Georgia Institute of Technology College of Sciences

A computationally expensive physical simulation is a huge bottleneck to advance in science and technology. Fortunately, many data-driven approaches have emerged to accelerate those simulations, thanks to the recent advancements in machine learning (ML) and artificial intelligence. For example, a well-trained 2D convolutional deep neural network can predict the solution of the complex Richtmyer–Meshkov instability problem with a speed-up of 100,000x [1]. However, the traditional black-box ML models do not incorporate existing governing equations, which embed underlying physics, such as conservation of mass, momentum, and energy. Therefore, the black-box ML models often violate important physics laws, which greatly concern physicists, and require big data to compensate for the missing physics information. Additionally, it comes with other disadvantages, such as non-structure-preserving, computationally expensive training phase, non-interpretability, and vulnerability in extrapolation. To resolve these issues, we can bring physics into the data-driven framework. Physics can be incorporated into different stages of data-driven modeling, i.e., the sampling stage and model-building stage. Physics-informed greedy sampling procedure minimizes the number of required training data for a target accuracy [2]. Physics-guided data-driven model better preserves the physical structure and is more robust in extrapolation than traditional black-box ML models. Numerical results, e.g., hydrodynamics [3,4], particle transport [5], plasma physics, and 3D printing, will be shown to demonstrate the performance of the data-driven approaches. The benefits of the data-driven approaches will also be illustrated in multi-query decision-making applications, such as design optimization [6,7].

Reference
[1] Jekel, Charles F., Dane M. Sterbentz, Sylvie Aubry, Youngsoo Choi, Daniel A. White, and Jonathan L. Belof. “Using Conservation Laws to Infer Deep Learning Model Accuracy of Richtmyer-meshkov Instabilities.” arXiv preprint arXiv:2208.11477 (2022).
[2] He, Xiaolong, Youngsoo Choi, William D. Fries, Jon Belof, and Jiun-Shyan Chen. “gLaSDI: Parametric Physics-informed Greedy Latent Space Dynamics Identification.” arXiv preprint arXiv:2204.12005 (2022).
[3] Copeland, Dylan Matthew, Siu Wun Cheung, Kevin Huynh, and Youngsoo Choi. “Reduced order models for Lagrangian hydrodynamics.” Computer Methods in Applied Mechanics and Engineering 388 (2022): 114259.
[4] Kim, Youngkyu, Youngsoo Choi, David Widemann, and Tarek Zohdi. “A fast and accurate physics-informed neural network reduced order model with shallow masked autoencoder.” Journal of Computational Physics 451 (2022): 110841.
[5] Choi, Youngsoo, Peter Brown, William Arrighi, Robert Anderson, and Kevin Huynh. “Space–time reduced order model for large-scale linear dynamical systems with application to Boltzmann transport problems.” Journal of Computational Physics 424 (2021): 109845.
[6] McBane, Sean, and Youngsoo Choi. “Component-wise reduced order model lattice-type structure design.” Computer methods in applied mechanics and engineering 381 (2021): 113813.
[7] Choi, Youngsoo, Gabriele Boncoraglio, Spenser Anderson, David Amsallem, and Charbel Farhat. “Gradient-based constrained optimization using a database of linear reduced-order models.” Journal of Computational Physics 423 (2020): 109787.

In this talk, we introduce a new method for minimizing analytic Morse functions over compact domains through the use of polynomial approximations. This is, in essence, an effective application of the Stone-Weierstrass Theorem, as we seek to extend a local method to a global setting, through the construction of polynomial approximants satisfying an arbitrary set precision in L-infty norm. The critical points of the polynomial approximant are computed exactly, using methods from computer algebra. Our Main Theorem states probabilistic conditions for capturing all local minima of the objective function $f$ over the compact domain. We present a probabilistic method, iterative on the degree, to construct the lowest degree possible least-squares polynomial approximants of $f$ which attains a desired precision over the domain. We then compute the critical points of the approximant and initialize local minimization methods on the objective function $f$ at these points, in order to recover the totality of the local minima of $f$ over the domain.

Surface bundles lie in the intersection of many areas of math: algebraic topology, 2–4 dimensional topology, geometric group theory, algebraic geometry, and even number theory! However, we still know relatively little about surface bundles, especially compared to vector bundles. In this interactive talk, I will present the general (and beautiful) fiber bundle theory, including characteristic classes, as a starting point, and you the audience will get to specialize the general theory to surface bundles, with rewards! The talk aims to be accessible to anyone who had exposure to algebraic topology. This is also part one of three talks about surface bundles I will give this semester.

We show that in Cartan-Hadamard manifolds M^n, n≥ 3, closed infinitesimally convex hypersurfaces S bound convex flat regions, if curvature of M^n vanishes on tangent planes of S. This encompasses Chern-Lashof characterization of convex hypersurfaces in Euclidean space, and some results of Greene-Wu-Gromov on rigidity of Cartan-Hadamard manifolds. It follows that closed simply connected surfaces in M^3 with minimal total absolute curvature bound Euclidean convex bodies, as stated by M. Gromov in 1985. The proofs employ the Gauss-Codazzi equations, a generalization of Schur comparison theorem to CAT(0) spaces, and other techniques from Alexandrov geometry outlined by A. Petrunin, including Reshetnyak’s majorization theorem, and Kirszbraun’s extension theorem.