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A PDE Perspective on Approximating Nonlocal Periodic Operators with Applications on Neural Networks for Critical SQG Equations

Elie Abdo, Ruimeng Hu, Quyuan Lin

TL;DR

This paper investigates the behavior of the periodic first-order fractional Laplacian and Riesz transform when acting on nonperiodic functions, thereby initiating a new PDE theory for approximating solutions to equations with nonlocalities using neural networks.

Abstract

Nonlocal periodic operators in partial differential equations (PDEs) pose challenges in constructing neural network solutions, which typically lack periodic boundary conditions. In this paper, we introduce a novel PDE perspective on approximating these nonlocal periodic operators. Specifically, we investigate the behavior of the periodic first-order fractional Laplacian and Riesz transform when acting on nonperiodic functions, thereby initiating a new PDE theory for approximating solutions to equations with nonlocalities using neural networks. Moreover, we derive quantitative Sobolev estimates and utilize them to rigorously construct neural networks that approximate solutions to the two-dimensional periodic critically dissipative Surface Quasi-Geostrophic (SQG) equation.

A PDE Perspective on Approximating Nonlocal Periodic Operators with Applications on Neural Networks for Critical SQG Equations

TL;DR

This paper investigates the behavior of the periodic first-order fractional Laplacian and Riesz transform when acting on nonperiodic functions, thereby initiating a new PDE theory for approximating solutions to equations with nonlocalities using neural networks.

Abstract

Nonlocal periodic operators in partial differential equations (PDEs) pose challenges in constructing neural network solutions, which typically lack periodic boundary conditions. In this paper, we introduce a novel PDE perspective on approximating these nonlocal periodic operators. Specifically, we investigate the behavior of the periodic first-order fractional Laplacian and Riesz transform when acting on nonperiodic functions, thereby initiating a new PDE theory for approximating solutions to equations with nonlocalities using neural networks. Moreover, we derive quantitative Sobolev estimates and utilize them to rigorously construct neural networks that approximate solutions to the two-dimensional periodic critically dissipative Surface Quasi-Geostrophic (SQG) equation.
Paper Structure (8 sections, 7 theorems, 94 equations)

This paper contains 8 sections, 7 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Functional Spaces
  3. Properties of the Operators $\tilde{\Lambda}$ and $\tilde{R}$
  4. Applications to the Critical SQG Equation
  5. Physics-Informed Neural Networks (PINNs)
  6. Error Estimates
  7. Conclusion
  8. Regularity of Solutions to the SQG Equation

Key Result

Theorem 3.1

The operator $\tilde{\Lambda}$ defined in def:tildelambda satisfies the following properties:

Theorems & Definitions (13)

  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Lemma 4.1: hu2023higher
  • Theorem 4.2
  • proof
  • Remark 1
  • Theorem 4.3
  • proof
  • ...and 3 more