Thursday, 30 October 2025
Note: This program includes links that allow direct access to detailed sections of the website (keynotes, posters, Gala dinner). Clicking a link will automatically take you to the relevant section, even if it is located on a different page.
09:15
Philip J. MORRISON
(The University of Texas at Austin, USA)
Auditorium Maupertuis
Geometric Methods in Thermodynamics (Session 1)
Chairman : François GAY-BALMAZ, Hiroaki YOSHIMURA
10:00 - Thermodynamic Functionality of Non-Detailed Balance Finite-Tape Information Ratchet (26) Lock Yue Chew
In this paper, we investigate into the functionality of the finite-tape information ratchet when its thermal transition is non-detailed balance. First, we construct an analytical framework of the information ratchet from stochastic thermodynamics by generalizing over that of [1] with detailed balance broken. This leads to special cases of the information processing first and second law stipulated by Semaan et. al. [2] with the appearance of housekeeping heat. Through the application of Kullback-Leibler divergence as a statistical distance, we observe theoretically the mathematical condition for the finite-tape information ratchet to serve as an heat engine: its cumulative change in entropy should exceed that of the reduction in statistical distance of its initial to stationary state. While this is true for both the equilibrium and nonequilibrium stationary state, the heat extraction from the latter is exacerbated by the flow of housekeeping heat. We demonstrate the validity of our results by a Markov transition model which displays statistical dynamics that is non-detailed balance.
10:20 - Entropy functionals and equilibrium states in mixed quantum-classical dynamics (32) Cesare Tronci
The computational challenges posed by many-particle quantum systems are often overcome by mixed quantum-classical (MQC) models in which certain degrees of freedom are treated as classical while others are retained as quantum. One of the fundamental questions raised by this hybrid picture involves the characterization of the information associated to MQC systems. Based on the theory of dynamical invariants in Hamiltonian systems, here we propose a family of hybrid entropy functionals that consistently specialize to the usual Rényi and Shannon entropies. Upon considering the MQC Ehrenfest model for the dynamics of quantum and classical probabilities, we apply the hybrid Shannon entropy to characterize equilibrium configurations for simple Hamiltonians. The present construction also applies beyond Ehrenfest dynamics.
10:40 - Variational approach to the stochastic thermodynamics of Langevin systems (53) Héctor Vaquero del Pino
In this paper, the first insights into a variational formulation of stochastic thermodynamics is presented for the finite-dimensional case of discrete systems. Following the variational approach in (Gay-Balmaz & Yoshimura 2019), the fundamental variational principle of classical mechanics is systematically extended to include irreversible and stochastic forces. By including thermodynamic entropy as an independent state variable in phase space, the conditions for thermodynamic consistency are derived extending the tools of stochastic thermodynamics, providing the fluctuation-dissipation relation. The method is illustrated with mechanical dissipative forces.
11:00 - Variational Principle for Stochastic Nonholonomic Systems Part I: Continuous-Time Formulation (57) Tianzhi Li
Nonholonomic mechanics has received considerable attention in dynamics and control area. However, due to a wide range of fluctuations in the physical world, the ideal mathematical models of mechanical systems with nonholonomic constraints suffer from issues of ignoring the real-world perturbations and physically difficult to realize. Motivated by recent developments in stochastic and constrained mechanics, here we present a stochastic variational formulation for mechanical systems with or without stochastic nonholonomic constraints. We give stochastic variational principles for both stochastically unconstrained and nonholonomic cases under the same framework by deriving the stochastic implicit Hamel equations. Moreover, an interesting example of the stochastic rolling disk is provided to illustrate the proposed method.
Room Vauban 1
Stochastic Geometric Dynamics
Chairman : Ana Bela CRUZEIRO, Jean-Claude ZAMBRINI, Stefania UGOLINI
10:00 - Stochastic perturbation of geodesics on the manifold of Riemannian metrics (81) Ali Suri
In this paper, using diffusion processes with values in the manifold of Riemannian metrics, we compute the evolution equation for the Lagrangian induced by the $L^2$ stochastic kinetic energy functional.
10:20 - Stochastic Maupertuis's principles and Jacobi's integration theorem (79) Qiao Huang
Building on stochastic geometric mechanics on Riemannian manifolds, we shall focus on extensions of the classical Maupertuis’s variational principle to a class of diffusion processes as extremals of a stochastic action functional preserving the expectation of energy. We shall also mention a recent and related stochastic Jacobi integration theorem, whose consequence will be analyzed elsewhere.
10:40 - Lagrangian averaging of singular stochastic actions for fluid dynamics (49) Ruiao Hu
We construct sub-grid scale models of incompressible fluids by considering expectations of semi-martingale Lagrangian particle trajectories. Our construction is based on the Lagrangian decomposition of flow maps into mean and fluctuation parts, and it is separated into the following steps. First, through Magnus expansion, the fluid velocity field is expressed in terms of fluctuation vector fields whose dynamics are assumed to be stochastic. Second, we use Malliavin calculus to give a regularised interpretation of the product of white noise when inserting the stochastic velocity field into the Lagrangian for Euler’s fluid. Lastly, we consider closures of the mean velocity by making stochastic analogues of Taylor’s frozen-in turbulence hypothesis to derive a version of the anisotropic Lagrangian averaged Euler equation.
11:00 - Continuous-time filtering in Lie groups: estimation via the Fréchet mean of solutions to stochastic differential equations (95) Magalie Bénéfice
We compute the Fréchet mean $\SE_t$ of the solution $X_t$ to a continuous-time stochastic differential equation in a Lie group. It provides an estimator with minimal variance of $X_t$. We use it in the context of Kalman filtering and more precisely to infer rotation matrices. In this paper, we focus on the prediction step between two consecutive observations. Compared to state-of-the-art approaches, our assumptions on the model are minimal.
Room Vauban 2
Optimization and learning on manifolds (Session 1)
Chairman : Cyrus MOSTAJERAN, Salem SAID
10:00 - Geometric design of the tangent term in landing algorithms for orthogonality constraints (101) Florentin Goyens
We propose a family a metrics over the set of full-rank $n\times p$ real matrices, and apply them to the landing framework for optimization under orthogonality constraints. The family of metrics we propose is a natural extension of the $\beta$-metric, defined on the Stiefel manifold.
10:20 - Role of Riemannian geometry in double-bracket quantum imaginary-time evolution (50) Marek Gluza
Double-bracket quantum imaginary-time evolution (DB-QITE) is a quantum algorithm which coherently implements steps in the Riemannian steepest-descent direction for the energy cost function. DB-QITE is derived from Brockett’s double-bracket flow which exhibits saddle points where gradients vanish. In this work, we perform numerical simulations of DB-QITE and describe signatures of transitioning through the vicinity of such saddle points. We provide an explicit gate count analysis using quantum compilation programmed in Qrisp.
10:40 - Numerical techniques for geodesic approximation in Riemannian shape optimization (55) Kathrin Welker
Shape optimization is commonly applied in engineering to optimize shapes with respect to an objective functional relying on PDE solutions. In this paper, we view shape optimization as optimization on Riemannian shape manifolds. We consider so-called outer metrics on the diffeomorphism group to solve PDE-constrained shape optimization problems efficiently. Commonly, the numerical solution of such problems relies on the Riemannian version of the steepest descent method. One key
difference between this version and the standard method is that iterates are updated via geodesics or retractions. Due to the lack of explicit expressions for geodesics, for most of the previously proposed metrics, very limited progress has been made in this direction. Leveraging the existence of explicit expressions for the geodesic equations associated to the outer metrics on the diffeomorphism group, we aim to study the viability of using such equations in the context of PDE-constrained shape optimization. However, solving geodesic equations is computationally challenging and often restrictive. Therefore, this paper discusses potential numerical approaches to simplify the numerical burden of using geodesics, making the proposed method computationally competitive with previously established methods.
11:00 - Geometric Gaussian Approximations of Probability Distributions (93) Nathaël Da Costa
Approximating complex probability distributions, such as Bayesian posterior distributions, is of central interest in many applications. We study the expressivity of geometric Gaussian approximations. These consist of approximations by Gaussian pushforwards through diffeomorphisms or Riemannian exponential maps. We first review these two different kinds of geometric Gaussian approximations. Then we explore their relationship to one another. We further provide a constructive proof that such geometric Gaussian approximations are universal, in that they can capture any probability distribution. Finally, we discuss whether, given a family of probability distributions, a common diffeomorphism can be found to obtain uniformly high-quality geometric Gaussian approximations for that family.
Auditorium Maupertuis
Geometric Methods in Thermodynamics (Session 2)
Chairman : François GAY-BALMAZ, Hiroaki YOSHIMURA
11:50 - Variational Principle for Stochastic Nonholonomic Systems Part II: Stochastic Nonholonomic Integrator (83) Tianzhi Li
Nonholonomic integrators are a class of geometric numerical integration schemes that are designed to simulate mechanical systems with nonholonomic constraints. To the best of our knowledge, so far there have been no variational integrators designed for stochastic systems with noisy nonholonomic constraints, which are extensively studied in robotics and control area. Based on the stochastic nonholonomic variational formulation introduced in Part I, we present a stochastic integrator for both stochastically unconstrained and stochastically nonholonomic systems under the same framework. The numerical integration scheme is obtained by deriving a discrete counterpart of the stochastic variational principle discussed in Part I.
12:10 - Interconnection and variational principles for fluid-bubble dynamics (78) François Gay-Balmaz
We consider the dynamics of a barotropic fluid interacting with a bubble filled with uniform gas from the perspective of system interconnection in Lagrangian mechanics. Extending the existing geometric framework to an infinite-dimensional setting requires careful consideration of the appropriate duality pairing underlying the relationship between interaction forces and distribution constraints. We address both inviscid and viscous cases, including surface tension, and consider both free-slip and no-slip interface conditions. This work represents a first step toward building the geometric foundations for Rayleigh-Plesset equation and its related models.
12:30 - Hamilton-Dirac formulation for thermodynamic systems with reaction and diffusion (84) Hiroaki Yoshimura
In this paper, we propose a Hamilton-Dirac formulation for non-simple nonequilibrium thermodynamic systems with chemical reactions and diffusion. A key feature of these systems is the degeneracy of their associated Lagrangian function. To address this, we build upon Dirac’s theory of constraints for degenerate Lagrangians and develop a Hamiltonian variational formulation for nonholonomic systems with nonlinear thermodynamic constraints, as well as primary constraints arising from the degeneracy. We introduce the constrained Hamiltonian on the primary constraint and clarify the underlying geometric structure using Dirac structures. Finally, we illustrate our Hamilton-Dirac formulation with an example of a membrane undergoing matter reaction and diffusion.
12:50 - On a generalisation of metriplectic systems (92) Jonas Kirchhoff
This note presents a notion of generalised metriplectic systems which includes not necessarily holonomic constraints using methods from port-Hamiltonian systems theory. Metriplectic systems can be written as particular dissipative Hamiltonian systems with the exergy as Hamiltonian. Within the generalised dissipative Hamiltonian systems, a generalisation of metriplectic systems is identified.
Room Vauban 1
Classical & Quantum Information, Geometry and Topology
Chairman : Florio M. CIAGLIA, FABIO DI COSMO, Pierre BAUDOT and Grégoire SERGEANT-PERTHUIS
11:50 - The category of non-commutative probabilities in Information Geometry (105) Laura Gonzalez Bravo
The category of non-commutative probabilities (NCP) is introduced to provide a categorical framework for classical and quantum information geometry. This framework enables the classification of field of covariances, as functors from NCP to Hilb, the category of Hilbert spaces and contractions, a description which is suited to both finite and infinite dimensional settings. Additionally, NCP sets an environment to
provide a detailed description of statistical models.
12:10 - Independent States are Orthogonal: a Categorical Framework to Treat Probability Geometrically (118) Paolo Perrone
Dagger categories (a.k.a. *-categories) can be seen as categories with a notion of « transpose », generalizing the transposition of matrices in linear algebra. This allows us to extend the ideas of orthogonality and orthogonal projector from Euclidean geometry and Hilbert space theory to a much more general and abstract context.
By means of a dagger category of probability spaces and transport plans, we show that this abstract notion of orthogonality can model exactly independence and conditional independence of random variables. Moreover, orthogonal projectors correspond exactly to conditioning, giving a unified description of « observations » for both quantum and classical experiments.
12:30 - Two-typed Tangent Vectors in Quantum Statistical Mechanics (31) Jan Naudts
The tangent space at a KMS state (Kubo Martin Schwinger state) can be decomposed into two subspaces in such a way that the time evolution, which is described by the modular automorphism group, satisfies the KMS condition w.r.t. each of these two subspaces. In particular, this implies that these subspaces are invariant for the time evolution and that they determine a discrete conserved quantity. The tangent space is said to be two-typed because each tangent vector is the sum of two vectors, one in each subspace. The pair of subspaces is non-unique. It is determined by the choice of an orthonormal basis diagonalizing the modular operator.
The paper is restricted to the finite-dimensional case. In this way the technicalities of handling unbounded operators are not exposed.
12:50 - A Historical Perspective on the Schützenberger-van~Trees Inequality: A Posterior Uncertainty Principle (109) Olivier Rioul
The Bayesian Cramér-Rao Bound (BCRB) is generally attributed to Van~Trees who published it in 1968. According to Stigler’s law of eponymy, no scientific discovery is named after its first discoverer. This is the case not only for the Cramér-Rao bound itself—due in particular to the French mathematicians Fréchet and Darmois—but also for the van Trees inequality: The French physician, geneticist, epidemiologist and mathematician Marcel-Paul (Marco) Schützenberger, in a paper of just fifteen lines written in 1956—more than a decade before van Trees—had not only demonstrated the BCRB but, as a close examination of his proof shows, used a very original approach based on the Weyl-Heisenberg uncertainty principle on the posterior distribution. This work reviews and extends Schützenberger’s approach to Fisher information matrices, which opens up new perspectives.
13:10 - Tree inference with varifold distances (136) Elodie Maignant
In this paper, we consider a tree inference problem motivated by the critical problem in single-cell genomics of reconstructing dynamic cellular processes from sequencing data. In particular, given a population of cells sampled from such a process, we are interested in the problem of ordering the cells according to their progression in the process. This is known as trajectory inference. If the process is differentiation, this amounts to reconstructing the corresponding differentiation tree. One way of doing this in practice is to estimate the shortest-path distance between nodes based on cell similarities observed in sequencing data. Recent sequencing techniques make it possible to measure two types of data: gene expression levels, and RNA velocity, a vector that predicts changes in gene expression. The data then consist of a discrete vector field on a Euclidean space of dimension equal to the number of genes under consideration. By integrating this velocity field, we recover for each single cell its trajectory from some initial stage to its current stage. Eventually, we assume that we have a faithful embedding of the tree in a Euclidean space, but which we only observe through the curves connecting the root to the nodes. Using varifold distances between such curves, we define a similarity measure between nodes which we prove approximates the shortest-path distance in a tree that is isomorphic to the target tree.
Room Vauban 2
Optimization and learning on manifolds (Session 2)
Chairman : Cyrus MOSTAJERAN, Salem SAID
11:50 - A probabilistic view on Riemannian machine learning models for SPD matrices (9) Thibault de Surrel
The goal of this paper is to show how different machine learning tools on the Riemannian manifold $\P_d$ of Symmetric Positive Definite (SPD) matrices can be united under a probabilistic framework. For this, we will need several Gaussian distributions that have been defined in $\P_d$. We will show how popular classifiers on $\P_d$ can be reinterpreted as Bayes Classifiers using these Gaussian distributions. These distributions will also be used for outlier detection and dimension reduction. By showing that those distributions are pervasive in the tools used on $\P_d$, we allow for other machine learning tools to be extended to $\P_d$.
12:10 - Efficiency of the Generalized Method of Moments from the Viewpoint of Differential Geometry (14) Hisatoshi Tanaka
The generalized method of moments (GMM) attains the semiparametric efficiency bound when the optimal weight matrix is chosen. In this study, we characterize the efficient choice of the weight matrix from the viewpoint of differential geometry. We induce a metric for the GMM manifold from a linear space in which the model set is embedded. Simultaneously, using the asymptotic normality of the GMM estimators, we define another metric of the manifold. In conclusion, we prove that the two metrics coincide when an optimal weight matrix is employed.
12:30 - p-Laplacians for Manifold-valued Hypergraphs (110) Jo Stokke
Hypergraphs extend traditional graphs by enabling the representation of N-ary relationships through higher-order edges. Akin to a common approach of deriving graph Laplacians, we define function spaces and corresponding symmetric products on the nodes and edges to derive hypergraph Laplacians. While this has been done before for Euclidean features, this work generalizes previous hypergraph Laplacian approaches to accommodate manifold-valued hypergraphs for many commonly encountered manifolds.
12:50 - Universal kernels via harmonic analysis on Riemannian symmetric spaces (59) Cyrus Mostajeran
The universality properties of kernels characterize the class of functions that can be approximated in the associated reproducing kernel Hilbert space and are of fundamental importance in the theoretical underpinning of kernel methods in machine learning. In this work, we establish fundamental tools for investigating universality properties of kernels in Riemannian symmetric spaces, thereby extending the study of this important topic to kernels in non-Euclidean domains. Moreover, we use the developed tools to prove the universality of several recent examples from the literature on positive definite kernels defined on Riemannian symmetric spaces, thus providing theoretical justification for their use in applications involving manifold-valued data.
15:00
Auditorium Maupertuis
Information Geometry, Delzant Toric Manifold & Integrable System
Chairman : Mathieu MOLITOR, Hajime FUJITA, Daisuke TARAMA and Frédéric BARBARESCO
16:30 - Adler-Kostant-Symes Theorem and Algebraic Complete Integrability of Information Geometry and Souriau Lie Groups Thermodynamics (19) Frederic Barbaresco
Y. Nakamura established that gradient systems defined on specific statistical manifolds, such as those associated with Gaussian and multinomial distributions, satisfy the conditions of Liouville complete integrability. Furthermore, he demonstrated that gradient flows on statistical manifolds may be linearised through the application of dual coordinates from information geometry, in a manner analogous to the action-angle coordinates employed in Hamiltonian mechanics to characterise integrable systems. We extend this line of inquiry to Souriau’s symplectic model of Information Geometry for Lie groups. Subsequently, we examine algebraic complete integrability in the sense of Adler and van Moerbeke, as well as the symplectic structure underlying Lax pairs, which can be formulated in terms of algebraic-geometric structures. This approach underlines the interplay between the analytical and group-theoretical methodologies in the study of integrable systems. Within this framework, we study the Adler-Kostant-Symes theorem as a principal tool for the construction of integrable systems, leveraging its capacity to establish algebraic integrability.
16:50 - Statistical transformation models of multivariate normal distributions and their α-geodesic flows (75) Daisuke Tarama
This paper deals with the geodesic flows of the $\alpha$-connections arising from the statistical transformation model for the multivariate normal distributions on $\mathbb{R}^d$.
The probability density functions are parameterized by the semi-direct product Lie group $GL_+\left(d,\mathbb{R}\right)\ltimes \mathbb{R}^d$.
The Fisher-Rao semi-definite metric and the Amari-Chentsov cubic tensor are left-invariant tensors on the Lie group.
One can then describe the geodesic flows of the $\alpha$-connections as left-invariant system.
It is interesting that the geodesic flow of the Fisher-Rao semi-definite metric can be formulated in terms of the subriemannian geometry.
In fact, the Fisher-Rao geodesic flow is a subriemannian geodesic flow for a step-two left-invariant subriemannian structure on the semi-direct product Lie group.
In this paper, an explicit formula of the Amari-Chentsov cubic tensor is obtained and consequently the equation for the $\alpha$-geodesic flows is found concretely.
As preliminaries of the current paper, the general framework of information geometry is briefly reviewed in relation to the Fisher-Rao metric and the Amari-Chentsov cubic tensor, particularly in the case of statistical transformation models.
17:10 - Geodesic flow of a statistical manifold associated to Souriau's thermodynamics (86) Jérémie Pierard de Maujouy
We investigate the relation between Souriau’s Lie group thermodynamics and statistical transformation models.
Souriau associated with a statistical mechanical system a Gibbs set that described the states of thermodynamical equilibrium. These Gibbs sets can be understood as statistical transformation models constructed through a general procedure that involves a representation of the symmetry group. As such, they have a Fisher-Rao metric which is consistent with the usual construction for group-parametrized statistical transformation models.
As an example, we consider the case of the Fisher distributions on the 2-sphere, viewed as a Hamiltonian SO(3)-manifold. We compute the Fisher-Rao metric, which is invariant under rotations. The geodesic flow on so(3) associated with the Fisher distributions on the 2-sphere is integrable and provides a first example of dynamical systems on Gibbs sets.
17:30 - Moment polytopes of toric exponential families (40) Mathieu Molitor
We show that the moment polytope of a Kähler toric mani-fold, constructed as the torification (in the sense of M. Molitor, « Kähler toric manifolds from dually flat spaces », arXiv:2109.04839) of an exponential family defined on a finite sample space, is the projection of a higher-dimensional simplex.
17:50 - A compactification of the orthogonal foliation via toric geometry (58) Hajime Fujita
We provide a compactification of the orthogonal foliation for the dually flat structure on the probability simplex. In particular we examine the orthogonality of the $e$-foliation and $m$-foliation on the boundary. We use a toric geometric aspect of the probability simplex.
18:10 - Torsion of α-connections on the density manifold (35) Lorenz Schwachhöfer
We study the torsion of the $\alpha$-connections defined on the density manifold in terms of a regular Riemannian metric.
In the case of the Fisher-Rao metric our results confirm the fact that all $\alpha$-connections are torsion free. For the $\alpha$-connections obtained by the
Otto metric, we show that, except for $\alpha = -1$, they are not torsion free.
Room Vauban 1
Geometric Green Learning on Groups and Quotient Spaces
Chairman : Alice Barbara TUMPACH, Diarra FALL and Levin MAIER
16:30 - Enhancing CNNs robustness to occlusions with bioinspired filters for border completion (56) Rita Fioresi
We exploit the mathematical modeling of the visual cortex mechanism for border completion to define custom filters for CNNs. We see a consistent improvement in performance, particularly in accuracy, when our modified LeNet 5 is tested with occluded MNIST images.
16:50 - A new model for natural groupings in high-dimensional data (71) Mireille Boutin
Clustering aims to divide a set of points into groups. The current paradigm assumes that the grouping is well-defined (unique) given the probability model from which the data is drawn.
Yet, recent experiments have uncovered several high-dimensional datasets that form different binary groupings after projecting the data to randomly chosen one-dimensional subspaces. This paper describes a probability model for the data that could explain this phenomenon. It is a simple model to serve as a proof of concept for understanding the geometry of high-dimensional data. Our construction makes it clear that one needs to make a distinction between « groupings » and « clusters » in the original space. It also highlights the need to interpret any clustering found in projected data as merely one among potentially many other groupings in a dataset.
17:10 - Information Geometry on the ℓ²-Simplex via the q-Root Transform (85) Levin Maier
In this paper, we introduce \emph{$\ell^p$-information geometry}, an infinite dimensional framework that shares key features with the geometry of the space of probability densities \( \mathrm{Dens}(M) \) on a closed manifold, while also incorporating aspects of measure-valued information geometry. We define the \emph{$\ell^2$-probability simplex} with a noncanonical differentiable structure induced via the \emph{$q$-root transform} from an open subset of the $\ell^p$-sphere. This structure renders the $q$-root map an \emph{isometry}, enabling the definition of \emph{Amari–Čencov $\alpha$-connections} in this setting.
We further construct \emph{gradient flows} with respect to the $\ell^2$ Fisher–Rao metric, which solve an infinite-dimensional linear optimization problem. These flows are intimately linked to an \emph{integrable Hamiltonian system} via a \emph{momentum map} arising from a Hamiltonian group action on the infinite-dimensional complex projective space.
\keywords{infinite-dimensional information geometry \and $\ell^p$-information geometry \and Amari–Čencov $\alpha$-connections \and integrable Hamiltonian systems \and infinite-dimensional linear programming}
17:30 - K-P Quantum Neural Networks (87) Elija Perrier
We present an extension of K–P time-optimal quantum control
solutions using global Cartan $KAK$ decompositions for geodesic-based solutions. Extending recent time-optimal \emph{constant–$\theta$} control results, we integrate Cartan methods into equivariant quantum neural network (EQNN) for quantum control tasks. We show that a finite-depth limited EQNN ansatz equipped with Cartan layers can
replicate the constant–$\theta$ sub-Riemannian geodesics for K–P problems. We demonstrate how for certain classes of control problem on Riemannian symmetric spaces, gradient-based training using an appropriate cost
function converges to certain global time-optimal solutions when satisfying simple regularity conditions. This generalises prior geometric control theory methods
and clarifies how optimal geodesic estimation can be performed in quantum
machine learning contexts.
\keywords{Quantum control \and K–P problem \and Equivariant QNN \and Cartan decomposition \and
Optimal geodesics \and Sub-Riemannian geometry \and Machine learning.
17:50 - Infinite-dimensional Siegel disc as symplectic and Kähler quotient (106) Alice Barbara Tumpach
In this paper, we construct the restricted infinite-dimensional Siegel disc as a Marsden-Weinstein symplectic reduced space and as Kähler quotient of a weak Kähler manifold. The obtained symplectic form is invariant with respect to the left action of the infinite-dimensional restricted symplectic group and coincides with the Kirillov-Kostant-Souriau symplectic form of the restricted Siegel disc obtained via the identification with an affine coadjoint orbit of the restricted symplectic group, or equivalently with a coadjoint orbit of the universal central extension of the restricted symplectic group.
18:10 - The hyperkähler marriage between the sphere and the hyperbolic space (107) Alice Barbara Tumpach
We review the construction of the hyperkähler metric on the complexification of the projective space which extends the Kähler metric of the 2-sphere. The hyperbolic space sits in this complexification. In this paper, we are interested in the complex structure inherited on the hyperbolic space by the hyperkähler extension of the 2-sphere. Contrary to what is generally believed, we show that it differs from the natural complex structure of the hyperbolic disc inherited from its embedding in C.
Room Vauban 2
Applied Geometry-Informed Machine Learning (Session 1)
Chairman : Pierre-Yves LAGRAVE, Santiago VALASCO-FORERO and Teodora PETRISOR
16:30 - Riemannian Integrated Gradients: A Geometric View of Explainable AI (6) Lachlan Simpson
We introduce Riemannian Integrated Gradients (RIG); an extension of Integrated Gradients (IG) to Riemannian manifolds. We demonstrate that RIG restricts to IG when the Riemannian manifold is Euclidean space. We show that feature attribution can be phrased as an eigenvalue problem where attributions correspond to eigenvalues of a symmetric endomorphism.
16:50 - A Geometric Deep Learning Approach to Forecast the Time Series of Covariance Matrices (10) Michele Palma
The forecasting approaches of time-varying covariance matrices often overlook the geometric properties of symmetric positive definite matrices, ignoring the fact that these are points on a Riemannian manifold. This may lead to suboptimal forecast accuracy and might result in overparameterized model, making it infeasible to work with high-dimensional matrices. This paper introduces an innovative approach to forecasting time series of covariance matrices using a deep learning method grounded in Riemannian optimization. In an application with simulated data, we show that when geometric properties of the predicted object are taken into account, the prediction accuracy significantly improves.
17:10 - Learning Riemannian Metrics for Interpolating Animations (25) Sarah Kushner
We leverage a family of Riemannian metrics to upsample low frame rate animations for creative design and compression applications in computer graphics. Our method interpolates animated characters’ bone orientations along various geodesics from a family of invariant Riemannian metrics on a product of SO(3) manifolds. For compression, an optimization step selects the best-fitting metric. We show that our approach outperforms existing techniques.
17:30 - Conditioning Surface Shape Processes with Neural Operators (143) Stefan Sommer
We present a novel method for simulating infinite-dimensional conditional stochastic processes governing surface shape evolution. Given boundary conditions represented as spherical functions, we consider a function-valued diffusion process X with initial state X_0, conditioned on X_T. To address the simulation challenge, we develop a neural operator architecture leveraging spherical harmonic transforms to approximate the intractable drift term arising from Doob’s h-transform. The proposed operator demonstrates discretization equivariance, enabling direct application to spherical meshes at arbitrary resolutions without architectural modifications or retraining. We validate our method on several synthetic shape evoluation scenarios.
17:50 - GNN-Enhanced TCN Algorithms for ECG Signal Quality Recognition (66) Ferdinando Zanchetta
Temporal Convolutional Networks (TCNs) are among the most effective algorithms to deal with time series dataTo a time series can be also given the structure of directed graph, opening the doors to the usage of Graph Neural Networks (GNNs) in this context. In this paper we develop two distinct Geometric Deep Learning models that merge the capabilities of ordinary TCNs with the ones of GNNs, a supervised
classifier and an autoencoder-like model that we apply to solve a quality detection problem on electrocardiogram signals.
18:10 - WAN2DNS-PM: Weak adversarial networks for solving 2D incompressible Navier-Stokes equations in porous media (51) Frederic CADET
The use of neural networks has shown significant potential to reduce the computational costs associated with the dynamics of industrial computational fluids. Weak adversarial networks (WAN) leverage weak solution theory to transform the problem of solving PDEs into a Min-Max optimization problem, which is then solved by training a generative adversarial network. Although this method has been successfully applied to two-dimensional (2D) Navier-Stokes (NS) equations, previous work says nothing about the NS equation in porous media. In this study, we first leverage stream function to introduce the biharmonic formulation of NS in porous media. Then, we extend the WAN framework to solve NS equations in porous media (WAN2DNS-PM) and provide free surface flow as a numerical experiment. Our results demonstrate the stability and accuracy of the proposed method, highlighting its advantages over the traditional Physic-Informed Neural Networks (PINNs) algorithm, particularly for problems lacking strong solutions. This work contributes to the growing research on AI-driven numerical methods for complex fluid dynamics problems, offering a promising approach for industrial applications.