Paper: arXiv 2508.14995

Authors: Anastasis Kratsios, Ariel Neufeld, Philipp Schmocker

Abstract

Neural operators (NOs) are a class of deep learning models designed to simultaneously solve infinitely many related problems by casting them into an infinite-dimensional space, whereon these NOs operate. A significant gap remains between theory and practice: worst-case parameter bounds from universal approximation theorems suggest that NOs may require an unrealistically large number of parameters to solve most operator learning problems, which stands in direct opposition to a slew of experimental evidence. This paper closes that gap for a specific class of {“NOs”}, generative {“equilibrium operators”} (GEOs), using (realistic) finite-dimensional deep equilibrium layers, when solving families of convex optimization problems over a separable Hilbert space $X$. Here, the inputs are smooth, convex loss functions on $X$, and outputs are the associated (approximate) solutions to the optimization problem defined by each input loss. We show that when the input losses lie in suitable infinite-dimensional compact sets, our GEO can uniformly approximate the corresponding solutions to arbitrary precision, with rank, depth, and width growing only logarithmically in the reciprocal of the approximation error. We then validate both our theoretical results and the trainability of GEOs on three applications: (1) nonlinear PDEs, (2) stochastic optimal control problems, and (3) hedging problems in mathematical finance under liquidity constraints.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 3.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper employs advanced infinite-dimensional functional analysis, convex optimization theory, and neural operator approximation theorems, resulting in a very high math complexity. The empirical validation, while present in three domains, is limited to theoretical claims and numerical illustrations without backtest-ready implementation details or statistical metrics, placing it in the Lab Rats quadrant.

Research Flowchart

  flowchart TD
  A["Research Goal:<br>Bridge Theory-Practice Gap in NOs<br>for Infinite Convex Programs"] --> B["Methodology:<br>Generative Equilibrium Operators GEOs"]
  B --> C["Data/Input:<br>Families of Smooth Convex Loss Functions<br>on Separable Hilbert Space X"]
  C --> D["Computational Process:<br>Finite-Dim Deep Equilibrium Layers<br>Log-Complexity Architecture"]
  D --> E["Validation Applications:<br>1. Nonlinear PDEs<br>2. Stochastic Optimal Control<br>3. Constrained Financial Hedging"]
  E --> F["Key Outcome:<br>Uniform Approximation of Solutions<br>Logarithmic Growth in Parameters vs Error"]

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4 of 87 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.