Seminar in EDP and Applied Mathematics
june 26, 2024 – 14h (Brazil)
Streaming: Youtube Channel | SEMINARIO DE EDP E MATEMATICA APLICADA
Event
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About Seminars
Our Online Seminar is one of the most important events in Brazil, it has been held since August 2020, every Wednesday at 2 pm, Brasília time, with a frequency of 14 days. Two 40-minute lectures are presented in each session. Our speakers are world-renowned mathematicians from Europe, the United States and South America.
Featured Talks & Speakers
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Giovanni Fantuzzi
Department of Mathematics, FAU Erlangen-Nürnberg,Germany
14:00h – 14:40h
Moment-Sum-of-Squares relaxations for variational problems
Moment-Sum-of-Squares (moment-SOS) relaxations are an established technique to compute converging sequences of lower bounds on the global minimum of finite-dimensional polynomial optimization problems. In this talk, I will discuss two recent extensions of moment-SOS relaxations to infinite-dimensional variational problems, where a (possibly nonconvex) integral functional is to be minimized over functions from a Sobolev space. The first extension optimizes so-called “null Lagrangian translations” and returns certified lower bounds on the global minimum of the variational problem. The second extension, instead, produces upper bounds by approximating minimizers of finite element discretizations of the variational problem. Conditions that ensure the convergence of these upper and lower bounds to the desired global minimum will be discussed, and current gaps between theory and practice will be illustrated by means of examples.
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Song Yongcun
Department of Mathematics, FAU Erlangen-Nürnberg,Germany
14:50h – 15:30h
Physics-informed neural networks for non-smooth PDE-constrained optimization problems
We study the application of well-known physics-informed neural networks (PINNs) for solving non-smooth PDE-constrained optimization problems. First, we consider a class of PDE-constrained optimization problems where additional nonsmooth regularization is employed for constraints on the control or design variables. For solving such problems, we combine the alternating direction method of multipliers (ADMM) and PINNs and propose the ADMM-PINNs algorithmic framework, which unties the PDE constraints and the nonsmooth regularization terms for iterations. Accordingly, at each iteration, one of the resulting subproblems is a smooth PDE-constrained optimization which can be efficiently solved by PINNs, and the other is a simple nonsmooth optimization problem which usually has a closed-form solution or can be efficiently solved by various standard optimization algorithms or pre-trained neural networks. Then, we consider the optimal control of PDEs with interfaces. We employ the recently developed discontinuity-capturing neural network to tackle the non-smoothness of the PDEs with interfaces and propose hard-constraint PINNs for solving such optimal control problems. The hard-constraint PINNs ensure both the boundary and interface conditions are satisfied strictly, and meanwhile, they are decoupled from the learning of the PDEs. All these PINNs methods are mesh-free, easy to implement, and scalable to different PDE settings. Various numerical results are reported to validate the effectiveness and efficiency of the proposed PINNs methods.
About Organization
Juan Limaco -UFF-Coordenador
Mauro Rincon – UFRJ – Brazil
Anna Doubova-U.Sevilla-Spain
Luz de Teresa-UNAM Mexico
Diego Souza – U Sevilla -Spain
Felipe Chaves-UFPB-Brazil
Roberto Capistrano – UFPE Brazil
Sandra Malta – LNCC – Brazil
Marcelo Cavalcanti – UEM – Brazil;
Rui Almeida – UBI – Portugal
Roxana Lopez – UNMSM – Peru;
Mauricio Sepulveda – UdeC -Chile;
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