From Theory to Practice: Mathematical Approaches to Scientific Machine Learning

Machine learning (ML) has achieved unprecedented empirical success in diverse applications. It has been applied to solve scientific and engineering problems and has emerged as a new research field: Scientific Machine Learning (SciML). However, many ML techniques are highly complex and sophisticated, often requiring extensive trial-and-error experimentation and problem-specific techniques to be implemented effectively. This complexity frequently poses significant challenges for scientific research, including reproducibility and rigor. This talk explores mathematical approaches, offering more principled and reliable methodologies for SciML. The first part will present recent efforts advancing the predictive power of physics-informed machine learning through robust training/optimization methods. This includes an effective training method for multivariate neural networks, namely, Active Neuron Least Squares (ANLS), and a two-step training method for deep operator networks. The second part is about how to embed the first principles of physics into neural networks. I will present a general framework for designing NNs that obey the first and second laws of thermodynamics. The framework not only provides flexible ways of leveraging available physics information but also results in expressive NN architectures. I will also present an intriguing phenomenon of this framework when it is applied to latent-space dynamics identification, where a correlation emerges between the entropy production rate in the latent space and the behavior of the full-state solution.
 

Bio
https://math.sciences.ncsu.edu/people/yshin8/