in it. If you're good at matrix algebra give it a read :