Carlos Uriarte's thesis, Postdoc Fellow at BCAM, has been selected by SEMA (Sociedad Española de Matemática Aplicada) for the ECCOMAS Awards for the Two Best PhD Theses in 2024 on Computational Methods in Applied Sciences and Engineering

• Carlos Uriarte, Postdoc Fellow at BCAM (Mathematical Design, Modelling, and Simulations), defended his thesis titled "Solving Partial Differential Equations using Artificial Neural Networks" at UPV/EHU, receiving international doctoral mention and the highest possible qualification ("sobresaliente cum laude").
• The thesis was supervised by professors David Pardo (Group Leader, BCAM - UPV/EHU Ikerbasque Research Professor, Mathematical Design, Modelling, and Simulations) and Elisabete Alberdi (UPV/EHU).
   

SEMA (Sociedad Española de Matemática Aplicada) has selected Carlos Uriarte's thesis for the ECCOMAS Awards for the Two Best PhD Theses in 2024 on Computational Methods in Applied Sciences and Engineering.

Carlos Uriarte, Postdoc Fellow at BCAM (Mathematical Design, Modelling, and Simulations), defended his thesis titled "Solving Partial Differential Equations using Artificial Neural Networks" at UPV/EHU, receiving international doctoral mention and the highest possible qualification ("sobresaliente cum laude"). The thesis was supervised by professors David Pardo (Group Leader, BCAM - UPV/EHU Ikerbasque Research Professor, Mathematical Design, Modelling, and Simulations) and Elisabete Alberdi (UPV/EHU).

The thesis explores the use of neural networks for solving Partial Differential Equations (PDEs). While traditional numerical methods such as finite differences or finite elements have proven effective, they face challenges in high-dimensional problems. In this regard, neural networks offer a promising solution. Carlos Uriarte's thesis makes three main contributions:

• Deep Finite Element Method (Deep FEM): Introduces an approach inspired by finite element methods, where the neural network architecture mimics refined mesh connectivity 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 improving integration accuracy without significantly increasing computational costs.

The thesis not only proposes new methodologies but also establishes solid mathematical foundations for future research at the intersection of neural networks and scientific computing. Its strong applied character makes it an excellent candidate for the award.

Best wishes from BCAM to Carlos Uriarte!