arXiv subjects

Sophie Moufawad

Publications and source records attributed to Sophie Moufawad.

5 recordsLinked to original sources

arXiv2026

A Two-Stage Learning PINN Approach for Solving the Inverse Problem of the 1D Porous Medium Equation

The Porous Medium Equation (PME), given by $u_t = Δ(u^m)$ for $m > 1$, is a degenerate nonlinear parabolic partial differential equation that arises in various physical applications such as fluid flow in porous media, heat transfer in plasmas, and population dynamics. It is known for its nonlinear diffusion and finite propagation speed. In this paper, we study numerical solutions of the one-dimensional direct and inverse PME using Physics-Informed Neural Networks (PINNs), and compare them with classical numerical methods and available analytical and manufactured solutions. While PINNs provide a flexible framework for solving both forward and inverse problems, we show that the standard inverse formulation suffers from a strong sensitivity to the initial guess, leading to only local convergence. To address this issue, we propose a novel two-stage PINN training framework for the inverse problem, which significantly improves convergence stability and allows reliable recovery of the unknown parameter even for poor initial guesses. Overall, the proposed approach demonstrates that PINNs are a flexible and accurate alternative to classical methods for the 1D PME, and the introduced two-stage training strategy substantially improves their robustness in inverse problems, providing a solid basis for extensions to more complex geometries and higher-dimensional cases.

math.OC↗

arXiv2026

Direct Problem for Gas Diffusion in Polar Firn with Variable Coefficients

We consider the mathematical model of gas trapping in deep polar ice (firns), which consists of a parabolic partial differential equation, that can degenerate at one boundary extreme. In [1], we considered all the coefficients to be constants, except the diffusion coefficient D(z) that is to be reconstructed. In this paper, we assume both the diffusion coefficient D(z) and the volume fraction f(z) are functions. The difficulty in this problem, both theoretically and computationally, arises from the fact that D(z) and f(z) may be zero at bottom of the firn. To handle such degeneracy, we defined appropriate weighted Sobolev spaces and used Lion's theorem to prove existence and uniqueness of the semi-variational formulation of the Firn PDE. A full discrete system is obtained through a P1 Finite element Galerkin procedure in space and an Euler-Implicit scheme in time. Sufficient conditions for the existence and uniqueness of the solution for the discrete system are obtained.

math.NA↗

arXiv2026

Solving Inverse PDE Problems using Minimization Methods and AI

Many physical and engineering systems require solving direct problems to predict behavior and inverse problems to determine unknown parameters from measurement. In this work, we study both aspects for systems governed by differential equations, contrasting well-established numerical methods with new AI-based techniques, specifically Physics-Informed Neural Networks (PINNs). We first analyze the logistic differential equation, using its closed-form solution to verify numerical schemes and validate PINN performance. We then address the Porous Medium Equation (PME), a nonlinear partial differential equation with no general closed-form solution, building strong solvers of the direct problem and testing techniques for parameter estimation in the inverse problem. Our results suggest that PINNs can closely estimate solutions at competitive computational cost, and thus propose an effective tool for solving both direct and inverse problems for complex systems.

math.NA↗

arXiv2021

A Finite-Element Model for the Hasegawa-Mima Wave Equation

In a recent work, two of the authors have formulated the non-linear space-time Hasegawa-Mima plasma equation as a coupled system of two linear PDEs, a solution of which is a pair $(u,w)$, with $w=(I-Δ)u$. The first equation is of hyperbolic type and the second of elliptic type. Variational frames for obtaining weak solutions to the initial value Hasegawa-Mima problem with periodic boundary conditions were also derived. Using the Fourier basis in the space variables, existence of solutions were obtained. Implementation of algorithms based on Fourier series leads to systems of dense matrices. In this paper, we use a finite element space-domain approach to semi-discretize the coupled variational Hasegawa-Mima model, obtaining global existence of solutions in $H^2$ on any time interval $[0,T]$ for all T. In the sequel, full-discretization using an implicit time scheme on the semi-discretized system leads to a nonlinear full space-time discrete system with a nonrestrictive condition on the time step. Tests on a semi-linear version of the implicit nonlinear full-discrete system are conducted for several initial data, assessing the efficiency of our approach.

math.NA↗

arXiv2018

Numerical Stability of s-step Enlarged Krylov Subspace Conjugate Gradient Methods

Recently, enlarged Krylov subspace methods, that consists of enlarging the Krylov subspace by a maximum of t vectors per iteration based on the domain decomposition of the graph of A, were introduced in the aim of reducing communication when solving systems of linear equations Ax=b. In this paper, the s-step enlarged Krylov subspace Conjugate Gradient methods are introduced, whereby s iterations of the enlarged Conjugate Gradient methods are merged in one iteration. The numerical stability of these s-step methods is studied, and several numerically stable versions are proposed. Similarly to the enlarged Krylov subspace methods, the s-step enlarged Krylov subspace methods have a faster convergence than Krylov methods, in terms of iterations. Moreover, by computing st basis vectors of the enlarged Krylov subspace $\mathscr{K}_{k,t}(A,r_0)$ at the beginning of each s-step iteration, communication is further reduced. It is shown in this paper that the introduced methods are parallelizable with less communication, with respect to their corresponding enlarged versions and to Conjugate Gradient.

math.NA↗