Pablo Herrera

I am a Researcher at the Basque Center for Applied Mathematics (BCAM) in Bilbao, Spain. My work lies at the intersection of traditional numerical analysis and deep learning, with a specific focus on developing robust and stable methods for variational problems.

Research Interests

My current research focuses on:

• Neural Networks for resolution of PDEs: Developing machine learning architectures, such as Ritz-Uzawa Neural Networks (RUNNs), to stably and accurately solve Partial Differential Equations, particularly those featuring low-regularity solutions where standard energy-based methods often fail.
• Discretization of Variational Problems: Designing variational formulations by rewriting the PDE as a sequence of minimizations over a space of trainable parameters.
• Spectral Bias Mitigation: Implementing data-driven frequency tuning strategies to allow neural networks to dynamically adapt their bandwidth and capture high-frequency components.
• Stochastic Integration: Employing unbiased stratified stochastic quadrature rules to strictly control integration bias and evaluate variance reduction in data-scarce regimes.
• Stochastic Optimization: Leveraging hybrid optimization schemes, such as combining Least Squares with Adam (LS/Adam), to efficiently decouple the training of linear and non-linear parameters and accelerate convergence.
• Initialization Techniques: Designing data-driven initialization methods to accelerate the training process.

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