Georgia Institute of Technology College of Sciences

This presentation will expound the challenges involved in the generation of digital twins (DT) as valuable tools for supporting innovation and providing informed decision support for the optimization of properties and/or performance of advanced material systems. This presentation will describe the foundational AI/ML (artificial intelligence/machine learning) concepts and frameworks needed to formulate and continuously update the DT of a selected material system. The central challenge comes from the need to establish reliable models for predicting the effective (macroscale) functional response of the heterogeneous material system, which is expected to exhibit highly complex, stochastic, nonlinear behavior. This task demands a rigorous statistical treatment (i.e., uncertainty reduction, quantification and propagation through a network of human-interpretable models) and fusion of insights extracted from inherently incomplete (i.e., limited available information), uncertain, and disparate (due to diverse sources of data gathered at different times and fidelities, such as physical experiments, numerical simulations, and domain expertise) data used in calibrating the multiscale material model. This presentation will illustrate with examples how a suitably designed Bayesian framework combined with emergent AI/ML toolsets can uniquely address this challenge.

Data assimilation techniques are crucial for correcting trajectories when modeling complex dynamical systems. The Latent Ensemble Score Filter (Latent-EnSF), our recently developed data assimilation method, has shown great promise in high-dimensional and nonlinear data assimilation problems with sparse observations. However, this method faces the challenge of high computational cost due to the expensive forward simulation. In this talk, we present Latent Dynamics EnSF (LD-EnSF), a novel methodology that evolves the neural dynamics in a low-dimensional latent space and significantly accelerates the data assimilation process.

To achieve this, we introduce a novel variant of Latent Dynamics Networks (LDNets) to effectively capture the system's dynamics within a low-dimensional latent space. Additionally, we propose a new method for encoding sparse observations into the latent space using recurrent neural networks. We demonstrate the robustness, accuracy, and efficiency of the proposed methods and their limitations for complex dynamical systems with highly sparse (in both space and time) and noisy observations, including shallow water wave propagation for tsunami modeling, FourCastNet in numerical weather prediction, and Kolmogorov flow that exhibits chaotic and turbulent phenomena.

What is the smallest number of edges that a graph can have if it contains all $D$-degenerate graphs on $n$ vertices as subgraphs? A counting argument shows that this number is at least of order $n^{2−1/D}$, assuming n is large enough. We show that this is tight up to a polylogarithmic factor.

Joint work with Peter Allen and Julia Böttcher.

Primitive integral Apollonian circle packings are fractal arrangements of tangent circles with integer curvatures.  The curvatures form an orbit of a 'thin group,' a subgroup of an algebraic group having infinite index in its Zariski closure.  The curvatures that appear must fall into a restricted class of residues modulo 24. The twenty-year-old local-global conjecture states that every sufficiently large integer in one of these residue classes will appear as a curvature in the packing. We prove that this conjecture is false for many packings, by proving that certain quadratic and quartic families are missed. The new obstructions are a property of the thin Apollonian group (and not its Zariski closure), and are a result of quadratic and quartic reciprocity, reminiscent of a Brauer-Manin obstruction. Based on computational evidence, we formulate a new conjecture.  This is joint work with Summer Haag, Clyde Kertzer, and James Rickards.  Time permitting, I will discuss some new results, joint with Rickards, that extend these phenomena to certain settings in the study of continued fractions.