Learning solution operator of dynamical systems with diffusion maps kernel ridge regression

Learning solution operator of dynamical systems with diffusion maps kernel ridge regression

by Prof. John Harlim

Formal & Physical Sciences AI — Models & Research UYQArtificial intelligence (AI)UYQMMachine learning
🎙 Prof. John Harlim 👥 2K 📅 August 13, 2026 ⏱ 69 min 👁 16 📄 original study 🧭 2026-08-15
Available in: English (current) Français

Keywords

diffusion mapskernel ridge regressionsolution operatordynamical systemsmanifold learning

Summary

The talk by Professor John Harlim presents a method for learning solution operators of dynamical systems using diffusion maps kernel ridge regression. The core idea is to exploit the low-dimensional manifold structure of high-dimensional data to construct a data-driven basis (via diffusion maps) that improves the efficiency of kernel ridge regression. The speaker begins by introducing diffusion maps as a dimension reduction technique based on the asymptotic expansion of a kernel integral operator, which approximates the Laplace-Beltrami operator on the manifold. He then discusses the convergence properties of diffusion maps and introduces a symmetrized version, the diffusion maps kernel, which converges to the heat kernel. This kernel is used in kernel ridge regression to learn the solution operator, mapping the current state to the next state. The talk emphasizes the importance of tuning the kernel bandwidth and regularization parameter, and proposes validation metrics based on long-term prediction accuracy, including valid prediction time for chaotic systems. Numerical examples are shown to illustrate the method’s performance.

164 words

Critical Evaluation

Value of the Information & Strength of the Argument

The talk provides a valuable contribution by connecting manifold learning (diffusion maps) with kernel methods for dynamical systems. The argumentation is solid, grounded in mathematical theory and supported by references to published results. The speaker clearly explains the theoretical foundations and practical considerations, such as the choice of validation metrics. The presentation is rigorous and addresses potential limitations, such as the need for careful hyperparameter tuning.

Scientific Rigor, Source Quality, Title Accuracy

The talk demonstrates high scientific rigor, with references to established results in diffusion maps and kernel methods. The speaker cites specific papers and results, and the presentation is hosted by the Isaac Newton Institute, adding credibility. The title accurately reflects the content. The talk is technical and assumes a specialized audience, but the methodology is clearly explained.

138 words

Title / Content Match

The title accurately reflects the content: the talk focuses on learning solution operators for dynamical systems using diffusion maps kernel ridge regression.

Quality & Reliability

8/10

Presentation of original research with rigorous mathematical derivations and references to published results. The speaker is a professor at a major university, and the talk is hosted by a prestigious research institute. The content is technical and assumes a specialized audience, but the methodology is clearly explained.

Key Moments

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

Cited Sources

Concurring Sources

Contribution & Novelties

The talk presents a novel approach to learning solution operators by leveraging diffusion maps to construct a data-driven kernel that approximates the heat kernel, thereby providing a principled basis for kernel ridge regression. This bridges manifold learning and operator learning, offering a theoretically grounded alternative to neural network-based methods. The speaker also introduces practical validation metrics for long-term prediction, addressing a common pitfall in dynamical system modeling.

Pour aller plus loin :

96 words

Radar Profile

The radar profile shows high scores in technical level and information quality, reflecting the advanced mathematical content and rigorous presentation. The lower score in quantity of information is due to the focused scope of the talk, which is appropriate for a seminar.

Reliability 8/10