Colloquium, Dept. of Mathematics and Applied Math. -- Γενικό Σεμινάριο

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Colloquium

Abstract: We will give a concise derivation of the Navier-Stokes equations, the principal models, and discuss key research methods and main results.

Viscoelastic Wave Propagation in Brain MR Elastography: A Stabilized Finite Element Approach

Abstract: Magnetic Resonance Elastography (MRE) is a non-invasive imaging technique that estimates the mechanical properties of soft tissues from their response to low-frequency harmonic excitation. Shear waves are induced in the tissue, the resulting displacement field is measured with phase-contrast MRI, and stiffness is reconstructed locally by an inversion procedure, known as Tissue Stiffness Reconstruction (TSR). In the brain, this stiffness quantification has emerged as sensitive markers of ageing and neurological disease. The talk begins with an introduction to brain MRE and the associated inverse problem, followed by an overview of time-harmonic wave propagation in viscoelastic media, the role of complex-valued moduli, and the numerical challenges posed by the near-incompressibility of brain tissue and by the need to account for presence of the cerebrospinal fluid (CSF). The main part of the talk presents a Variational Multiscale (VMS) finite element formulation for the forward problem. Brain tissue is described as a nearly incompressible viscoelastic solid and CSF as an incompressible Stokes fluid, leading to a coupled fluid– solid problem of saddle-point type in the frequency domain. The VMS stabilization circumvents the inf- sup condition, allowing equal-order linear interpolation of the primary variables and pressure, and thus efficient simulations on three-dimensional geometries. The formulation is verified using manufactured solutions and applied to an anatomically realistic brain geometry, where it reveals how the presence of CSF alters the displacement fields measured in MRE. The talk concludes with a discussion of the role of accurate forward models in generating synthetic data and supporting the development of MRE inversion methods.

Abstract: Metric (and discrete) Poincaré inequalities are estimates for the variation of functions from the discrete interval $\{1,\dots,n\}$ into a metric space $\mathcal{M}=(M,d)$. These estimates involve “discrete derivatives” on regular graphs (usually expanders). They have numerous applications in metric geometry (and, in particular, in the theory of metric embeddings), geometric group theory, nonlinear geometric functional analysis, as well as theoretical computer science.

We will explain the significance of these inequalities and their applications, and discuss some open problems. Several of the results presented are part of joint work with D. J. Altschuler, K. Tyros, and K. Tikhomirov.

Physics-Informed, Physics-Free, and Numerical Analysis-Informed Machine Learning for Complex/Multiscale Dynamical Systems

Abstract: Physics-Informed Neural Networks (PINNs) and DeepONets have opened new directions in scientific machine learning, particularly for dynamical systems governed by partial differential equations (PDEs). Yet these methods rely on high-dimensional training problems that can be NP-hard, making them especially costly for high-dimensional multiscale and complex systems.

In this talk, I will present recent work on the data-driven modeling and efficient numerical analysis of emergent spatiotemporal dynamics. Drawing on numerical analysis, nonlinear manifold learning, random-feature embeddings, machine learning, and the Equation-Free multiscale methodology, I will discuss four approaches: (i) physics-free and physics-informed methods for learning invariant manifolds from data and constructing reduced-order models, a fundamental task in studying complex systems; (ii) data driven bifurcation and stability analysis using local neural operators from sparse spatio-temporal data, including the detection of critical points that mark the onset of irreversible shifts; (iii) a PDE-free framework that learns solution operators directly from data, bypassing the computationally demanding task to learn explicit governing PDEs with neural operators; and (iv) Fredholm Neural Networks, a numerical-analysis-informed approach to designing and training deep networks for forward and inverse PDE problems highlighting how numerical-analysis principles can guide (deep) network architecture, and training.