Georgia Institute of Technology College of Sciences

Machine learning (ML) has achieved unprecedented empirical success in diverse applications. It now has been applied to solve scientific and engineering problems, which has become an emerging field, Scientific Machine Learning (SciML). However, many ML techniques are highly complex and sophisticated, often requiring extensive trial-and-error experimentation and specialized problem-dependent tricks to implement effectively. This complexity frequently leads to significant challenges, such as reproducibility and rigorness, for scientific research. This talk explores mathematical approaches, offering more principled and reliable methodologies in 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 in the context of latent space dynamics identification where a correlation appears between an entropy production rate in the latent space and the behaviors of the full-state solution.