# Quantitative Approximation for Neural Operators in Nonlinear Parabolic Equations

arXiv: https://arxiv.org/abs/2410.02151 · HTML: https://mlsanity.com/paper/2410.02151/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 50 references that are in the map.
- First posted on arXiv 3 Oct 2024.

## Builds on (most cited first, 15 of 15)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [PDEBENCH: An Extensive Benchmark for Scientific Machine Learning](https://mlsanity.com/paper/2210.07182/) (2022)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [Multiwavelet-based Operator Learning for Differential Equations](https://mlsanity.com/paper/2109.13459/) (2021)
- [Spherical Fourier Neural Operators: Learning Stable Dynamics on the Sphere](https://mlsanity.com/paper/2306.03838/) (2023)
- [Error bounds for approximations with deep ReLU neural networks in Ws,p norms](https://mlsanity.com/paper/1902.07896/) (2019)
- [Operator Learning: Algorithms and Analysis](https://mlsanity.com/paper/2402.15715/) (2024)
- [Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equations](https://mlsanity.com/paper/2112.08125/) (2021)
- [Operator learning with PCA-Net: upper and lower complexity bounds](https://mlsanity.com/paper/2303.16317/) (2023)
- [Physics informed WNO](https://mlsanity.com/paper/2302.05925/) (2023)
- [Deep Equilibrium Based Neural Operators for Steady-State PDEs](https://mlsanity.com/paper/2312.00234/) (2023)
- [Deep Operator Learning Lessens the Curse of Dimensionality for PDEs](https://mlsanity.com/paper/2301.12227/) (2023)
- [Mixtures of Neural Operators Reduce Active Complexity in Operator Learning](https://mlsanity.com/paper/2404.09101/) (2024)

## Built on by (most cited first, 5 of 5)

- [One model to solve them all: 2BSDE families via neural operators](https://mlsanity.com/paper/2511.01125/) (2025)
- [Reformulating Neural Operators in d+1 Dimensions for Embedding Evolution](https://mlsanity.com/paper/2505.11766/) (2025)
- [Why and When Deep is Better than Shallow: Implementation-Agnostic State-Transition Model of Deep Learning](https://mlsanity.com/paper/2505.15064/) (2025)
- [Generalization Error Bounds for Picard-Type Operator Learning in Nonlinear Parabolic PDEs](https://mlsanity.com/paper/2605.10277/) (2026)
- [Approximation Theory of Laplacian-Based Neural Operators for Reaction-Diffusion System](https://mlsanity.com/paper/2605.12025/) (2026)

## 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.