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Discovery of Unstable Singularities

AMS Seminar Series

Speaker: Prof. Javier Gómez-Serrano, Department of Mathematics, Brown University

Presentation

Seminar Slide Deck (PDF-4.9MB)

Abstract

In this talk, I will explain how to construct numerically several new unstable singularities to certain equations in fluids (CCF, IPM, Boussinesq) using machine learning methods. Our approach combines curated machine learning architectures and training schemes with a high-precision Gauss-Newton optimizer, achieving accuracies that significantly surpass previous work across all discovered solutions, reaching near double-float machine precision, attaining a level of accuracy constrained only by the round-off errors of the GPU hardware, potentially leading to rigorous mathematical validation via computer-assisted proofs.

Biography

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Javier Gómez-Serrano is Associate Professor of Mathematics at Brown University whose interests include partial differential equations, fluid mechanics, computer-assisted proofs, machine learning, spectral geometry, and analysis. He received his Ph.D. from the Universidad Autonoma de Madrid in 2013 and held positions at Princeton, the Universitat de Barcelona, and CRM Montreal before joining Brown in 2022. Along with Terrence Tao of UCLA, he has been working with a team at Google DeepMind to help develop AlphaEvolve, a Gemini-powered coding agent for designing advanced algorithms. He is the recipient of a Vicent Caselles Mathematical Research Award from the Spanish Royal Mathematical Society and the BBVA Foundation, an Antoino Valle Prize, and the Antoino Ambrosetti Medal.