Data-driven operator learning: Dictionary optimization and randomized neural nets

Data-driven operator learning: Dictionary optimization and randomized neural nets

by Mohammad Tabish

Formal & Physical Sciences AI — Models & Research UYQArtificial intelligence (AI)UYQNNeural networks and fuzzy systems
🎙 Mohammad Tabish 👥 3K 📅 December 1, 2025 ⏱ 50 min 👁 148 📄 original study 🧭 2026-08-16
Available in: English (current) Français

Keywords

Koopman operatorEDMDSINDyPDE-FINDrandomized neural networkstransfer operatorsspectral decompositiondictionary optimizationgradient descentuncertainty quantification

Summary

Mohammad Tabish presents his PhD research on data-driven operator learning for dynamical systems. He introduces transfer operators, specifically the Perron-Frobenius and Koopman operators, which describe the evolution of densities and observables, respectively. He highlights the challenge of selecting appropriate basis functions (dictionaries) for methods like EDMD, SINDy, and PDE-FIND. To address this, he proposes a gradient descent-based framework for optimizing parametric dictionaries, demonstrating its effectiveness on a triple-well potential and a modified Chua circuit. He then introduces RaNNDy, a randomized neural network approach for learning transfer operators and their spectral decompositions. In RaNNDy, hidden-layer weights are randomly initialized and fixed, while only the output layer is trained, reducing computational cost and avoiding common deep learning issues. The method provides closed-form solutions for eigenfunctions and enables uncertainty quantification via ensemble learning. Applications include stochastic dynamical systems, protein folding, and the quantum harmonic oscillator. The talk concludes with numerical examples showing the accuracy and efficiency of the proposed methods.

157 words

Critical Evaluation

Value of the Information & Strength of the Argument

The presentation offers significant value by addressing a key limitation in data-driven dynamical systems analysis: the selection of basis functions. The proposed dictionary optimization framework is a novel contribution that improves the accuracy of EDMD, SINDy, and PDE-FIND by learning interpretable basis functions from data. The introduction of RaNNDy provides a computationally efficient alternative to deep learning approaches, with closed-form solutions and uncertainty quantification. The argumentation is solid, grounded in mathematical principles such as variational principles and supported by numerical experiments on diverse systems. The speaker clearly explains the motivation and potential impact, making a compelling case for the methods.

Scientific Rigor, Source Quality, Title Accuracy

The talk demonstrates scientific rigor through clear mathematical formulations and reproducible numerical examples. The speaker references established methods (EDMD, SINDy, PDE-FIND) and builds upon prior work by Frank Noé and others. However, the presentation does not include explicit citations to specific papers or external sources, relying instead on general knowledge. The title accurately reflects the content, and the talk is well-structured. The speaker does not mention any external sources or provide a reference list, which limits the verifiability of the claims. Nevertheless, the methods are presented with sufficient detail to be assessed.

207 words

Title / Content Match

The title accurately reflects the content, focusing on data-driven operator learning with dictionary optimization and randomized neural networks.

Quality & Reliability

8/10

The presentation is based on original research, with clear mathematical formulations and numerical examples. The methods are well-motivated and the results are presented with appropriate caveats. However, the talk is a seminar presentation and does not provide full peer-reviewed details or extensive validation.

Key Moments

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

Contribution & Novelties

The talk presents two novel contributions: a gradient descent-based framework for optimizing dictionaries in operator learning, and RaNNDy, a randomized neural network approach for learning transfer operators. The dictionary optimization improves the accuracy of existing methods by learning interpretable basis functions from data, addressing a key limitation. RaNNDy offers a computationally efficient alternative to deep learning, with closed-form solutions and uncertainty quantification. These methods are demonstrated on diverse systems, showing broad applicability.

Pour aller plus loin :

  • Koopman operator — Provides background on the Koopman operator and its applications.
  • Dynamic mode decomposition — Related to EDMD, a foundational method for data-driven dynamical systems.
  • Sparse identification of nonlinear dynamics (SINDy) — The SINDy method for discovering governing equations.
  • Extreme learning machines — A type of randomized neural network similar to RaNNDy.

130 words

Radar Profile

The radar profile shows high scores across all dimensions, indicating a well-rounded presentation with strong technical depth, reliable information, and substantial content. The balance between quantity and quality suggests a comprehensive and rigorous talk.

Reliability 8/10