Mathematical artificial data for operator learning

2025-07-09

AI Summary

In operator learning for differential equations, data-driven methods rely on costly labeled data, while model-driven approaches struggle to balance computational efficiency and solution accuracy. To address these limitations, this paper proposes the Mathematical Artificial Data (MAD) framework. MAD leverages the intrinsic physical structure of differential equations—employing symbolic computation to generate parametric analytical solutions—and constructs physically consistent synthetic function datasets, eliminating dependence on experimental or numerical simulation data entirely. By tightly integrating data-driven learning with physics-informed modeling, MAD establishes a novel data-free training paradigm. Evaluated on challenging 2D multi-parameter problems featuring functional boundary conditions and source terms, MAD demonstrates significantly improved generalization capability and computational efficiency, achieving superior operator approximation accuracy compared to state-of-the-art baseline methods.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Learning to Search
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

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.

Problem

• Overcoming costly labeled data needs in differential equation solving
• Bridging efficiency-accuracy gaps in model-driven operator learning
• Enabling scalable physics-informed operator discovery without experimental data

Innovation

• Integrates physical laws with data-driven learning
• Generates physics-embedded analytical synthetic data
• Eliminates dependence on experimental training data