A research fellow position is available at University of Pavia in Italy (contract starting next March 2025). Deadline for application: January 30, 2025.
Postgraduate (Master by Research / PhD) under Swinburne-National Road Safety Action Grants Program (NRSAGP) funded by Australia’s Department of Infrastructure, Transport, Regional Development, Communications and the Arts
Variational formulation based on duality to solve partial differential equations: Use of B-splines and machine learning approximants
N. Sukumar Amit Acharya
Many partial differential equations (PDEs) such as Navier–Stokes equations in fluid mechanics, inelastic deformation in solids, and transient parabolic and hyperbolic equations do not have an exact, primal variational structure. Recently, a variational principle based on the dual (Lagrange multiplier) field was proposed. The essential idea in this approach is to treat the given PDE as constraints, and to invoke an arbitrarily chosen auxiliary potential with strong convexity properties to be optimized. This leads to requiring a convex dual functional to be minimized subject to Dirichlet boundary conditions on dual variables, with the guarantee that even PDEs that do not possess a variational structure in primal form can be solved via a variational principle. The vanishing of the first variation of the dual functional is, up to Dirichlet boundary conditions on dual fields, the weak form of the primal PDE problem with the dual-to-primal change of variables incorporated. We derive the dual weak form for the linear, one-dimensional, transient convection diffusion equation. A Galerkin discretization is used to obtain the discrete equations, with the trial and test functions chosen as linear combination of either RePU activation functions (shallow neural network) or B-spline basis functions; the corresponding stiffness matrix is symmetric. For transient problems, a space-time Galerkin implementation is used with tensor-product B-splines as approximating functions. Numerical results are presented for the steady-state and transient convection-diffusion equation, and transient heat conduction. The proposed method delivers sound accuracy for ODEs and PDEs and rates of convergence are established in the L2 norm and H1 seminorm for the steady-state convection-diffusion problem.
In this presentation, I provide a brief overview of Physics-Informed Neural Networks (PINNs) while highlighting the fundamental issues.
Link to the slides: https://www.researchgate.net/publication/386336316_A_brief_overview_of_…
Your feedback is welcome.
Title: “Experimental characterization of micropatterned interfaces subjected to dynamic loading” (PhD research project)
Main Research Project: SURFACE (ERC-2021-Startting Grant, https://doi.org/10.3030/101039198)
Scholarship: ≈20 k€/year (depending on taxes according to Italian regulations)
Duration: 3 years, starting Q1 2025
Hello iMechanica Community, We are organizing a mini symposium (MS) titled "226 - Machine Learning-Based Modeling, Prediction, and Optimization in Advanced Manufacturing and Multiphysical Properties of Materials" at the 18th U.S. National Congress on Computational Mechanics (USNCCM18), to be held in Chicago, Illinois, from July 20-24, 2025.
