Prof. Javier Gómez-Serrano | AI-Driven Mathematical Discovery: Singularities, Algorithms, and Beyond

Prof. Javier Gómez-Serrano | AI-Driven Mathematical Discovery: Singularities, Algorithms, and Beyond

by Javier Gómez-Serrano

Formal & Physical Sciences Mathematics PBMathematicsPBWApplied mathematics
🎙 Javier Gómez-Serrano 👥 8K 📅 April 20, 2026 ⏱ 54 min 👁 310 📄 expert opinion 🧭 2026-08-15
Available in: English (current) Français

Keywords

singularityPINNAlphaFoldcomputer-assisted proofmachine learning

Summary

In this seminar talk, Professor Javier Gómez-Serrano presents recent advances in using artificial intelligence for mathematical discovery, with a focus on singularities in fluid dynamics and the potential of generative AI. He outlines two complementary approaches: small models (physics-informed neural networks, PINNs) for solving specific problems, and large models (like AlphaFold) for broader exploration. The talk is structured as the first part of a two-part series, with the second part to be given by Bogdan. Gómez-Serrano begins by motivating the study of singularities in the 3D Euler and Navier-Stokes equations, highlighting the open problem of finite-time blowup. He explains the self-similar ansatz and how it transforms the problem into finding stationary solutions in a rescaled frame, making it amenable to numerical methods. He then details his work using PINNs to discover self-similar blowup profiles for the 2D Boussinesq equations, achieving unprecedented accuracy (residuals as low as 10^-13). He discusses techniques to improve PINN performance, such as incorporating asymptotic behavior, compactifying the domain, and using multi-stage training. The second part of the talk shifts to large models, specifically AlphaFold, which uses genetic algorithms and LLMs to explore the space of algorithms for mathematical problems. He emphasizes the potential of these methods to assist in mathematical discovery, though he notes that rigorous proofs still require classical analysis and computer-assisted techniques. The talk concludes with a Q&A session where he addresses questions about error measurement and the feasibility of computer-assisted proofs.

238 words

Critical Evaluation

Value of the Information & Strength of the Argument

The talk provides valuable insights into the application of AI in mathematics, particularly in the context of singularity formation in PDEs. The speaker presents concrete examples and results, demonstrating the effectiveness of PINNs in discovering new solutions with high precision. The argumentation is solid, as he explains the methodology and the improvements over previous work, and he acknowledges the limitations and open questions. The discussion of AlphaFold’s potential for mathematical discovery is forward-looking and thought-provoking, though it is less detailed than the PINN part. Overall, the talk offers a balanced perspective on the current state and future possibilities of AI in mathematics.

Scientific Rigor, Source Quality, Title Accuracy

The talk is scientifically rigorous, with the speaker referencing specific papers and results, and he mentions a recent Quanta article for broader context. The sources cited include his own work and that of others, and he clearly distinguishes between established results and ongoing research. The title accurately reflects the content, covering both the singularity problem and the algorithmic approaches. The talk is given at a reputable institution (Isaac Newton Institute), which adds to its credibility. The speaker also engages with audience questions, clarifying technical details and acknowledging uncertainties, which further demonstrates scientific integrity.

210 words

Title / Content Match

The title accurately reflects the content: the speaker discusses AI-driven mathematical discovery, focusing on singularities in PDEs and the use of algorithms, including small models (PINNs) and large models (AlphaFold).

Quality & Reliability

8/10

The talk is given by a recognized researcher (Brown University) at the Isaac Newton Institute, presenting recent results and methods. The content is technical and appears rigorous, with references to published work and ongoing research. However, as a seminar talk, it lacks peer-reviewed detail and is partly based on unpublished results.

Key Moments

Markers derived by PSI from the transcript: the creator did not define chapters.

Cited Sources

Concurring Sources

Contribution & Novelties

The talk presents recent advances in using AI for mathematical discovery, particularly in the context of singularity formation in fluid dynamics. The speaker’s work on PINNs achieves unprecedented accuracy in finding self-similar blowup profiles, potentially enabling computer-assisted proofs. The discussion of AlphaFold’s application to mathematics is novel and highlights the potential of large language models in exploring algorithmic spaces. The talk contributes to the growing field of AI-assisted mathematics by demonstrating both small and large model approaches.

Pour aller plus loin :

130 words

Radar Profile

The radar profile shows high scores in technical level and information quality, reflecting the advanced and rigorous nature of the talk. The quantity of information is also high, but the global reliability is slightly lower due to the informal setting and reliance on unpublished results. The overall balance indicates a technically dense and informative presentation.

Reliability 8/10

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