  There has been a lot of recent interest in adopting machine learning methods
for scientific and engineering applications. This has in large part been
inspired by recent successes and advances in the domains of Natural Language
Processing (NLP) and Image Classification (IC). However, scientific and
engineering problems have their own unique characteristics and requirements
raising new challenges for effective design and deployment of machine learning
approaches. There is a strong need for further mathematical developments on the
foundations of machine learning methods to increase the level of rigor of
employed methods and to ensure more reliable and interpretable results. Also as
reported in the recent literature on state-of-the-art results and indicated by
the No Free Lunch Theorems of statistical learning theory incorporating some
form of inductive bias and domain knowledge is essential to success.
Consequently, even for existing and widely used methods there is a strong need
for further mathematical work to facilitate ways to incorporate prior
scientific knowledge and related inductive biases into learning frameworks and
algorithms. We briefly discuss these topics and discuss some ideas proceeding
in this direction.
