Noticias en español
Error bounds for Physics Informed Neural Networks in Nonlinear Schrödinger equations placed on unbounded domains.
Abstract: We consider the subcritical nonlinear Schrödinger (NLS) in dimension one posed on the unbounded real line. Several previous works have considered the deep neural network approximation of NLS solutions from the numerical and theoretical point of view in the case of bounded domains. In this paper, we introduce a PINNs method to treat the case of unbounded domains and show rigorous bounds on the associated approximation error in terms of the energy and Strichartz norms, provided reasonable integration schemes are available. Applications to traveling waves, breathers and solitons, as...
Read MoreNonexistence and uniqueness of breathers for modified Zakharov-Kuznetsov models.
Abstract: In this talk we will consider the (focusing) modi ed Zakharov-Kuznetsov (mZK) in dimension N ≥ 1: ut + (∆u + 2u3)x1 = 0,for a given real-valued function u = u(t, x), where t ∈ R and x ∈ RN . This equation is a specialcase of the completely integrable modi ed Korteweg-de Vries (mKdV) equation ut + (uxx +2u3)x = 0. During this talk we will present results related to existence and nonexistence of quasimonochromatic breathers solution for the mZK equation, depending on the dimnesion N . Additionally we will show how the famous breather solution of the mKdV equation...
Read More“Hamilton-Jacobi-Bellman Solution Approximation with Machine Learningfor the Synthesis of Optimal Feedbacks”
Abstract: The design of optimal feedbacks for control problems is a challenging task. The classical method for tackling this problem is based on dynamic programming. This involves finding the value function of the control problem by solving the Hamilton-Jacobi-Bellman (HJB) equation. However, this equation suffers from the “curse of dimensionality”, i.e., the computational cost of solving it grows exponentially with the dimension of the underlying control problem. For this reason, several methods based on machine learning have been proposed to solve HJB. Although numerical...
Read MoreA stochastic differential Colonel Blotto game in a Stackelberg contract theory setting.
Abstract: The Colonel Blotto game is a resource allocation game where players decide where to focus their forces between different battlefields. We extend the standard Blotto game to a dynamic stochastic setting, in a time-continuous, two-player, zero-sum game. Using the dynamic programming principle, we explicitly characterize some Nash equilibrium strategies as well as the value of the game through a Hamilton-Jacobi-Bellman equation admitting a smooth solution. We formulate the game generally enough to allow for various rewards, as well as various drivers of randomness. We also present an...
Read MoreCanonical colourings in random graphs.
Abstract: The canonical Ramsey theorem of Erdős and Rado impliesthat for a given graph H, if n is sufficiently large then any colouring of the edges of K_n gives rise to copies of H that exhibit certain colour patterns, namely monochromatic, rainbow or lexicographic. I will discuss recent results on the threshold at which the random graph G(n,p) inherits the canonical Ramsey properties of K_n.
Read MoreOn the Asymptotic Stability of Solitary Wave Solutions to the Boussinesq Model in the Energy Space.
Abstract: The Good Boussinesq (GB) model is known to admit solitary wave solutions with speeds in the range −1<c<1. In this talk, we revisit existing results and present new findings on the asymptotic stability of solitary wave solutions to the GB equation with power-type nonlinearity and general initial data in the energy space H1xL2. These new result complete the orbital stability stability result established by Bona and Sachs (1988). The proof employs a novel set of virial estimates specifically tailored to the GB system in a moving frame. In particular, we introduce a...
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