Galerkin Methods for Optimal Control || Foundation Models & Information Retrieval || Oct 9, 2026

Value of the Information & Strength of the Argument

The value of the information is high: the first talk solves a fundamental theoretical gap in operator learning by establishing universal approximation with derivatives under suitable topologies. The argumentation is rigorous, building from classical results and clearly identifying why direct generalizations fail. The introduction of Bastiani differentiability is elegant and well motivated, and the connection to numerical methods (Deep Hilbert-Galerkin) demonstrates practical relevance. The second talk, while more applied, provides a structured framework (representation-modeling-system) for integrating foundation models into IR systems, with concrete examples and a clear thesis. Both speakers support their claims with published work and explicit references.

Scientific Rigor, Source Quality, Title Accuracy

The scientific rigor is high: de Feo’s talk is based on two arXiv preprints (2605.15285 and 2603.19463) and explicitly cites Hornik (1991) and other operator learning theorems. He acknowledges limitations and engages in a technical Q&A, showing depth. Jia’s talk references recent work in IR and foundation models, though specific citations are not given in the provided transcription. The title is fully adequate, matching the two talks. The public discussion is scholarly, indicating a well-informed audience. No external contradictory sources were identified in the provided material.

The title accurately reflects the two talks: the first on Galerkin methods for optimal control (de Feo), the second on adapting foundation models for information retrieval (Jia). It is concise and informative.

8/10

The seminar presents rigorous mathematical proofs and results, referencing specific arXiv preprints and classical theorems (Hornik 1991). The arguments are logically structured and rely on established functional analysis and approximation theory. Some aspects, like the generality of the architectures, are debatable but well argued.

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

• 00:00Introduction to the seminar and first speaker Filippo de Feo.
• 03:00De Feo outlines the talk: derivative-informed operator learning, motivations, state-of-the-art, and theory.
• 10:00Motivations: higher-order accuracy, efficient training, and applications in constrained optimization and optimal control.
• 15:00State-of-the-art review: classical deep learning theorems (Hornik 1991) and existing results for operator learning, noting limited coverage of derivatives.
• 20:00Failure of universal approximation under operator norms due to finite-rank derivatives; discussion of the structural obstruction.
• 25:00Introduction of Bastiani differentiability and joint evaluation map to restore universal approximation.
• 30:00Presentation of main theorems: density of encoder-decoder architectures in Bastiani compact-open topologies and weighted Sobolev spaces.
• 40:00Application of the theory to Deep Hilbert-Galerkin Methods for HJB PDEs from optimal control of stochastic PDEs.
• 50:00Q&A session on architecture assumptions (e.g., FNOs) and relation to other works.
• 1:00:00Transition to second speaker Pengyue Jia, discussing foundation models for information retrieval.

• Universal Approximation Theorems for Nonlinear Operators and Their Derivatives and Deep Hilbert-Galerkin Methods — De Feo's first paper on approximation and deep Hilbert-Galerkin methods, cited at the beginning.
• Deep Galerkin Methods for Infinite-Dimensional PDEs — De Feo et al. paper on Deep Hilbert-Galerkin methods, mentioned in the abstract.
• Hornik (1991) - Universal approximation of functions and their derivatives — Classical finite-dimensional result generalized in de Feo's work.
• Chen & Chen - Universal approximation for operator learning — Foundational operator learning approximation theorem referenced in the state-of-the-art.
• Yao et al. - For Neural Operators — Related work on derivative-informed operator learning with C1 Fréchet differentiability, discussed critically.