[Todos] Coloquio especial conjunto DF-DC-DM-IC - Martin Arjovsky - Miercoles 10/12, 15 h

[Nahuel Andrés]

Estimadas/os,

Les invitamos al coloquio especial conjunto de los departamentos de Física,
Computación, Matemática y el Instituto de Cálculo a cargo de,

*Martín Arjovsky* (DeepMind)
*Discovery of Unstable Singularities in Partial Differential Equations
using Machine Learning*

Abstract: Whether singularities can form in fluids remains a foundational
unanswered question in mathematics. This phenomenon occurs when solutions
to governing equations, such as the 3D Euler equations, develop infinite
gradients from smooth initial conditions. Historically, numerical
approaches have primarily identified stable singularities. However, these
are not expected to exist for key open problems, such as the boundary-free
Euler and Navier-Stokes cases, where unstable singularities are
hypothesized to play a crucial role. Here, we present the first systematic
discovery of new families of unstable singularities. A stable singularity
is a robust outcome, forming even if the initial state is slightly
perturbed. In contrast, unstable singularities are exceptionally elusive;
they require initial conditions tuned with infinite precision, being in a
state of instability whereby infinitesimal perturbations immediately divert
the solution from its blow-up trajectory. In particular, we present
multiple new, unstable self-similar solutions for the incompressible porous
media equation and the 3D Euler equation with boundary, revealing a simple
empirical asymptotic formula relating the blow-up rate to the order of
instability. 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. For specific solutions, we reach near double-float
machine precision, attaining a level of accuracy constrained only by the
round-off errors of the GPU hardware. This level of precision meets the
requirements for rigorous mathematical validation via computer-assisted
proofs. This work provides a new playbook for exploring the complex
landscape of nonlinear partial differential equations (PDEs) and tackling
long-standing challenges in mathematical physics.

Bio: Martin Arjovsky is a Research Scientist at DeepMind. Prior to that, he
was a postdoc in SIERRA with Francis Bach, and before a PhD student at New
York University, advised by Léon Bottou. He is from Buenos Aires, Argentina
and received the BSc and MSc degrees from the University of Buenos Aires.
He also spent time in different places (including Google, Facebook,
Microsoft, Université de Montréal, and DeepMind). His master’s thesis
advisor was Yoshua Bengio, who also advised him during his stay at UdeM. In
general, he’s interested in the intersection between learning and
mathematics, how we can ground the different learning processes that are
involved in different problems, and leverage this knowledge to develop
better algorithms. Along these lines, he’s worked in many different areas
of machine learning, including optimization, unsupervised learning, out of
distribution generalization, and exploration in reinforcement learning.

El coloquio será en *español *a las *15 h* en el *Aula 1402 - Pabellón*
*0+infinito *- Ciudad Universitaria - CABA.
Comisión organizadora de los coloquios del DF 2025.
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