Workshop
Mathematical Perspectives in Scientific Modeling
Rome, May 28-30, 2025
The workshop aims to promote interaction and collaboration between different areas of Mathematics by presenting the primary role of pure and applied mathematics in scientific modeling. It is intended to address a wide audience and it is framed in the activities of the Excellence Project 2023-2027 of the Department of Mathematics of Rome Tor Vergata.
Machine-learning enhanced (polytopal) Finite Element methods
Paola Francesca Antonietti (Polytechnic University of Milan, Italy)
In this talk, we discuss the integration Machine Learning (ML) techniques into (polytopal) Finite Element Methods to enhance their accuracy and flexibility while addressing the complexities encountered in practical applications such as computational neuroscience and geoscience. In the first part of the talk, we present innovative ML-driven mesh agglomeration strategies in polytopal Finite Element Methods. We introduce a novel algorithm based on Graph Neural Networks to process the connectivity graph of mesh elements and the physical properties of the model under analysis simultaneously, thereby agglomerating mesh elements and ensuring the generation of high-quality coarse grids. The resulting agglomerated meshes can be employed to reduce the computational burden while maintaining a detailed representation of the geometry and constructing efficient grid hierarchies for geometric Multigrid methods, demonstrating significant improvements in computational efficiency. In the second part of the talk, we present a novel method based on deep learning to accelerate the convergence of Algebraic Multigrid (AMG) techniques. Specifically, ANNs are trained to predict the optimal strong connection parameter that governs the sequence of coarsened matrix problems within the AMG algorithm, thereby effectively reducing the time to a solution. We examine diverse differential problems and discretisation schemes to validate the proposed methodologies, including Virtual Element Methods (VEMs) and polytopal Discontinuous Galerkin (PolyDG) methods. This highlights the potential of ML to advance numerical simulations in complex physical domains.
Canonical metrics on complex manifolds
Claudio Arezzo (ICTP Trieste, Italy)
I will give a general overview on existence of new constant (Ricci or Scalar) curvature metrics on compact and non-compact Kahler manifolds. I will discuss in particular the existence and geometric properties of some of these metrics in the cases of compact manifolds or pseudo-convex domains minus points or divisors with various asymptotics at infinity. Various ongoing projects and open problems will be discussed.
Optimal dimensionality reduction
Albert Cohen (Sorbonne University, France)
Understanding how to optimally approximate general compact sets by finite dimensional spaces is of central interest for designing efficient numerical methods in forward simulation or inverse problems. While the concept of n-width, introduced in 1936 by Kolmogorov, is well taylored to linear methods, finding the correct analogous concept for nonlinear approximation (which typically occurs when using adaptive methods or neural networks) is still the object of current research. In this talk, we shall discuss a general framework that allows to embrace various concepts of linear and nonlinear widths, present some recent results and relevant open problems.
μ-ellipticity and nonautonomous integrals
Cristiana De Filippis (University of Parma, Italy)
μ-ellipticity is a degenerate form of ellipticity, typical of fundamental integrals in geometric analysis or nonlinear elasticity. The regularity theory for autonomous models, e.g., the area functional, is classical after Bombieri & De Giorgi & Miranda (ARMA '69), while the presence of external ingredients may lead to disturbing anomalies, cf. Ladyzhenskaya & Ural'tseva (CPAM '70) and Giaquinta & Modica & Souček (Comm. Univ. Carolinae '79). I will discuss recent advances on nonautonomous, μ-elliptic problems focusing on sharp results and borderline configurations for the validity of Schauder theory. From recent, joint work with Filomena De Filippis (Parma), Giuseppe Mingione (Parma) and Mirco Piccinini (Pisa).
Complex dynamics and elliptic curves
Laura DeMarco (Harvard University, USA)
In holomorphic dynamical systems, one studies maps on the Riemann sphere (or other complex manifolds) with focus on their Julia sets and invariant measures. From this point of view, the Lattès maps — those that are quotients of maps on elliptic curves — are rather uninteresting; their dynamical features are well understood. But viewed algebraically, there are still many unanswered questions. I'll begin the talk with some history of these maps. Then I'll describe a recent question about the geometry of torsion points on elliptic curves and how it has led to interesting complex-dynamical questions about other families of maps and, in turn, new perspectives on the arithmetic side.
Wassertein Sobolev functions and their numerical approximations
Massimo Fornasier (Technical University of Munich, Germany)
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:
- Solving a finite number of optimal transport problems and computing the corresponding Wasserstein potentials.
- Employing empirical risk minimization with Tikhonov regularization in Wasserstein Sobolev spaces.
- Addressing the problem through the saddle point formulation that characterizes the weak form of the Tikhonov functional's Euler-Lagrange equation.
Modelling families of complex curves and minimal surfaces
Franc Forstnerič (University of Ljubljana, Slovenia)
A complex manifold is called an Oka manifold if every holomorphic map from a convex set in a Euclidean space to the manifold is a limit of entire maps. We show that every Oka manifold admits families of noncompact complex curves of a given topological type, endowed with a variable but prescribed family of conformal structures, with approximation on compact Runge subsets. A similar result holds for families of minimal surfaces in Euclidean spaces.
Necessary mechanisms for super AI and stopping hallucinations: The consistent reasoning paradox and the ‘I don't know’ function
Anders C. Hansen (University of Cambridge, UK)
Creating Artificial Super Intelligence (ASI) (AI that surpasses human intelligence) is the ultimate challenge in AI research. This is, as we show, fundamentally linked to the problem of avoiding hallucinations (wrong, yet plausible answers) in AI. We discover a key mechanism that must be present in any ASI. This mechanism is not present in any modern chatbot and we establish that without it, ASI will never be achievable. Moreover, we reveal that AI missing this mechanism will always hallucinate. The mechanism we introduce is the computation of what we call an ‘I don't know’ function. An ‘I don't know’ function determines when an AI is correct and when it will not be able to answer with 100% confidence. The root to these findings is the Consistent Reasoning Paradox (CRP) that we discover, which is a paradox in logical reasoning. The CRP shows that the above mechanism must be present, and that — surprisingly — the weaker concept of being ‘almost sure’ is impossible. In particular, an ASI cannot be ‘almost sure’ (say 90% sure). It will compute an ‘I don't know’ function and either be correct with 100% confidence, or it will not be more than 50% sure. The CRP addresses a long-standing issue that stems from Turing's famous statement that infallible AI cannot be intelligent, where he questions how much intelligence may be displayed if an AI makes no pretence at infallibility. The CRP reveals the answer — consistent reasoning requires fallibility — and thus marks a necessary fundamental shift in AI design if ASI is to ever be achieved and hallucinations to be stopped.
Infinite-dimensional Bayesian inference for time evolution PDEs
Richard Nickl (University of Cambridge, UK)
We will discuss recent progress in our understanding of Gaussian process based inference methods for parameters or states of time evolution phenomena modelled by non-linear partial differential equations (PDEs) such as Navier Stokes, McKean Vlasov, and reaction diffusion systems. We will show that posteriors can deliver consistent solutions in the ‘informative’ large data/small noise limit, discuss probabilistic approximations to the fluctuations of such posterior measures in infinite dimensions, and how such results can be used to show that the non-convex problem of computation of the associated ‘filtering’ distributions are polynomial time problems.
Hodge symmetries of singular varieties
Mihnea Popa (Harvard University, USA)
The Hodge diamond of a smooth projective complex variety contains essential topological and analytic information, including fundamental symmetries provided by Poincaré and Serre duality. I will describe recent progress on understanding how much symmetry there is in the analogous Hodge-Du Bois diamond of a singular variety, and the concrete ways in which this symmetry reflects the singularity types. In the process, we will see how invariants from commutative algebra and higher dimensional geometry influence the topology of an algebraic variety, for instance by means of new weak Lefschetz theorems.
Unbalanced Optimal Transport and the Hellinger-Kantorovich metric
Giuseppe Savaré (Bocconi University, Italy)
Unbalanced Optimal Transport extends classical Optimal Transport theory to account for measures with varying total mass. The Hellinger-Kantorovich (HK) metric provides a natural distance in this setting: it emerges from an optimal entropy-transport problem, where the strict mass preservation constraint is relaxed, allowing for applications in many applied fields. The HK distance can be interpreted as an optimal transport problem on a cone space, leading to a rich geometric structure and well-defined geodesics. Its dual formulation provides insights into regularity properties and solvability conditions, with implications for gradient flows and geodesic convexity. The talk will give a brief overview of the topic and recent progress, which stems from joint work with Matthias Liero, Alexander Mielke and Giacomo Sodini.
Partially hyperbolic dynamics
Marcelo Viana (IMPA Rio de Janeiro, Brazil)
The concept of a hyperbolic dynamical system was introduced by S. Smale, D. Anosov and Ya. Sinai in the 1960's. One goal was to characterize (structurally) stable systems, that is, whose qualitative behavior is not affected by small perturbations of the system. That was eventually achieved in a tour de force by R. Mañé and others. Another goal was to provide a paradigm for the behavior of ‘most’ dynamical systems. This turned out to be too optimistic and, in response, various generalizations of hyperbolicity have been proposed. Among them, the notion of partial hyperbolicity proved to be particularly fruitful, and has been at the heart of the research in the field for the last 30 years or so. I will briefly discuss some of the surprising features exhibited by partially hyperbolic systems, especially concerning rigidity, Lyapunov exponents and pathological foliations.
Control and Machine Learning
Enrique Zuazua (University of Erlangen-Nuremberg, Germany)
Systems control, or cybernetics — a term first coined by Ampère and later popularized by Norbert Wiener — refers to the science of control and communication in animals and machines. The pursuit of this field dates back to antiquity, driven by the desire to create machines that autonomously perform human tasks, thereby enhancing freedom and efficiency. The objectives of control systems closely parallel those of modern Artificial Intelligence (AI), illustrating both the profound unity within Mathematics and its extraordinary capacity to describe natural phenomena and drive technological innovation. In this lecture, we will explore the connections between these mathematical disciplines and their broader implications. We will also discuss our recent work addressing two fundamental questions: Why does Machine Learning perform so effectively? And how can data-driven insights be integrated into the classical applied mathematics framework, particularly in the context of Partial Differential Equations (PDE) and numerical methods? This effort is leading us to a new emerging field of PDE+D(ata) in parallel to the development of new Digital Twins technologies.