Publications

2025

• Méthode hybride de simulation de champs ultrasonores dans une grande structure stratifiée avec des objets au contact
  
  • Kubecki Romain
  • Ducasse Eric
  • Bonnet Marc
  • Deschamps Marc
  , 2025. Ce travail a pour objectif de simuler la propagation d'ultrasons dans une structure stratifiée de grande taille comportant des objets au contact (de type traducteur, raidisseur, ou autre), dans un contexte de contrôle non destructif. La taille modérée des objets permet leur simulation par éléments finis, qui est par contre prohibitive pour la structure stratifiée de base. Si cette dernière est de géométrie canonique (plane ou tubulaire à symétrie de révolution), les champs peuvent en revanche être calculés par une méthode semi-analytique rapide utilisant des transformées de Laplace en temps et de Fourier par rapport aux coordonnées « longitudinales » (plan de la plaque ou positions axiale et azimutale dans le tube). En effet, dans le domaine <latex>(k,r,s)</latex> (<latex>k~vecteur</latex> d'onde, <latex>r~position</latex> dans l'épaisseur, <latex>s~variable</latex> de Laplace), le problème de propagation peut être résolu de manière exacte, et massivement parallélisable. Pour exploiter les atouts des deux méthodes, nous proposons une approche itérative de couplage par <i>décomposition de domaine</i> (DDM), reposant sur une suite de problèmes de propagation dans chaque sous-domaine comprenant sur leur interface commune des conditions aux limites dépendant des solutions de l'itération précédente. La littérature montre que le choix de conditions de Robin (de type impédance) entre deux domaines couplés garantit dans beaucoup de situations la convergence des itérations de couplage. Nous prouvons que cette convergence a bien lieu pour notre contexte particulier et présentons une validation numérique préliminaire en configuration 2D. Le caractère spatialement non-local du traitement semi-analytique de la structure stratifiée nous conduit ensuite à construire des fonctions de base négligeables en-dehors d'un voisinage de l'interface et à développer un protocole spécifique pour leur couplage avec les éléments finis. Ces deux aspects constituent les principaux ingrédients de la méthode hybride proposée ici. <latex>\medskip\hspace20mm</latex><i>Ce travail est financé par la DGA-AID et le CEA-List.</i>
• Optimisation par méthode adjointe discrète du bruit tonal d'une hélice estimé par la formulation fréquentielle de Hanson et Parzych
  
  • Mohammedi Yacine
  • Daroukh Majd
  • Buszyk Martin
  • Hajczak Antoine
  • Salah El-Din Itham
  • Bonnet Marc
  , 2025. Ce travail est consacré à l'optimisation à visée aéroacoustique de la forme d'une pale d’hélice en utilisant la méthode adjointe discrète. Cette dernière sera appliquée aux équations de Navier-Stokes stationnaires ainsi qu'à la formulation intégrale fréquentielle de Hanson et Parzych destinée au calcul du bruit tonal de rotor. Les sensibilités de la pression acoustique sont obtenues par dérivation analytique de la formulation intégrale. Ainsi, les sensibilités de toute fonction objectif exprimée en fonction de la pression acoustique peuvent être calculées. Ensuite, un solveur adjoint discret des équations de Navier-Stokes avec moyenne de Reynolds fournit les gradients de la fonction objectif en fonction des paramètres de forme. Ces derniers sont validés par comparaison avec une estimation par différences finies précise à l'ordre deux. Enfin, une optimisation multidisciplinaire et multi-objectifs est effectuée sur une hélice tripale subsonique isolée en condition de vol de croisière.
• Roadmap on metamaterial theory, modelling and design
  
  • Davies Bryn
  • Szyniszewski Stefan
  • Dias Marcelo
  • de Waal Leo
  • Kisil Anastasia
  • P Smyshlyaev Valery
  • Cooper Shane
  • Kamotski Ilia
  • Touboul Marie
  • Craster Richard
  • Capers James
  • Horsley Simon
  • Hewson Robert
  • Santer Matthew
  • Murphy Ryan
  • Thillaithevan Dilaksan
  • Berry Simon
  • Conduit Gareth
  • Earnshaw Jacob
  • Syrotiuk Nicholas
  • Duncan Oliver
  • Kaczmarczyk Łukasz
  • Scarpa Fabrizio
  • Pendry John
  • Martí-Sabaté Marc
  • Guenneau Sébastien
  • Torrent Daniel
  • Cherkaev Elena
  • Wellander Niklas
  • Alù Andrea
  • Madine Katie
  • Colquitt Daniel
  • Sheng Ping
  • Bennetts Luke
  • Krushynska Anastasiia
  • Zhang Zhaohang
  • Mirzaali Mohammad
  • Zadpoor Amir
  Journal of Physics D: Applied Physics, IOP Publishing, 2025, 58 (20), pp.203002. This Roadmap surveys the diversity of different approaches for characterising, modelling and designing metamaterials. It contains articles covering the wide range of physical settings in which metamaterials have been realised, from acoustics and electromagnetics to water waves and mechanical systems. In doing so, we highlight synergies between the many different physical domains and identify commonality between the main challenges. The articles also survey a variety of different strategies and philosophies, from analytic methods such as classical homogenisation to numerical optimisation and data-driven approaches. We highlight how the challenging and many-degree-of-freedom nature of metamaterial design problems call for techniques to be used in partnership, such that physical modelling and intuition can be combined with the computational might of modern optimisation and machine learning to facilitate future breakthroughs in the field. (10.1088/1361-6463/adc271)
  DOI : 10.1088/1361-6463/adc271
• A Rellich-type theorem for the Helmholtz equation in a junction of stratified media
  
  • Al Humaikani Sarah
  • Bonnet-Ben Dhia Anne-Sophie
  • Fliss Sonia
  • Hazard Christophe
  , 2025. <div><p>We prove that there are no non-zero square-integrable solutions to a two-dimensional Helmholtz equation in some unbounded inhomogeneous domains which represent junctions of stratified media. More precisely, we consider domains that are unions of three half-planes, where each half-plane is stratified in the direction orthogonal to its boundary. As for the well-known Rellich uniqueness theorem for a homogeneous exterior domain, our result does not require any boundary condition. Our proof is based on half-plane representations of the solution which are derived through a generalization of the Fourier transform adapted to stratified media. A byproduct of our result is the absence of trapped modes at the junction of open waveguides as soon as the angles between branches are greater than π/2.</p></div>
• An inverse problem related to an elasto-plastic beam
  
  • Bourgeois Laurent
  • Mercier Jean-François
  Inverse Problems, IOP Publishing, 2025, 41 (10). We consider an elasto-plastic beam and address the following inverse problem: external forces have created some plastic strains in this beam, which therefore shows a residual observable deformation once the structure is load-free. Can we retrieve the loading history from this observation, or at least the plastic strains ? After proving the well-posedness of the forward problem, we show that the solution can be described in a semi-explicit way in the pure bending case, so that the forward problem amounts to a one dimensional non linear problem. Such problem is smooth enough for us to solve the inverse problem by using a classical least square method, which is illustrated with the help of some numerical examples. (10.1088/1361-6420/ae0e49)
  DOI : 10.1088/1361-6420/ae0e49
• Explicit T-coercivity for the Stokes problem: a coercive finite element discretization
  
  • Ciarlet Patrick
  • Jamelot Erell
  , 2024, pp.137-159. Using the T -coercivity theory as advocated in Chesnel-Ciarlet [Numer. Math., 2013], we propose a new variational formulation of the Stokes problem which does not involve nonlocal operators. With this new formulation, unstable finite element pairs are stabilized. In addition, the numerical scheme is easy to implement, and a better approximation of the velocity and the pressure is observed numerically when the viscosity is small (10.1016/j.camwa.2025.03.028)
  DOI : 10.1016/j.camwa.2025.03.028
• Stability of time stepping methods for discontinuous Galerkin discretizations of Friedrichs' systems
  
  • Imperiale Sébastien
  • Joly Patrick
  • Rodríguez Jerónimo
  , 2025. In this work we study new various energy-based theoretical results on the stability of s-stages, s-th order explicit Runge-Kutta integrators as well as a modified leap-frog scheme applied to discontinuous Galerkin discretizations of transient linear symmetric hyperbolic Friedrichs' systems. We restrict the present study to conservative systems and Cauchy problems.
• An entropy penalized approach for stochastic optimization with marginal law constraints. Complete version
  
  • Bourdais Thibaut
  • Oudjane Nadia
  • Russo Francesco
  , 2025. This paper focuses on stochastic optimal control problems with constraints in law, which are rewritten as optimization (minimization) of probability measures problem on the canonical space. We introduce a penalized version of this type of problems by splitting the optimization variable and adding an entropic penalization term. We prove that this penalized version constitutes a good approximation of the original control problem and we provide an alternating procedure which converges, under a so called "Stability Condition", to an approximate solution of the original problem. We extend the approach introduced in a previous paper of the same authors including a jump dynamics, non-convex costs and constraints on the marginal laws of the controlled process. The interest of our approach is illustrated by numerical simulations related to demand-side management problems arising in power systems.
• A Production Routing Problem with Mobile Inventories
  
  • Lefgoum Raian
  • Afsar Sezin
  • Carpentier Pierre
  • Chancelier Jean-Philippe
  • de Lara Michel
  , 2025. <div><p>Hydrogen is an energy vector, and one possible way to reduce CO 2 emissions. This paper focuses on a hydrogen transport problem where mobile storage units are moved by trucks between sources to be refilled and destinations to meet demands, involving swap operations upon arrival. This contrasts with existing literature where inventories remain stationary. The objective is to optimize daily routing and refilling schedules of the mobile storages. We model the problem as a flow problem on a time-expanded graph, where each node of the graph is indexed by a time-interval and a location and then, we give an equivalent Mixed Integer Linear Programming (MILP) formulation of the problem. For small to medium-sized instances, this formulation can be efficiently solved using standard MILP solvers. However, for larger instances, the computational complexity increases significantly due to the highly combinatorial nature of the refilling process at the sources. To address this challenge, we propose a two-step heuristic that enhances</p></div>
• Efficient Quantum Circuits for Non-Unitary and Unitary Diagonal Operators with Space-Time-Accuracy trade-offs
  
  • Zylberman Julien
  • Nzongani Ugo
  • Simonetto Andrea
  • Debbasch Fabrice
  ACM Transactions on Quantum Computing, ACM, 2025. Unitary and non-unitary diagonal operators are fundamental building blocks in quantum algorithms with applications in the resolution of partial differential equations, Hamiltonian simulations, the loading of classical data on quantum computers (quantum state preparation) and many others. In this paper, we introduce a general approach to implement unitary and non-unitary diagonal operators with efficient-adjustable-depth quantum circuits. The depth, i.e. the number of layers of quantum gates of the quantum circuit, is reducible with respect either to the width, i.e. the number of ancilla qubits, or to the accuracy between the implemented operator and the target one. While exact methods have an optimal exponential scaling either in terms of size, i.e. the total number of primitive quantum gates, or width, approximate methods prove to be efficient for the class of diagonal operators depending on smooth, at least differentiable, functions. Our approach is general enough to allow any method for diagonal operators to become adjustable-depth or approximate, decreasing the depth of the circuit by increasing its width or its approximation level. This feature offers flexibility and can match with the hardware limitations in coherence time or cumulative gate error. We illustrate these methods by performing quantum state preparation and non-unitary-real-space simulation of the diffusion equation: an initial Gaussian function is prepared on a set of qubits before being evolved through the non-unitary evolution operator of the diffusion process. (10.1145/3718348)
  DOI : 10.1145/3718348
• A complex-scaled boundary integral equation for the embedded eigenvalues and complex resonances of the Neumann-Poincaré operator on domains with corners
  
  • Maltez Faria Luiz
  • Monteghetti Florian
  , 2025. The adjoint of the harmonic double-layer operator, also known as the Neumann-Poincaré (NP) operator, is a boundary integral operator whose real eigenvalues are associated with surface modes that find applications in e.g. photonics. On 2D domains with corners, the NP operator looses its compactness, as each corner induces a bounded interval of essential spectrum, and can exhibit both embedded eigenvalues and complex resonances. This work introduces a non-self-adjoint boundary integral operator whose discrete spectrum contains both embedded eigenvalues and complex resonances of the NP operator. This operator is obtained using a Green's function that is complex-scaled at each corner of the boundary. Numerical experiments using a Nyström discretization on a graded mesh demonstrates the accuracy of the method and its advantage over a 2D finite element discretization implementing the same complex scaling technique. (10.1016/j.camwa.2025.08.012)
  DOI : 10.1016/j.camwa.2025.08.012
• Mathematical and numerical analysis of the modes of a heterogeneous electromagnetic waveguide.
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Chesnel Lucas
  • Fliss Sonia
  • Parigaux Aurélien
  , 2025. In the homogeneous case, i.e. with constant epsilon and mu, the modes (E_n, H_n, \beta_n) are easily obtained by solving scalar problems in the section S of the guide and are pairwise orthogonal in L^2(S). They are either propagating (\beta in R) or purely evanescent (\beta in iR) and they have phase and group velocities of the same sign. For heterogeneous guides, i.e. with varying epsilon and mu in the section, these properties are generally not true and the mathematical analysis of the modes is much more delicate. In this talk, we present different formulations to study them and discuss their respective advantages. For strong variations of epsilon and/or mu, we show numerically that inverse modes, with group and phase velocities of opposite sign, can exist. Such cases for which PMLs fail to capture the outgoing solution are one of the reasons why we develop modal transparent conditions.
• Adaptive mesh refinement on Cartesian meshes applied to the mixed finite element discretization of the multigroup neutron diffusion equations
  
  • Ciarlet Patrick
  • Do Minh-Hieu
  • Madiot François
  , 2025. <div><p>The multigroup neutron diffusion equations are often used to model the neutron density at the nuclear reactor core scale. Classically, these equations can be recast in a mixed variational form.This chapter presents an adaptive mesh refinement approach based on a posteriori estimators. We focus on refinement strategies on Cartesian meshes, since such structures are common for nuclear reactor core applications.</p><p>This preprint corresponds to the Chapter 19 of volume 60 in AAMS, Advances in Applied Mechanics (to appear).</p></div>
• Not all sub-Riemannian minimizing geodesics are smooth
  
  • Chitour Yacine
  • Jean Frédéric
  • Monti Roberto
  • Rifford Ludovic
  • Sacchelli Ludovic
  • Sigalotti Mario
  • Socionovo Alessandro
  , 2025. A longstanding open question in sub-Riemannian geometry is the following: are sub-Riemannian length minimizers smooth? We give a negative answer to this question, exhibiting an example of a C 2 but not C 3 length-minimizer of a real-analytic (even polynomial) sub-Riemannian structure.
• Stochastic transport by Gaussian noise with regularity greater than 1/2
  
  • Flandoli Franco
  • Russo Francesco
  , 2025. Diffusion with stochastic transport is investigated here when the random driving process is a very general Gaussian process, including Fractional Brownian motion. The purpose is the comparison with a deterministic PDE, which in certain cases represents the equation for the mean value. From this equation we observe a reduced dissipation property for small times and an enhanced diffusion for large times, with respect to delta correlated noise when regularity is higher than the one of Brownian motion, a fact interpreted qualitatively here as a signature of the modified dissipation observed for 2D turbulent fluids due to the inverse cascade. We give results also for the variance of the solution and for a scaling limit of a two-component noise input.
• Solving numerically the two-dimensional time harmonic Maxwell problem with sign-changing coefficients
  
  • Chaaban Farah
  • Ciarlet Patrick
  • Rihani Mahran
  , 2025. We are investigating the numerical solution to the 2D time-harmonic Maxwell equations in the presence of a classical medium and a metamaterial, that is with sign-changing coefficients. As soon as the problem has a unique solution, we are able to build a converging numerical approximation based on the finite element method, for which there is no constraint on the meshes related to the sign-changing behavior. Numerical examples illustrate the theory.
• Alpha Mesh Swc: automatic and robust surface mesh generation from the skeleton description of brain cells
  
  • Mcsweeney-Davis Alex
  • Fang Chengran
  • Caruyer Emmanuel
  • Kerbrat Anne
  • Li Jing-Rebecca
  Briefings in Bioinformatics, Oxford University Press (OUP), 2025, 26 (3). In recent years, there has been a significant increase in publicly available skeleton descriptions of real brain cells from laboratories all over the world. In theory, this should make it possible to perform large scale realistic simulations on brain cells. However, currently there is still a gap between the skeleton descriptions and high quality simulation-ready surface and volume meshes of brain cells. We propose and implement a tool called {\it Alpha\_Mesh\_Swc} to generate automatically and efficiently triangular surface meshes that are optimized for finite elements simulations. We use an Alpha Wrapping method with an offset parameter on component surface meshes to efficiently generate a global watertight mesh. Then mesh simplification and re-meshing are used to produce an optimal surface mesh. Our methodology limits the number of surface triangles while preserving geometrical accuracy, permits cutting and gluing of cell components, is robust to imperfect skeleton descriptions, and allows mixed cell descriptions (surface meshes combined with skeletons). We compared the robustness, performance and accuracy of {\it Alpha\_Mesh\_Swc} against existing tools and found significant improvement in terms of mesh accuracy. We show, on average, we can generate fully automatically a brain cell (neurons or glia) surface mesh in a couple of minutes on a laptop computer resulting in a simplified surface mesh with only around 10k nodes. The resulting meshes were used to perform diffusion MRI simulations in neurons and microglia. The code and a number of sample brain cell surface meshes have been made publicly available. (10.1093/bib/bbaf258)
  DOI : 10.1093/bib/bbaf258
• A Two-Timescale Decision-Hazard-Decision Formulation for Storage Usage Values Calculation
  
  • Martinez Parra Camila
  • de Lara Michel
  • Chancelier Jean-Philippe
  • Carpentier Pierre
  • Janin Jean-Marc
  • Ruiz Manuel
  , 2024. The penetration of renewable energies requires additional storages to deal with intermittency. Accordingly, there is growing interest in evaluating the opportunity cost (usage value) associated with stored energy in large storages, a cost obtained by solving a multistage stochastic optimization problem. Today, to compute usage values under uncertainties, an adequacy resource problem is solved using stochastic dynamic programming assuming a hazard-decision information structure. This modelling assumes complete knowledge of the coming week uncertainties, which is not adapted to the system operation as the intermittency occurs at smaller timescale. We equip the twotimescale problem with a new information structure considering planning and recourse decisions: decision-hazard-decision. This structure is used to decompose the multistage decision-making process into a nonanticipative planning step in which the on/off decisions for the thermal units are made, and a recourse step in which the power modulation decisions are made once the uncertainties have been disclosed. In a numerical case, we illustrate how usage values are sensitive as how the disclosure of information is modelled. (10.48550/arXiv.2408.17113)
  DOI : 10.48550/arXiv.2408.17113
• Differentiable Optimisation: Theory and Algorithms -- Part II: Algorithms
  
  • Simonetto Andrea
  , 2025. This course follows naturally OPT201, which covers the theory part of continuous optimisation. OPT201 focuses on optimality conditions, convexity, and duality. In OPT202, we will look at how to use these notions to build algorithms that solve the problems. In particular, the aim of the course is to be able to answer the questions, 1. Given an optimisation problem, which algorithm do I use to solve it? 2. Which properties and theoretical guarantees does the algorithm that I have chosen have? 3. Conversely, if I want to use a certain algorithm, which characteristics does the optimisation problem need to have? In order to answer to these three questions, we will need to build a theory of algorithms, and ultimately understand what we really mean by solving an optimisation problem.
• Energy stable and linearly well-balanced numerical schemes for the non-linear Shallow Water equations with Coriolis force
  
  • Audusse Emmanuel
  • Dubos Virgile
  • Gaveau Noémie
  • Penel Yohan
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2025, 47 (01), pp.A1-A23. We analyse a class of energy-stable and linearly well-balanced numerical schemes dedicated to the nonlinear Shallow Water equations with Coriolis force. The proposed algorithms rely on colocated finite-difference approx- imations formulated on cartesian geometries. They involve appropriate diffusion terms in the numerical fluxes, expressed as discrete versions of the linear geostrophic equilibrium. We show that the resulting methods ensure semi-discrete energy estimates. Among the proposed algorithms a colocated finite-volume scheme is described. Numerical results show a very clear improvement around the nonlinear geostrophic equilibrium when compared to those of classic Godunov-type schemes. (10.1137/22M1515707)
  DOI : 10.1137/22M1515707
• Global solution of Quadratic Problems by Interval Methods and Convex Relaxations
  
  • Elloumi Sourour
  • Lambert Amélie
  • Neveu Bertrand
  • Trombettoni Gilles
  Journal of Global Optimization, Springer Verlag, 2025, 91 (2), pp.331–353. Interval branch-and-bound solvers provide reliable algorithms for handling non-convex optimization problems by ensuring the feasibility and the optimality of the computed solutions, i.e. independently from the floating-point rounding errors. Moreover, these solvers deal with a wide variety of mathematical operators. However, these solvers are not dedicated to quadratic optimization and do not exploit nonlinear convex relaxations in their framework. We present an interval branch-andbound method that can efficiently solve quadratic optimization problems. At each node explored by the algorithm, our solver uses a quadratic convex relaxation which is as strong as a semi-definite programming relaxation, and a variable selection strategy dedicated to quadratic problems. The interval features can then propagate efficiently this information for contracting all variable domains. We also propose to make our algorithm rigorous by certifying firstly the convexity of the objective function of our relaxation, and secondly the validity of the lower bound calculated at each node. In the non-rigorous case, our experiments show significant speedups on general integer quadratic instances, and when reliability is required, our first results show that we are able to handle medium-sized instances in a reasonable running time. (10.1007/s10898-024-01370-8)
  DOI : 10.1007/s10898-024-01370-8
• An operator approach to the analysis of electromagnetic wave propagation in dispersive media. Part 1: general results.
  
  • Cassier Maxence
  • Joly Patrick
  , 2025. In this chapter, we investigate the mathematical models for electromagnetic wave propagation in dispersive isotropic passive linear media for which the dielectric permittivity $\varepsilon$ and magnetic permeability $\mu$ depend on the frequency. We emphasize the link between physical requirements and mathematical properties of the models. A particular attention is devoted to the notions of causality and passivity and its connection to the existence of Herglotz functions that determine the dispersion of the material. We consider successively the cases of the general passive media and the so-called local media for which $\varepsilon$ and $\mu$ are rational functions of the frequency. This leads us to analyse the important classes of non-dissipative and dissipative generalized Lorentz models. In particular, we discuss the connection between mathematical and physical properties of models through the notions of stability, energy conservation, dispersion and modal analyses, group and phase velocities and energy decay in dissipative systems.
• A hybridizable discontinuous Galerkin method with transmission variables for time-harmonic acoustic problems in heterogeneous media
  
  • Pescuma Simone
  • Gabard Gwenael
  • Chaumont-Frelet Théophile
  • Modave Axel
  Journal of Computational Physics, Elsevier, 2025, 534, pp.114009. We consider the finite element solution of time-harmonic wave propagation problems in heterogeneous media with hybridizable discontinuous Galerkin (HDG) methods. In the case of homogeneous media, it has been observed that the iterative solution of the linear system can be accelerated by hybridizing with transmission variables instead of numerical traces, as performed in standard approaches. In this work, we extend the HDG method with transmission variables, which is called the CHDG method, to the heterogeneous case with piecewise constant physical coefficients. In particular, we consider formulations with standard upwind and general symmetric fluxes. The CHDG hybridized system can be written as a fixed-point problem, which can be solved with stationary iterative schemes for a class of symmetric fluxes. The standard HDG and CHDG methods are systematically studied with the different numerical fluxes by considering a series of 2D numerical benchmarks. The convergence of standard iterative schemes is always faster with the extended CHDG method than with the standard HDG methods, with upwind and scalar symmetric fluxes. (10.1016/j.jcp.2025.114009)
  DOI : 10.1016/j.jcp.2025.114009
• Analysis of time-harmonic electromagnetic problems with elliptic material coefficients
  
  • Ciarlet Patrick
  • Modave Axel
  Mathematical Methods in the Applied Sciences, Wiley, 2025. We consider time-harmonic electromagnetic problems with material coefficients represented by elliptic fields, covering a wide range of complex and anisotropic material media. The properties of elliptic fields are analyzed, with particular emphasis on scalar fields and normal tensor fields. Time-harmonic electromagnetic problems with general elliptic material fields are then studied. Well-posedness results for classical variational formulations with different boundary conditions are reviewed, and hypotheses for the coercivity of the corresponding sesquilinear forms are investigated. Finally, the proposed framework is applied to examples of media used in the literature: isotropic lossy media, geometric media, and gyrotropic media. (10.1002/mma.70318)
  DOI : 10.1002/mma.70318
• Machine Learning for Scientific Computing and Numerical Analysis
  
  • Montanelli Hadrien
  , 2025. These notes are for third-year students at École Polytechnique (Palaiseau, France), who have completed two years of classes préparatoires, making the material equivalent to a graduate-level course in the UK or US. The development of a Scientific Machine Learning (SciML) course for third-year students bridges the gap between scientific computing and machine learning. SciML, an emerging field, enables efficient methods for solving complex problems in science and engineering, like partial differential equations (PDEs). This course offers a new, valuable specialization, helping students stand out in the evolving job market, where demand for ML experts with specialized skills is growing.