mlsanityData to 7 Oct 2026

Quantitative Approximation for Neural Operators in Nonlinear Parabolic Equations

arXiv 2410.02151 ↗First posted 3 Oct 2024cs.LG cs.NA math.NAcited by 5

Not on the map. Why?

Abstract (arXiv)

Neural operators serve as universal approximators for general continuous operators. In this paper, we derive the approximation rate of solution operators for the nonlinear parabolic partial differential equations (PDEs), contributing to the quantitative approximation theorem for solution operators of nonlinear PDEs. Our results show that neural operators can efficiently approximate these solution operators without the exponential growth in model complexity, thus strengthening the theoretical foundation of neural operators. A key insight in our proof is to transfer PDEs into the corresponding integral equations via Duahamel's principle, and to leverage the similarity between neural operators and Picard's iteration, a classical algorithm for solving PDEs. This approach is potentially generalizable beyond parabolic PDEs to a range of other equations, including the Navier-Stokes equation, nonlinear Schr"odinger equations and nonlinear wave equations, which can be solved by Picard's iteration.

Key facts. Cited by 5 papers in the map (data through 7 Oct 2026). Builds on 15 of its 50 references that are in the map. First posted on arXiv 3 Oct 2024.

Builds on

15 of 50 references.

References · 15

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  2. 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 722
  3. 2019Deep Equilibrium Modelscited by 603
  4. 2022PDEBENCH: An Extensive Benchmark for Scientific Machine Learningcited by 396
  5. 2020Model Reduction and Neural Networks for Parametric PDEscited by 257
  6. 2021Multiwavelet-based Operator Learning for Differential Equationscited by 229
  7. 2023Spherical Fourier Neural Operators: Learning Stable Dynamics on the Spherecited by 225
  8. 2019Error bounds for approximations with deep ReLU neural networks in Ws,p normscited by 95
  9. 2024Operator Learning: Algorithms and Analysiscited by 84
  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. 2023Physics informed WNOcited by 26
3 more
  1. 2023Deep Equilibrium Based Neural Operators for Steady-State PDEscited by 19
  2. 2023Deep Operator Learning Lessens the Curse of Dimensionality for PDEscited by 18
  3. 2024Mixtures of Neural Operators Reduce Active Complexity in Operator Learningcited by 16

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2025: 3; 2026: 2.

5 papers.

  1. 2025One model to solve them all: 2BSDE families via neural operatorscited by 3
  2. 2025Reformulating Neural Operators in d+1 Dimensions for Embedding Evolutioncited by 0
  3. 2025Why and When Deep is Better than Shallow: Implementation-Agnostic State-Transition Model of Deep Learningcited by 0
  4. 2026Generalization Error Bounds for Picard-Type Operator Learning in Nonlinear Parabolic PDEscited by 0
  5. 2026Approximation Theory of Laplacian-Based Neural Operators for Reaction-Diffusion Systemcited by 0