On universal approximation and error bounds for Fourier Neural Operators (2107.07562v1)

Abstract: Fourier neural operators (FNOs) have recently been proposed as an effective framework for learning operators that map between infinite-dimensional spaces. We prove that FNOs are universal, in the sense that they can approximate any continuous operator to desired accuracy. Moreover, we suggest a mechanism by which FNOs can approximate operators associated with PDEs efficiently. Explicit error bounds are derived to show that the size of the FNO, approximating operators associated with a Darcy type elliptic PDE and with the incompressible Navier-Stokes equations of fluid dynamics, only increases sub (log)-linearly in terms of the reciprocal of the error. Thus, FNOs are shown to efficiently approximate operators arising in a large class of PDEs.

• The paper establishes that FNOs serve as universal approximators for continuous operators on Sobolev spaces.
• The paper derives explicit error bounds, showing that network size increases only sub-logarithmically with decreasing error.
• The paper demonstrates FNOs' computational efficiency in solving PDEs, including Darcy-type and Navier-Stokes equations.

Overview of the Paper "On Universal Approximation and Error Bounds for Fourier Neural Operators"

This paper meticulously investigates the theoretical foundation of Fourier Neural Operators (FNOs), presenting them as a framework for learning operators in infinite-dimensional spaces. Unlike traditional neural networks, which primarily focus on mappings between finite-dimensional spaces, FNOs are designed to handle mappings such as those between functional spaces typically encountered in the context of partial differential equations (PDEs).

Universal Approximation and Error Bounds

The authors establish that FNOs are universal approximators for continuous operators that map between infinite-dimensional spaces. The paper begins with a proof of the universal approximation theorem for FNOs, showing their capability to approximate any continuous operator on compact subsets of a Sobolev space $H^s$. Through detailed mathematical derivations, the authors show that FNOs leverage the Fourier space's properties to efficiently approximate functions, making the universal approximation property extend beyond finite-dimensional mappings.

Additionally, the paper addresses the computational efficiency of FNOs by deriving explicit error bounds for their application in solving PDEs, specifically a Darcy type elliptic PDE and the incompressible Navier-Stokes equations of fluid dynamics. The authors demonstrate that the network size required to achieve a particular approximation error only increases sub-logarithmically with the reciprocal of the error, highlighting the efficiency of FNOs for these types of problems.

Implications for PDEs and Computational Efficiency

The paper provides a bridge between the theoretical capabilities of FNOs and their practical utility in scientific computing. The analysis of FNOs emulating pseudo-spectral methods offers insights into how they can be effectively utilized for efficient numerical solutions of PDEs, such as those constituting models of fluid dynamics.

The implications of this research are profound for fields where the modeling and simulation of natural phenomena via PDEs are integral. The theoretical results open up pathways for more computationally feasible simulations that rely on learning operators from data, thus pushing the performance boundaries of simulations in engineering, physics, and applied sciences.

Comparison with DeepONets and Future Directions

The paper also compares FNOs to DeepONets, another operator learning framework, illustrating how FNOs can be perceived as specialized DeepONets that harness Fourier representations to achieve their operator learning task. This specialized approach provides a more structured and potentially computationally lean method for operator approximation, suggesting that FNOs may offer advantages in cases where the basis functions that can exploit underlying periodicity are suitable.

As the field moves forward, the focus on leveraging FNOs beyond PDE-centric applications could further extend their theoretical and practical utility. Researchers could explore the synergy between FNOs and data-driven approaches or investigate their adaptability in non-scientific domains such as image and signal processing. This paper lays a foundational work that could inspire future advancements in the field of operator learning.