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.

The 2025 Atlanta Science Festival Mathapalooza! will take place in an art gallery at the opening of an exhibition, The Art of Math.  The show will begin with magic by Matt Baker, followed by hands-on art construction, circus acts, "Math Court" skits, and 4-dimensional dance, before ending with a performance by Tracy Woodard of Bach's Crab Canon, with a discussion of its symmetry algebra.  Find out more and get tickets at https://atlantasciencefestival.org/events-2025/1113-mathapalooza-at-the-gallery/

Recent advances in deep learning have introduced numerous innovative frameworks for solving high-dimensional mean-field games (MFGs). However, these methods are often limited to solving single-instance MFGs and require extensive computational time for each instance, presenting challenges for practical applications.

In this talk, I will present our recent work on a novel framework for learning the MFG solution operator. Our model takes MFG instances as input and directly outputs their solutions in a single forward pass, significantly improving computational efficiency. Our method offers two key advantages: (1) it is discretization-free, making it particularly effective for high-dimensional MFGs, and (2) it can be trained without requiring supervised labels, thereby reducing the computational burden of preparing training datasets common in existing operator learning methods. If time permits, I will also explore connections between this framework and in-context learning, highlighting its broader implications and potential for further advancements.

Modern Artificial Intelligence (AI) systems, such as ChatGPT, rely on artificial neural networks (ANNs), which are historically inspired by the human brain. Despite this inspiration, the similarity between ANNs and biological neural networks is largely superficial. For instance, the foundational McCulloch-Pitts-Rosenblatt unit of ANNs drastically oversimplifies the complexity of real neurons.Recognizing the intricate temporal dynamics in biological neurons and the ubiquity of feedback loops in natural networks, we suggest reimagining neurons as feedback controllers. A practical implementation of such controllers within biological systems is made feasible by the recently developed Direct Data-Driven Control (DD-DC). We find that DD-DC neuron models can explain various neurophysiological observations, affirming our theory.