Titles and Abstracts

Machine Learning–Enhanced Polytopal Finite Element Methods with Applications to Neurodegenerative Disease Modelling

This talk explores integrating Machine Learning (ML) techniques into polytopal Finite Element Methods for the numerical discretization of Partial Differential Equations to improve accuracy, flexibility, and computational efficiency while addressing the challenges inherent in complex real-world applications, especially in computational neuroscience.

We present several ML-driven enhancement strategies at different stages of the computational pipeline. In the first part of the talk, we introduce novel ML-based algorithms for mesh agglomeration and refinement in polytopal FEMs. Specifically, we propose novel algorithms based on Graph Neural Networks for efficiently constructing high-quality, agglomerated (coarse) grids that preserve geometric accuracy while significantly reducing computational costs. The key feature is the ability to jointly process the mesh’s connectivity graph and the model's underlying physical properties. Such agglomerated grids also serve as the basis for constructing grid hierarchies in geometric multigrid solvers, leading to notable improvements in overall performance. In the second part of the talk, we introduce a novel deep learning framework to accelerate the convergence of Algebraic Multigrid iterative solvers. The proposed approach predicts the optimal strong-connection parameter and selects the optimal smoother, thereby decreasing the overall time to solution.

All the proposed methods are validated across various differential problems and discretization schemes, including Virtual Element Methods and polytopal Discontinuous Galerkin methods.

Finally, we illustrate the practical performance of the proposed approaches within an application in computational neuroscience—specifically, the simulation of neurodegenerative disease dynamics—highlighting advances in numerical modeling for complex physical systems.

A new look at distributional regression: Wassertein Sobolev functions and their numerical approximations

We start the talk by presenting general results of strong density of sub-algebras of bounded Lipschitz functions in metric Sobolev spaces. We apply such results to show the density of smooth cylinder functions in Sobolev spaces of functions on the Wasserstein space P2 endowed with a finite positive Borel measure. As a byproduct, we obtain the infinitesimal Hilbertianity of Wassertein Sobolev spaces.  By taking advantage of these results, we further address the challenging problem of the numerical approximation of Wassertein Sobolev functions defined on probability spaces. Our particular focus centers on the Wasserstein distance function, which serves as a relevant example. In contrast to the existing body of literature focused on approximating efficiently pointwise evaluations, we chart a new course to define functional approximants by adopting three machine learning-based approaches:

1. Solving a finite number of optimal transport problems and computing the corresponding Wasserstein potentials.

2. Employing empirical risk minimization with Tikhonov regularization in Wasserstein Sobolev spaces.

3. Addressing the problem through the saddle point formulation that characterizes the weak form of the Tikhonov functional's Euler-Lagrange equation.

As a theoretical contribution, we furnish explicit and quantitative bounds on generalization errors for each of these solutions. In the proofs, we leverage the theory of metric Sobolev spaces introduced above and we combine it with techniques of optimal transport, variational calculus, and  large deviation bounds. In our numerical implementation, we harness appropriately designed neural networks to serve as basis functions. Consequently, our constructive solutions significantly enhance at equal accuracy the evaluation speed, surpassing that of state-of-the-art methods by several orders of magnitude. The talk presents a collection of results with Pascal Heid, Giuseppe Savaré, and Giacomo Sodini.

The Geometry of Random Neural Networks

We study the geometric properties of random neural networks by investigating the boundary volumes of their excursion sets for different activation functions, as the depth increases. 

More specifically, we show that, for activations which are not very regular (e.g., the Heaviside step function), the boundary volumes exhibit fractal behavior, with their Hausdorff dimension monotonically increasing with the depth. 

On the other hand, for activations which are more regular (e.g., ReLU, logistic and tanh), as the depth increases, the expected boundary volumes can either converge to zero, remain constant or diverge exponentially, depending on a single spectral parameter which can be easily computed. Our theoretical results are confirmed in some numerical experiments based on Monte Carlo simulations.

Modeling with Purpose: Robust and Interpretable AI 

From Clinical Use Cases to Mapping Interdisciplinarity and Research Impact

Modern AI systems, and deep learning in particular, can achieve striking accuracy on many tasks. However, if we care about purposeful deployment in high-stakes domains, accuracy alone is not enough. In this talk I will first outline a compact view of AI, ML and DL, explaining why these methods work so well today and where their fundamental limitations lie: overfitting, domain shift, lack of interpretability, and fragile generalization in small-sample,

high-dimensional regimes that are typical of real-world applications, together with ethical concerns around bias, transparency and accountability.

I will then focus on how mathematical structure can turn generic learning machines into robust and interpretable models. Examples from clinical data — including imaging, electronic health records and biological measurements — will illustrate how regularization, sparsity,

network-based modeling and uncertainty quantification help to stabilize predictions, encode prior knowledge and make outputs more meaningful for domain experts. I will also comment on the role of data visualization in conveying these complex models and their uncertainties in a form that clinicians can actually use.

In the final part, I will show how the same families of methods and visual strategies can be used to model the scientific ecosystem itself. My current work on data visualization and the evaluation of interdisciplinarity and research impact treats publications, collaborations and topics as a high-dimensional, evolving network, where again we must design structured models and visual representations that remain robust, interpretable and honest about their limitations. This parallel helps to frame clinical AI and science-of-science analytics as two instances of a common challenge: making complex, data-driven models understandable and useful to domain experts.

Efficient Certifying Algorithms for Linear Classification

We apply the idea of certifying algorithms in the context of machine learning problems. An efficient certifying algorithm is proposed that, given a set of n points in R^d with binary labels, either returns a hyperplane separating the points, or identifies d+2 of the labeled points that cannot be separated by any hyperplane.  

The existence of such d+2 points in the inseparable case is known to be guaranteed by Kirchberger's theorem in combinatorial geometry; we show how to compute these points efficiently. We then propose a dimension-free and constructive extension of Kirchberger's theorem, where for any e> 0 one finds either a separating hyperplane, or O(1/e^2) of the labeled points that cannot be separated with normalized margin by any hyperplane.

 Our algorithms are based on solving one primal-dual pair of linear programs with d primal and n dual variables, and at most n-d linear equation systems with O(d) equations and O(d) unknowns. 

The Replica Method as a Tool for High-Dimensional Inference and Learning

The replica method, originally developed in the physics of disordered systems, has proved to be a powerful heuristic for analysing random models in high dimensions. This talk will show how it can yield typical-case predictions for two modern problems in inference and learning.

 First, in spiked sparse matrix models, we use the replica formalism to study low-rank signal recovery when the noise matrix has finite connectivity. The method provides analytical expressions for the typical top eigenvalue, eigenvector statistics, and overlap observables in terms of recursive distributional equations, extending the classical Baik–Ben Arous–Péché transition beyond dense settings. 

Second, in high-dimensional classification with heavy-tailed features, we analyse a superstatistical data model with randomly fluctuating variances. The replica computation gives asymptotic generalisation and training errors of empirical risk minimisation with generic convex losses and regularisers, and identifies when Gaussian universality breaks down. 

Together, these examples illustrate how the replica method offers a consistent heuristic framework for understanding phase transitions and asymptotic performance limits in random inference problems, even where rigorous tools are lacking.