Quantitative Approximation for Neural Operators in Nonlinear Parabolic Equations
Authors: Takashi Furuya, Koichi Taniguchi, Satoshi Okuda
ICLR 2025 | Venue PDF | Archive PDF | Plain Text | LLM Run Details
| Reproducibility Variable | Result | LLM Response |
|---|---|---|
| Research Type | Theoretical | 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. |
| Researcher Affiliation | Academia | 1Doshisha University, EMAIL 2Shizuoka University, EMAIL 3Rikkyo University, EMAIL |
| Pseudocode | No | The paper provides mathematical definitions, theorems, and proofs but does not include any structured pseudocode or algorithm blocks. |
| Open Source Code | No | This paper does not address the implementation of neural operators, so we briefly comment it here. Our proof based on Picard s iteration is constructive, and this constructive approach may offer valuable insights for future experimental studies, particularly when incorporating constraints specific to the architecture of PDE tasks (for instance, weight-tied architecture discussed in Remark 3). Note that, due to the form of integral equation (P ), our neural operators need to include the integral with respect to time, which may be computationally expensive. However, the previous work Kovachki et al. (2023, Section 7.3) employed the Fourier transforms with respect to time for computing FNOs, whose techniques might be useful for computing our neural operators. We leave the details of experimental studies of our theory in the future works. |
| Open Datasets | No | This paper is a theoretical work focusing on approximation theorems for neural operators and does not involve empirical studies or the use of datasets. |
| Dataset Splits | No | This paper is a theoretical work and does not involve empirical studies or the use of datasets, therefore no dataset splits are mentioned. |
| Hardware Specification | No | This paper is a theoretical work and does not describe any experiments that would require specific hardware specifications. |
| Software Dependencies | No | This paper is a theoretical work and does not describe any implementation details or specific software dependencies with version numbers. |
| Experiment Setup | No | This paper is a theoretical work and does not describe any experiments, training processes, or specific hyperparameters. |