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Using an abstract perturbation result established by Keller and Liverani, we obtain the H\"older continuity of the invariant density and entropy of the physical measures for some families of piecewise expanding maps. We apply these results to a family of two-dimensional tent maps., Comment: 16 pages, 1 figure

The recent rise of deep learning has led to numerous applications, including solving partial differential equations using Physics-Informed Neural Networks. This approach has proven highly effective in several academic cases. However, their lack of physical invariances, coupled with other significant weaknesses, such as an inability to handle complex geometries or their lack of generalization capabilities, make them unable to compete with classical numerical solvers in industrial settings. In this work, a limitation regarding the use of automatic differentiation in the context of physics-informed learning is highlighted. A hybrid approach combining physics-informed graph neural networks with numerical kernels from finite elements is introduced. After studying the theoretical properties of our model, we apply it to complex geometries, in two and three dimensions. Our choices are supported by an ablation study, and we evaluate the generalisation capacity of the proposed approach.

We provide a construction method for biharmonic submanifolds in cohomogeneity one manifolds. In particular, we give new examples of biharmonic submanifolds and study the normal index of these submanifolds. We use this strategy to construct metrics on the sphere admitting biharmonic non-minimal hypersurfaces with three distinct principal curvatures. Finally, we perform a similar study of $r$-harmonic submanifolds of cohomogeneity one manifolds., Comment: 22 pages, no figures. v2: generalized construction method, addressed corrections, modified exposition

We study semiflows generated via impulsive perturbations of Lorenz flows. We prove that such semiflows admit a finite number of physical measures. Moreover, if the impulsive perturbation is small enough, we show that the physical measures of the semiflows are close, in the weak* topology, to the unique physical measure of the Lorenz flow. A similar conclusion holds for the entropies associated with the physical measures.

Modern Fortran is a standardized language that includes object-oriented and parallel programming paradigms. The Fortran-lang community, created at the end of 2019, is actively working to modernize its ecosystem. New compilers are under development. And the fourth Fortran standard of the 21st century is due to be published in autumn 2023., Comment: This article is the official English translation of the French article published in Bulletin de la Soci{\'e}t{\'e} Informatique de France (November 2023)

Classic results by L.-S. Young show that the decay of correlations for systems that admit inducing schemes can be obtained through the recurrence rates of the inducing scheme. Reciprocal results were obtained for non-invertible systems (without contracting directions). Here, we obtain reciprocal results also for invertible systems (with contracting directions)., Comment: Title correction

We characterize homogeneous hypersurfaces in complex space forms which arise as critical points of a higher order energy functional. As a consequence, we obtain existence and non-existence results for $\mathbb{CP}^n$ and $\mathbb{CH}^n$, respectively. Moreover, we study the stability of biharmonic hypersurfaces and compute the normal index for a large family of solutions., Comment: 16 pages, 0 figures. v2: new references, Corollary 5.3 has been improved, and some minor changes. v3: improved exposition

Since the seminal work of [9] and their Physics-Informed neural networks (PINNs), many efforts have been conducted towards solving partial differential equations (PDEs) with Deep Learning models. However, some challenges remain, for instance the extension of such models to complex three-dimensional geometries, and a study on how such approaches could be combined to classical numerical solvers. In this work, we justify the use of graph neural networks for these problems, based on the similarity between these architectures and the meshes used in traditional numerical techniques for solving partial differential equations. After proving an issue with the Physics-Informed framework for complex geometries, during the computation of PDE residuals, an alternative procedure is proposed, by combining classical numerical solvers and the Physics-Informed framework. Finally, we propose an implementation of this approach, that we test on a three-dimensional problem on an irregular geometry.

We study $ { \mathrm{ SU } ( p + 1 ) \times \mathrm{ SU } ( n - p ) } $-equivariant maps between complex projective spaces. For every $ { n, p \in \mathbb{ N } } $ with $ { 0 \leq p < n } $, we construct two explicit families of uncountable many harmonic self-maps of $ \mathbb{ CP }^n $, one given by holomorphic maps and the other by maps that are neither holomorphic nor antiholomorphic. We prove that each solution is equivariantly weakly stable and explicitly compute the equivariant spectrum for some specific maps in both families., Comment: 15 pages, 1 figure. v2: section 5 has been expanded

We prove strong statistical stability of a large class of one-dimensional maps which may have an arbitrary finite number of discontinuities and of non-degenerate critical points and/or singular points with infinite derivative, and satisfy some expansivity and bounded recurrence conditions. This generalizes known results for maps with critical points and bounded derivatives and in particular proves statistical stability of Lorenz-like maps with critical points and singularities studied in [S. Luzzatto and W. Tucker. Non-uniformly expanding dynamics in maps with singularities and criticalities. Inst. Hautes Etudes Sci. Publ. Math., (89):179-226, 1999]. We introduce a natural metric on the space of maps with discontinuities which does not seem to have been used in the literature before., Comment: 22 pages. arXiv admin note: text overlap with arXiv:0709.1723

Current simulation of metal forging processes use advanced finite element methods. Such methods consist of solving mathematical equations, which takes a significant amount of time for the simulation to complete. Computational time can be prohibitive for parametric response surface exploration tasks. In this paper, we propose as an alternative, a Graph Neural Network-based graph prediction model to act as a surrogate model for parameters search space exploration and which exhibits a time cost reduced by an order of magnitude. Numerical experiments show that this new model outperforms the Point-Net model and the Dynamic Graph Convolutional Neural Net model.

In this article we study random tower maps driven by an ergodic automorphism. We prove quenched exponential correlations decay for tower maps admitting exponential tails. Our technique is based on constructing suitable cones of functions, defined on the random towers, which contract with respect to the Hilbert metric under the action of appropriate transfer operators. We apply our results to obtain quenched exponential correlations decay for several non-iid random dynamical systems including small random perturbations of Lorenz maps and Axiom A attractors., Comment: Revised version. To appear in Nonlinearity

We obtain entropy formulas for SRB measures with finite entropy given by inducing schemes. In the first part of the work, we obtain Pesin entropy formula for the class of noninvertible systems whose SRB measures are given by Gibbs-Markov induced maps. In the second part, we obtain Pesin entropy formula for invertible maps whose SRB measures given by Young sets, taking into account a classical compression technique along the stable direction that allows a reduction of the return map associated with a Young set to a Gibbs-Markov map. In both cases, we give applications of our main results to several classes of dynamical systems with singular sets, where the classical results by Ruelle and Pesin cannot be applied. We also present examples of systems with SRB measures given by inducing schemes for which Ruelle inequality does not hold., Comment: New application and new references added