BCAM’S Postdoc Fellow Carlos Uriarte’s Doctoral Thesis, triple awarded

• The researcher’s thesis, titled “Solving Partial Differential Equations using Artificial Neural Networks”, has been selected as the best for the ECCOMAS awards, as well as for the prizes granted annually by SEMNI (Spanish Society of Numerical Methods in Engineering) and SEMA (Spanish Society of Applied Mathematics).
• Defended on February 23, 2024, at the Faculty of Informatics of UPV/EHU in Leioa, it also received the international doctorate mention and the highest possible grade (“sobresaliente cum laude”) at the Faculty of Informatics of UPV/EHU in Leioa.

Carlos Uriarte, Postdoc Fellow (Mathematical Design, Modelling and Simulations) at BCAM, has been triple awarded. His thesis has been considered the best for the ECCOMAS awards and those granted annually by SEMNI (Spanish Society of Numerical Methods in Engineering) and SEMA (Spanish Society of Applied Mathematics).

Uriarte’s doctoral thesis, titled “Solving Partial Differential Equations using Artificial Neural Networks” and supervised by Prof. David Pardo (BCAM) and Prof. Elisabete Alberdi (UPV/EHU), was defended on February 23, 2024, at the Faculty of Informatics of UPV/EHU in Leioa and received the international doctorate mention and the highest possible grade (“sobresaliente cum laude”).

This work explores the use of neural networks to solve Partial Differential Equations (PDEs). While traditional numerical methods, such as finite difference or finite element methods, have proven effective, they face challenges in high-dimensional problems. In this context, the use of neural networks can offer an effective solution. In his thesis, Carlos Uriarte makes three main contributions:

• Deep Finite Element Method (Deep FEM): A novel approach inspired by the finite element method, where the neural network architecture mimics the connectivity of refined meshes to solve parametric problems.
• Deep Double Ritz Method (D2RM): A residual minimization scheme employing two neural networks to approximate solutions with enhanced numerical stability.
• Memory-Based Monte Carlo Integration: A strategy that improves integration accuracy without significantly increasing computational cost.

The thesis not only proposes new methodologies but also establishes a solid mathematical foundation for future research at the intersection of neural networks and scientific computing.

The ECCOMAS, SEMA, and SEMNI awards recognize not only the academic excellence of Uriarte’s research but also its potential impact on the development of advanced computational methods to solve complex problems in engineering and applied sciences.

“I began my research in the field of artificial neural networks for solving partial differential equations somewhat skeptical. Without the support of my supervisors and research colleagues, I believe none of this would have been possible. This award also belongs to all of them,” says Uriarte.

From BCAM, we congratulate Carlos on this impressive achievement and recognize his exceptional dedication and talent. Zorionakl, Carlos!