Last Thursday, June 6, at the Faculty of Physical and Mathematical Sciences of the Universidad de Concepción (UdeC), the seminar 'Combining Galerkin schemes and neural networks to solve partial differential equations' was held, organized by Dr. Manuel E. Solano, academic of that department and researcher of the Center for Research in Mathematical Engineering, CI²MA, Dr. Manuel E. Solano.   

The speakers at the seminar were two academics from the Institute of Mathematics of the Pontificia U. Católica de Valparaíso: Dr. Sergio Rojas Hernández and Dr. Ignacio Muga Urquiza, and their presentations were complementary to the Colloquium of the Department of Mathematical Engineering of the UdeC that took place earlier in the day, whose talk was given by the researcher of the U. of Krakow AGH Paweł Maczuga.   

In this regard, the director of CI²MA, Dr. Rodolfo Araya Durán, noted that "it is very interesting to host this type of initiative, particularly on a topic that is trending topic in applied mathematics. We hope that these visits can lead to a new scientific development associated with our center".   

Dr. Solano, also a member of the Center for Mathematical Modeling at the University of Chile, explained that "originally, Pawel was going to come to WONAPDE 2024, but for health reasons he had to postpone the trip. Then a couple of months ago his guiding professor wrote to me so that Pawel could visit CI²MA and we could explore collaboration topics together".   

New techniques for better solutions  

"While I don't work directly on his research topic which is related to Neural Networks to solve partial differential equations (PDEs)," Solano explained, "I found it interesting to learn what he does to apply these techniques to our problems. In this same direction, Sergio Rojas and Ignacio Muga, as far as I know, are the only researchers in Chile working on this topic. That is why I invited them, in order to enrich the discussions we have had during the week".    

Robust Variational Physics Informed Neural Networks' is the title of the talk presented by Rojas. In it, explains the PUCV researcher, "I presented a new methodology to define robust cost functionals based on a neural network approximation technique known as Variational Physics-Informed Neural Networks".   

"The main objective of this visit", explains Dr. Rojas, "was to visit CI²MA in order to initiate collaborations with Professors Manuel Solano and Gabriel Gatica, as well as to continue an existing collaboration with Dr. Julio Careaga".   

Meanwhile, his colleague Ignacio Muga presented 'Neural Control of Discrete Weak Formulations of PDEs' and explained that "with Sergio and Pawel we are collaborating to solve equations using finite elements, which is basically what is done here, in addition to finite volumes, but we are also introducing neural networks, which are behind everything that is artificial intelligence, machine learning and that is basically what we came to show".   

In his talk, Muga presented on "how to enhance a finite element method using neural networks, for example, if you find the solution to a differential equation in a discrete approximation of the differential equation and you are not 100% satisfied and you want to improve certain attributes of the discrete solution, then we devised a way to do it by introducing a control that is based on a neural network".