Georgia Institute of Technology College of Sciences

One essential challenge in the computational modeling of multiscale systems is the availability of reliable and interpretable closures that faithfully encode the micro-dynamics. For systems without clear scale separation, there generally exists no such a simple set of macro-scale field variables that allow us to project and predict the dynamics in a self-determined way. We introduce a machine-learning (ML) based approach that enables us to reduce high-dimensional multi-scale systems to reliable macro-scale models with low-dimensional variational structures that preserve canonical degeneracies and symmetry constraints. The non-Newtonian hydrodynamics of polymeric fluids is used as an example to illustrate the essential idea. Unlike our conventional wisdom about ML modeling that focuses on learning the PDE form, the present approach directly learns the energy variational structure from the micro-model through an end-to-end process via the joint learning of a set of micro-macro encoder functions. The final model, named the deep non-Newtonian model (DeePN2), retains a multi-scale nature with clear physical interpretation and strictly preserves the frame-indifference constraints. We show that DeePN2 can capture the broadly overlooked viscoelastic differences arising from the specific molecular structural mechanics without human intervention.

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.

Thin liquid films flowing down vertical fibers spontaneously exhibit complex interfacial dynamics, leading to irregular wavy patterns and traveling liquid droplets. Such droplet dynamics are fundamental components in many engineering applications, including mass and heat exchangers for thermal desalination, as well as water vapor and particle capture. Recent experiments demonstrate that critical flow regime transitions can be triggered by varying inlet geometries and external fields. Similar interacting droplet dynamics have also been observed on hydrophobic substrates, arising from interfacial instabilities in volatile liquid films. In this talk, I will describe lubrication and weighted residual models for falling droplets. The coarsening dynamics of condensing droplets will be discussed using a lubrication model. I will also present our recent results on developing optimal boundary control and mean-field control for droplet dynamics. 

Reservoir computing is a branch of neuromorphic computing, which is usually realized in the form of ESNs (Echo State Networks). In this talk, I will present some fundamentals of reservoir computing from both the mathematical and the computational points of view. While reservoir computing was designed for sequential/time-series data, we recently observed its great performances in dealing with static image data once the reservoir is set to process certain image features, not the images themselves. Hence, I will discuss possible applications and open questions in reservoir computing.