Approximation Rates in Fréchet Metrics: Barron Spaces, Paley-Wiener Spaces, and Fourier Multipliers.

Area of Science:

• Numerical Analysis
• Machine Learning
• Partial Differential Equations

Background:

• Operator learning offers a novel approach to simulating partial differential equations (PDEs) using neural networks.
• This method aims to learn the solution operator of a PDE, creating a neural network that approximates the mapping in infinite-dimensional spaces.

Purpose of the Study:

• To investigate the general approximation capabilities of neural networks for linear differential operators.
• To approximate the operator's symbol in the Fourier domain and analyze the approximation error.

Main Methods:

• Approximation of linear differential operators by their symbols in the Fourier domain.
• Utilizing a topology induced by a sequence of semi-norms, analogous to Hörmander symbols.
• Measuring approximation error using a Fréchet metric.

Main Results:

• Identified sufficient conditions for achieving a predefined approximation error for linear differential operators.
• Extended the main theorem by reducing assumptions on the sequence of seminorms.
• Presented a concrete example of symbols that can be well-approximated, leveraging existing Barron space results.

Conclusions:

• The study provides theoretical foundations for operator learning in simulating PDEs.
• Sufficient conditions for accurate approximation of linear differential operators are established.
• The findings pave the way for more efficient and accurate neural network-based PDE solvers.