  Although deep learning has achieved remarkable success in various scientific
machine learning applications, its opaque nature poses concerns regarding
interpretability and generalization capabilities beyond the training data.
Interpretability is crucial and often desired in modeling physical systems.
Moreover, acquiring extensive datasets that encompass the entire range of input
features is challenging in many physics-based learning tasks, leading to
increased errors when encountering out-of-distribution (OOD) data. In this
work, motivated by the field of functional data analysis (FDA), we propose
generalized functional linear models as an interpretable surrogate for a
trained deep learning model. We demonstrate that our model could be trained
either based on a trained neural network (post-hoc interpretation) or directly
from training data (interpretable operator learning). A library of generalized
functional linear models with different kernel functions is considered and
sparse regression is used to discover an interpretable surrogate model that
could be analytically presented. We present test cases in solid mechanics,
fluid mechanics, and transport. Our results demonstrate that our model can
achieve comparable accuracy to deep learning and can improve OOD generalization
while providing more transparency and interpretability. Our study underscores
the significance of interpretable representation in scientific machine learning
and showcases the potential of functional linear models as a tool for
interpreting and generalizing deep learning.
