Department of Mathematics and Statistics, University of Nevada, Reno
Title
Analytical exact solution in the form of power series to the porous medium equation
Abstract
The porous medium equation (PME) is a nonlinear diffusion equation, where the diffusivity is a power-law function of the unknown quantity.
In hydrological applications, it will be the hydraulic head. We consider the case of a one-dimensional reservoir, which is initially dry, and is of a
semi-infinite extent. For certain classes of boundary conditions, it is possible to introduce similarity variables and reduce initial-boundary value problem
for PME to a boundary value problem for a nonlinear ordinary differential equation. We show how to construct a solution in the form of a power series
for that nonlinear ODE and obtain the recurrence relation for the coefficients of the series. Also, we comment on the convergence of the series.
Institute of Mathematical Sciences, Ewha Womans University, Seoul, South Korean
Title
Deep Variational EIT via Coupled Neural Potentials and Stream Functions
Abstract
We propose a neural mixed variational formulation for electrical impedance tomography (EIT) that couples the electric potential and current density
fields within a unified energy framework. The approach is based on a quadratic functional that is strictly convex under physically consistent
boundary and divergence constraints. Unlike strong-form PINN formulations, it preserves the intrinsic variational structure of the forward model,
and yields improved numerical stability. Numerical experiments on synthetic EIT reconstructions show enhanced robustness and improved reconstruction
quality compared with strong-form PINN baselines.
Title
AI and Quantum Approaches to Solving Combinatorial Optimization Problems in Space and Missile Defense
Abstract
Satellite maneuver planning in contested space environments leads to large-scale combinatorial optimization problems with nonlinear dynamics, discrete decisions,
and hard resource constraints. This talk presents two complementary approaches to such problems: reinforcement learning (RL) and quadratic optimization methods
compatible with quantum annealing. Using weapon-engagement-zone (WEZ) avoidance in low Earth orbit as a case study, we show that deep RL can learn fast,
reactive maneuver policies that balance threat avoidance and fuel usage over short time horizons. To address long-horizon planning and explicit constraints,
we reformulate the problem as a Quadratic Unconstrained Binary Optimization (QUBO) and Constrained Quadratic Model (CQM), enabling hybrid classical-quantum
solvers to identify compact, globally optimized maneuver sets. The results highlight complementary mathematical regimes for learning-based control and
constrained combinatorial optimization.
