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.

Stabilized Finite Element Method for Coupled Fluid–Solid Wave Propagation in Brain MR

Abstract: Magnetic Resonance Elastography (MRE) is a non-invasive imaging modality for characterizing the mechanical properties of soft tissues by measuring their response to low-frequency mechanical excitation. In brain MRE, the viscoelastic behaviour of brain tissue and its interaction with cerebrospinal fluid (CSF) affect the propagation of mechanical waves and, consequently, the measured displacement fields. Experimental evidence indicates that CSF motion can affect these fields near fluid–solid interfaces, while its dynamics are often simplified or neglected in numerical models. In this talk, a computational framework for the numerical simulation of wave propagation in brain MRE will be presented. Brain tissue is described as a nearly incompressible viscoelastic solid, while CSF is represented as an incompressible Stokes fluid, leading to a coupled fluid–solid problem formulated in the frequency domain. A Variational Multiscale (VMS) formulation of the Finite Element Method (FEM) is employed to stabilize equal-order finite element approximations and enable efficient simulations on three-dimensional geometries. The numerical framework is examined through verification problems and subsequently applied to an anatomically realistic brain geometry to investigate the influence of CSF on viscoelastic wave propagation. The resulting model provides a computational basis for studying how fluid–solid interactions affect brain displacement fields, as well as an efficient tool for generating accurate synthetic data required for inverse analysis.

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.

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.