Mathematical artificial data for operator learning

TL;DR

The paper introduces MAD, a physics-embedded data generation framework that enhances operator learning for differential equations by eliminating the need for costly training data, thus improving efficiency and accuracy.

Contribution

MAD integrates physical laws with data-driven methods to generate synthetic solutions, enabling scalable and rigorous operator learning without experimental data.

Findings

Demonstrates MAD's effectiveness on 2D parametric problems

Shows superior efficiency and accuracy over existing methods

Validates generalizability across various differential equation scenarios

Abstract

Machine learning has emerged as a transformative tool for solving differential equations (DEs), yet prevailing methodologies remain constrained by dual limitations: data-driven methods demand costly labeled datasets while model-driven techniques face efficiency-accuracy trade-offs. We present the Mathematical Artificial Data (MAD) framework, a new paradigm that integrates physical laws with data-driven learning to facilitate large-scale operator discovery. By exploiting DEs' intrinsic mathematical structure to generate physics-embedded analytical solutions and associated synthetic data, MAD fundamentally eliminates dependence on experimental or simulated training data. This enables computationally efficient operator learning across multi-parameter systems while maintaining mathematical rigor. Through numerical demonstrations spanning 2D parametric problems where both the boundary values and source term are functions, we showcase MAD's generalizability and superior efficiency/accuracy across various DE scenarios. This physics-embedded-data-driven framework and its capacity to handle complex parameter spaces gives it the potential to become a universal paradigm for physics-informed machine intelligence in scientific computing.