Schedule

Physics-Preserving Finite Elements  for Multiphysics Subsurface Systems

Sanghyun Lee, FSU

Abstract: Subsurface energy applications (oil and gas production, geothermal energy) demand models that are both physics-faithful and computationally dependable across tightly coupled, multiscale processes. We present physics-preserving finite element formulations for subsurface flow and coupled processes that enforce local conservation and remain robust against oscillations, inf–sup pathologies, and locking, delivering optimal-order accuracy without excessive degrees of freedom. These high-fidelity solvers provide reliable pressure and flux fields under strong heterogeneity and coupling. 

Completeness of reparametrization-invariant Sobolev metrics on the space of surfaces

Martin Bauer, FSU

Abstract: The space of immersions of a closed manifold M into Euclidean space is among the most important infinite dimensional manifolds. The most natural Riemannian metrics on this space are reparametrization-invariant Sobolev metrics; these form a hierarchy of metrics, based on their order—the number of derivatives they "see”. They arise as natural extensions of right-invariant metrics on diffeomorphism groups, central to Arnold’s geometric formulation of hydrodynamical equations. Beyond their theoretical significance, they play a crucial role in mathematical shape analysis and geometric data science, where they enable meaningful and robust comparisons between shapes modeled as curves or surfaces.

In 2013, David Mumford conjectured that for orders larger than dim(M)/2 + 1, these geometries are complete. Note, that by the seminal work of Ebin and Marsden a similar statement is known to be true for diffeomorphism groups. In the context of immersions this conjecture was shown to be true in the case of immersed curves in the work of Bruveris, Michor and Mumford. In this talk I will present the first construction of complete metrics on immersions of two-dimensional surfaces, discuss the context and techniques, as well as possible extensions to higher dimensions. 

Based on joint work with Cy Maor and Benedikt Wirth  (https://arxiv.org/abs/2512.01566).

Generic properties of viscosity solutions to Hamilton-Jacobi equations

Khai Nguyen, NCSU

Abstract: This talk presents recent results on the generic properties of conjugate points in viscosity solutions of first-order Hamilton–Jacobi equations. For terminal costs belonging to a countable intersection of open dense subsets, the set of conjugate points is closed and has locally bounded (d−2)-dimensional Hausdorff measure. As a consequence, the solution is continuously differentiable on an open, dense subset of $\R^d$.

Sloshing of Liquid Hydrogen in a Spherical Tank in Microgravity: Modal Analysis and Deep Learning Prediction

Mark Sussman, FSU

Abstract: This study investigates the sloshing dynamics of liquid hydrogen (LH$_2$) within a spherical tank under microgravity conditions in order to characterize sensitivity to changes in excitation strength and fluid volume; particularly in regimes where the Bond number is low and the gas–liquid interface adopts a nearly spherical configuration. A series of high-fidelity simulations are performed in which the tank is oscillated with fixed frequency and varying amplitudes, across different fill ratios. Proper Orthogonal Decomposition (POD) is employed to extract the dominant spatial flow structures and to quantify the energy content across sloshing modes. To enable future-state prediction of unsteady sloshing behavior, a Long Short-Term Memory (LSTM) deep learning model is trained on the temporal coefficients of the POD modes. The resulting data-driven model demonstrates robust short-term forecasting accuracy across a range of operating conditions. This integrated approach — combining physical simulation, reduced-order modeling, and deep learning — provides a new pathway for understanding and predicting cryogenic sloshing in low-gravity environments, with implications for propellant management in next-generation spacecraft and orbital depots.

Machine Learning seminar (Room Love 106)

A variational approach to studying dimension reduction algorithms

Ryan Murray, NCSU

Abstract: Dimension reduction algorithms, such as principal component analysis (PCA), multidimensional scaling (MDS), and stochastic neighbor embeddings (SNE and tSNE), are an important tool for data exploration, visualization, and subgroup identification. While these algorithms see broad application across many scientific fields, our theoretical understanding of non-linear dimension reductionalgorithms remains limited. This talk will describe new results that identify large data limits for MDS and tSNE using tools from the Calculus of Variations. We'll highlight connections with Gromov-Wasserstein distances, manifold learning, and Perona-Malik diffusion. Along the way, we will showcase situations where standard libraries give outputs that are misleading, and propose new computational algorithms to mitigate these issues and improve efficiency.

Colloquium (Room Love 101)

Analysis and Control in Poroelastic Systems with Applications to Biomedicine

Lorena Bociu, NCSU

Abstract: In biomechanics, local phenomena, such as tissue perfusion, are strictly related to the global features of the surrounding blood circulation. We propose a heterogeneous model where a local, accurate, 3D description of fluid flows through deformable porous media by means of poroelastic systems is coupled with a systemic 0D lumped model of the remainder of the circulation. This represents a multiscale strategy, which couples an initial boundary value problem to be used in a specific region with an initial value problem for the rest of the circulatory system.  We present new results on wellposedness analysis, optimal control and solution methods for this nonlinear multiscale interface coupling of PDEs and ODEs. Our results have applications in biomedicine and bioengineering, including tissue perfusion, fluid flow inside cartilages and bones, and design of bioartificial organs.

A Score-based Diffusion Model Approach for Adaptive Learning of Stochastic Partial Differential Equation Solutions

Feng Bao, FSU

Abstract: We shall present a novel framework for adaptively learning the time-evolving solutions of stochastic partial differential equations (SPDEs) using score-based diffusion models within a recursive Bayesian inference setting. SPDEs play a central role in modeling complex physical systems under uncertainty, but their numerical solutions often suffer from model errors and reduced accuracy due to incomplete physical knowledge and environmental variability. To address these challenges, we encode the governing physics into the score function of a diffusion model using simulation data and incorporate observational information via a likelihood-based correction in a reverse-time stochastic differential equation. This enables adaptive learning through iterative refinement of the solution as new data becomes available. To improve computational efficiency in high-dimensional settings, we introduce the ensemble score filter, a training-free approximation of the score function designed for real-time inference. Numerical experiments on benchmark SPDEs demonstrate the accuracy and robustness of the proposed method under sparse and noisy observations.  

Well-posedness and Optimal Control in Poroelasticity

Ala’ Alalabi, NCSU

Abstract: Fluid flows through deformable, porous media arise in numerous applications, ranging from geophysics to biomedicine. Such processes are described using poroelastic models which couple parabolic-elliptic partial differential equations (PDEs), describing the interaction between fluid transport and mechanical deformation. Mathematically, these coupled systems belong to the class of implicit, degenerate evolution equations. While the existence and regularity theory for such systems is well-established, optimal control problems constrained by poroelastic dynamics, especially relevant in biomedical applications, have only recently been considered. These control problems are challenging due to the degeneracy of the equations and the implicit nature of the coupling. In this work, we analyze the system using radiality theory - a generalization of the classical Hille-Yosida framework for non-explicit evolution equations. This abstract formulation yields new insights into well-posedness theory and enables a treatment of linear control problems constrained by degenerate evolution dynamics.

Strong and Weak Solutions for a Multiscale Interface Coupling of Poroelasticity and Lumped Hydraulic Systems

David Hernandez, NCSU

Abstract: We consider a heterogeneous system which consists of a (locally accurate) 3D descriptive model of fluid flowing through deformable porous media equations, coupled with a systemic 0D lumped model of the remainder of the circulation, where the fluid flow through a vascular network is described via its analogy with a current flowing through an electric circuit. In this talk, we present new results on wellposedness of strong and weak solutions for this multiscale interface coupling.

Generalized Rossby-Haurwitz waves in 2D and 3D

Stephen Preston, CUNY

Abstract: Rossby-Haurwitz waves are explicit exact nonsteady solutions of the 2D Euler equation on the sphere, arising from a perturbation around steady rotation. We show how to obtain other solutions to the Euler equations on a variety of other manifolds, particularly in three dimensions, including explicit formulas on the three-sphere. The main idea is to work with velocity fields which generate isometries and whose vorticity fields also generate isometries, and we give a classification theorem for these as well. This is joint work with Patrick Heslin.

Optimal Transport as a Transform for Scalar Conservation Laws

Rocio Diaz Martin, FSU

Abstract: We will view one-dimensional optimal transport as a nonlinear change of variables for scalar conservation laws. Motivated by the role of Fourier analysis in linear PDEs, we interpret optimal transport as a transform that reveals low-dimensional structure in transport-dominated dynamics. We show that, for one-dimensional conservation laws, a small number of transport-based modes can accurately describe the solution dynamics. This perspective offers a harmonic-analysis viewpoint on optimal transport and suggests new tools for analysis, approximation, and reduced-order modeling of PDEs. This is ongoing joint work with the groups led by Prof. Gustavo Rohde (University of Virginia) and by Prof. Harbir Antil (George Mason University).

A supervised learning scheme for computing Hamilton-Jacobi equation via density coupling

Shu Liu, FSU

Abstract: We propose a supervised learning scheme for the first-order Hamilton–Jacobi PDEs in high dimensions. The scheme is designed using the structure of a Hamiltonian flow on the probability manifold via a density coupling strategy. It is equivalently posed as a regression problem using the Bregman divergence, which provides the loss function in learning while the data is generated through the associated Hamiltonian system. The proposed scheme can be used to describe the behaviors of Hamilton–Jacobi PDEs beyond the singularity formations on the support of coupling density. It is also applicable to optimal control problems with terminal distribution constraint. Several numerical examples with different Hamiltonians will be demonstrated to support our findings.

When Complexity Simplifies: The Paradox of More Random, More Tractable

Malbor Asllani, FSU

Abstract: Heterogeneity is intrinsic to real networked systems but fundamentally challenges the classical Master Stability Function (MSF), which relies on identical node dynamics and modal decoupling. When local dynamics vary across nodes, instabilities may emerge without a clear identification of which node is responsible. We introduce a localization-based generalized MSF that restores interpretability in heterogeneous networks. The approach exploits the tendency of Laplacian eigenvectors in large random networks to localize on small subsets of nodes. Each non-uniform mode can then be approximated by a reduced Jacobian governed by the local dynamics of the node on which the eigenvector is concentrated, yielding a heterogeneous MSF that predicts instability onsets while preserving a modal viewpoint. The uniform mode, which is not localized, is treated conservatively by evaluating node-wise Jacobians across the network. Crucially, the framework enables node-level attribution: when a small number of nodes are modified, the heterogeneous MSF identifies which specific node drives the instability. For a simple illustration, we demonstrate this in a heterogeneous Brusselator network, where three perturbed nodes coexist with a large homogeneous background. Although all perturbed nodes contribute to spectral splitting, only one is correctly identified as responsible for the instability, in agreement with direct numerical simulations.