mlsanityData to 14 Oct 2026

Polynomial Scaling is Possible For Neural Operator Approximations of Structured Families of BSDEs

arXiv 2410.14788 ↗First posted 18 Oct 2024math.OC cs.LG cs.NAcited by 2

Not on the map. Why?

Abstract (arXiv)

Neural operator (NO) architectures learn nonlinear maps between infinite-dimensional function spaces and are widely used to accelerate simulation and enable data-driven model discovery. While universality results ensure expressivity, they do not address complexity: for broad operator classes described only through regularity (e.g. uniform continuity or Cr-regularity), information-theoretic lower bounds imply that minimax-optimal NO approximation rates scale exponentially in the reciprocal accuracy 1/varepsilon. This has shifted the focus of NO theory toward identifying additional problem-specific structure, beyond regularity, under which suitably tailored NO architectures can leverage to unlock polynomial scaling in 1/varepsilon. We exhibit the first polynomial-scaling regime for NO approximations of solution operators in stochastic analysis; by identifying structured families of non-Markovian BSDEs with randomized terminal condition parameterized by the Sobolev-regular terminal condition and by Sobolev-regular additive nonlinear perturbations of the generator. We prove that their solution operator can be approximated (uniformly over the family) by a tailored NO whose number of trainable parameters grows polynomially in 1/varepsilon. We unlock this polynomial scaling regime by informing the NO's inductive bias by factoring out the singular part of the associated semilinear elliptic PDE Green's function and by incorporating the Dol'eans--Dade exponential of the BSDE's common non-Markovian factor into the NO's decoding layers. As a byproduct, we extend polynomial-scaling guarantees from families of linear elliptic PDEs on regular domains to the semilinear setting.

Key facts. Cited by 2 papers in the map (data through 14 Oct 2026). Builds on 33 of its 76 references that are in the map. First posted on arXiv 18 Oct 2024.

Builds on

33 of 76 references.

References · 33

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  2. 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 527
  3. 2022Fourier Neural Operator with Learned Deformations for PDEs on General Geometriescited by 427
  4. 2023Convolutional Neural Operators for robust and accurate learning of PDEscited by 214
  5. 2021Policy Gradient and Actor-Critic Learning in Continuous Time and Space: Theory and Algorithmscited by 101
  6. 2022q-Learning in Continuous Timecited by 91
  7. 2016Kernels for sequentially ordered datacited by 84
  8. 2018Signature moments to characterize laws of stochastic processescited by 56
  9. 2022Deep Nonparametric Estimation of Operators between Infinite Dimensional Spacescited by 45
  10. 2021Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equationscited by 44
  11. 2023Operator learning with PCA-Net: upper and lower complexity boundscited by 44
  12. 2023Nonlocality and Nonlinearity Implies Universality in Operator Learningcited by 43
21 more
  1. 2021Deep Network Approximation: Achieving Arbitrary Accuracy with Fixed Number of Neuronscited by 41
  2. 2020nPINNs: nonlocal Physics-Informed Neural Networks for a parametrized nonlocal universal Laplacian operator. Algorithms and Applicationscited by 36
  3. 2023Out-of-distributional risk bounds for neural operators with applications to the Helmholtz equationcited by 32
  4. 2020Hamilton-Jacobi Deep Q-Learning for Deterministic Continuous-Time Systems with Lipschitz Continuous Controlscited by 31
  5. 2023The Parametric Complexity of Operator Learningcited by 28
  6. 2021Two-layer neural networks with values in a Banach spacecited by 26
  7. 2020Numerically Solving Parametric Families of High-Dimensional Kolmogorov Partial Differential Equations via Deep Learningcited by 22
  8. 2022Designing Universal Causal Deep Learning Models: The Geometric (Hyper)Transformercited by 22
  9. 2022Designing Universal Causal Deep Learning Models: The Case of Infinite-Dimensional Dynamical Systems from Stochastic Analysiscited by 22
  10. 2022Neural and spectral operator surrogates: unified construction and expression rate boundscited by 21
  11. 2023Global universal approximation of functional input maps on weighted spacescited by 19
  12. 2023Deep Equilibrium Based Neural Operators for Steady-State PDEscited by 19
  13. 2023Globally injective and bijective neural operatorscited by 13
  14. 2024Optimal deep learning of holomorphic operators between Banach spacescited by 12
  15. 2022Singular Value Perturbation and Deep Network Optimizationcited by 11
  16. 2024Operator Learning of Lipschitz Operators: An Information-Theoretic Perspectivecited by 11
  17. 2016A tale of a Principal and many many Agentscited by 7
  18. 2015Dynamic programming approach to principal-agent problemscited by 5
  19. 2024Universal approximation theorem for neural networks with inputs from a topological vector spacecited by 4
  20. 2014Moral Hazard in Dynamic Risk Managementcited by 2
  21. 2016Contracting theory with competitive interacting agentscited by 1

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2026: 2.

2 papers.

  1. 2026Generalization Error Bounds for Picard-Type Operator Learning in Nonlinear Parabolic PDEscited by 0
  2. 2026NeuralChaos: Optimal Adapted Approximation of Square Integrable Predictable Processescited by 0