Importance of the Mathematical Foundations of Machine Learning Methods for Scientific and Engineering Applications
This work identifies a critical gap for researchers and practitioners in scientific and engineering fields, but it is incremental as it calls for further development rather than presenting new solutions.
The paper addresses the challenge of applying machine learning to scientific and engineering problems by highlighting the need for enhanced mathematical foundations to improve rigor, reliability, and interpretability, emphasizing the importance of incorporating domain-specific inductive biases.
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.