# Importance of the Mathematical Foundations of Machine Learning Methods for Scientific and Engineering Applications

arXiv: https://arxiv.org/abs/1808.02213 · HTML: https://mlsanity.com/paper/1808.02213/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Aug 2018.

## Built on by (most cited first, 4 of 4)

- [Physics-informed neural networks for PDE-constrained optimization and control](https://mlsanity.com/paper/2205.03377/) (2022)
- [GD-VAEs: Geometric Dynamic Variational Autoencoders for Learning Nonlinear Dynamics and Dimension Reductions](https://mlsanity.com/paper/2206.05183/) (2022)
- [MLMOD: Machine Learning Methods for Data-Driven Modeling in LAMMPS](https://mlsanity.com/paper/2107.14362/) (2021)
- [Transferable Foundation Models for Geometric Tasks on Point Cloud Representations: Geometric Neural Operators](https://mlsanity.com/paper/2503.04649/) (2025)

## Abstract (arXiv)

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.