mlsanityData to 14 Oct 2026

Generative Neural Operators of Log-Complexity Can Simultaneously Solve Infinitely Many Convex Programs

arXiv 2508.14995 ↗First posted 20 Aug 2025cs.LG cs.NA math.NAcited by 7

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Abstract (arXiv)

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.

Key facts. Cited by 7 papers in the map (data through 14 Oct 2026). Builds on 40 of its 70 references that are in the map. First posted on arXiv 20 Aug 2025.

Builds on

40 of 70 references.

References · 40

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28 more
  1. 2019A Theoretical Analysis of Deep Neural Networks and Parametric PDEscited by 74
  2. 2023Neural Inverse Operators for Solving PDE Inverse Problemscited by 65
  3. 2016Deep Learning Approximation for Stochastic Control Problemscited by 55
  4. 2019Deep splitting method for parabolic PDEscited by 53
  5. 2022Generative Adversarial Neural Operatorscited by 52
  6. 2021Seismic wave propagation and inversion with Neural Operatorscited by 51
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  8. 2023Operator learning with PCA-Net: upper and lower complexity boundscited by 44
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Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2025: 1; 2026: 6.

7 papers.

  1. 2025One model to solve them all: 2BSDE families via neural operatorscited by 4
  2. 2026Weighted universal approximation of differentiable maps on infinite-dimensional manifoldscited by 2
  3. 2026Learning to Control: The iUzawa-Net for Nonsmooth Optimal Control of Linear PDEscited by 1
  4. 2026Adaptivity Under Realizability Constraints: Comparing In-Context and Agentic Learningcited by 1
  5. 2026Algorithmic Foundations of Deep Learning: Complexity-Theoretic Rates and a Characterization of Universal Approximationcited by 1
  6. 2026NeuralChaos: Optimal Adapted Approximation of Square Integrable Predictable Processescited by 0
  7. 2026Neural operators approximate strongly continuous convex monotone semigroupscited by 0