Differential Equations

Dirichlet-to-Neumann and Calderon operator via deep learning techniques.

Event Date: Jan 17, 2023 in Differential Equations, Seminars

Abstract: In this talk we consider the Dirichlet-to-Neumann operator and the Calderón mapping appearing in Calderon’s inverse problem. Using deep learning techniques, we prove that these maps are rigorously approximated by infinite-dimensional neural networks.

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Long time asymptotics of large data in the Kadomtsev-Petviashvili models and geometrical aspects of its dynamics.

Event Date: Jan 10, 2023 in Differential Equations, Seminars

Abstract: In this talk we consider the Kadomtsev-Petviashvili equations posed on R2. For both models, we provide sequential in time asymptotic descriptions of solutions obtained from arbitrarily large initial data, inside and far regions of the plane not containing lumps or line solitons, and under minimal regularity assumptions. A geometrical description of the dynamics will be given in terms of parabolic regions.

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On traveling waves for the Gross-Pitaevskii equations.

Event Date: Jan 03, 2023 in Differential Equations, Seminars

Abstract: In this talk, we will discuss some properties of traveling waves solutions for some variants of the classical Gross-Pitaevskii equation in the whole space, in order to include new physical models in Bose-Einstein condensates and nonlinear optics. We are interested in the existence of finite energy localized traveling waves solutions with nonvanishing conditions at infinity, i.e. dark solitons. After a review of the state of the art in the classical case, we will show some results for a family of Gross-Pitaevskii equations with nonlocal interactions in the potential energy, obtained...

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Neural Implicit Surface Evolution using Differential Equations.

Event Date: Dec 20, 2022 in Differential Equations, Seminars

Abstract: In this talk, we present a machine learning framework that uses smooth neural networks to model dynamic variations of implicit surfaces under partial differential equations. Examples include evolving an initial surface towards vector fields, smoothing and sharpening using the mean curvature equation, and interpolations of implicit surfaces regularized by specific differential equations.

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Cosmología Primordial.

Event Date: Dec 20, 2022 in Differential Equations, Seminars

Abstract: Nuestro entendimiento del origen del universo ha cambiado dramáticamente durante los últimos 40 años. Hoy sabemos que la estructura de gran escala del universo (compuesto por galaxias) debe su existencia a pequeñas fluctuaciones del espacio y el tiempo -fluctuaciones primordiales- ya presentes durante el Big-Bang. La teoría más aceptada para explicar el origen de estas fluctuaciones sostiene que ellas se deben a procesos cuánticos ocurridos antes del Big-Bang, durante una época conocida como inflación cósmica. Este cuadro lo hemos forjado utilizando observaciones basadas en...

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Existence of solutions on the critical hyperbola for a pure Lane-Emden system with Neumann boundary conditions.

Event Date: Dec 06, 2022 in Differential Equations, Seminars

Abstract: I will present some recent results obtained in collaboration with A. Pistoia and H. Tavares for a Lane-Emden system on a bounded regular domain with Neumann boundary conditions and critical nonlinearities. We show that, under suitable conditions on the exponents in the nonlinearities, least-energy (sign-changing) solutions exist. In the proof we exploit a dual variational formulation which allows to deal with the strong indefinite character of the problem, and we establish a compactness condition which is based on a new Cherrier type inequality. We then prove such condition by...

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