Seminario de Matemática Aplicada: Variational Physics-Informed Neural Networks optimized with least squares and adaptivity in the test space

Conferenciante: David Pardo Zubiaur. Centro Vasco de Matemáticas Aplicadas y Universidad del País Vasco. This presentation addresses challenges encountered by Robust Variational Physics-Informed Neural Networks (RVPINNs), particularly related to optimizer convergence issues like those seen with the gradient descent (GD) based solver ADAM. It proposes interpreting neurons in the final hidden layer as a discrete trial space basis and employing a least-squares (LS) solver to enhance optimizer performance. The talk discusses a hybrid GD/LS solver and an ultraweak variational formulation, which eliminates the need for automatic differentiation in assembling the least-squares system, leading to faster performance. Additionally, it explores the Deep Fourier Residual (DFR) method within RVPINNs, presenting an extension for adaptive strategies on polygonal domains. Numerical examples in 1D and 2D demonstrate the efficacy of the ultraweak DFR loss function with a hybrid Adam/LS solver, showcasing improvements in convergence speed, computational cost, and mesh refinement across various problem scenarios.