Many scientific studies in recent years have been collecting data at a high frequency, which can be considered as functional data. When both the response variable to be modelled and the covariates are functions, we provide a novel and easy-to-implement method addressing function-on-function linear modelling and obtain interpretable parameters. Two main types of models are considered: the concurrent model which explains the response curve $Y_i(t)$ at time $t$ from the values at same time $t$ of the covariates $X_i^l(t)$; the (feed-forward) integral model which explains $Y_i(t)$ based on the values of covariate curves $X_i^l(s)$ observed at any times $s\leq t$. A regularized inference approach is proposed, which accurately selects an appropriate set of basis functions that can be used for functional data reconstruction and at the same time provides smooth and interpretable functional parameters. Numerical studies on simulated data with different scenarios illustrate the good performance our the method to capture the relationship between covariates and response. The method is finally applied to the well-known data in order to compare it to some existed competitors. On Canadian weather data with the problem of predicting precipitations from temperature measurements and on Hawaii Ocean data for predicting ocean salinity from temperature, oxygen, chloropigments and density measurements, our method made significant improvements on prediction error.