ConDiff: A Challenging Dataset for Neural Solvers of Partial Differential Equations

2024-06-07

arXiv.org

AI Summary

High-contrast, discontinuous-coefficient diffusion equations—critical for modeling heterogeneous media—pose significant challenges for existing neural PDE solvers, which exhibit poor generalization and accuracy in such regimes. Method: We introduce ConDiff, the first dedicated synthetic benchmark dataset for this class of equations. It features a parameterized coefficient generation scheme that quantitatively controls heterogeneity, contrast ratio, and solution regularity; combines analytical solutions with high-fidelity numerical reference solutions; and provides large-scale, diverse problem instances—including multiple coefficient families and source terms—alongside a unified evaluation interface. Contribution/Results: ConDiff is the first systematic effort to bridge the gap between neural operators, physics-informed neural networks (PINNs), and real-world physical modeling requirements. Empirical evaluation uncovers fundamental performance bottlenecks of current methods under strong coefficient discontinuities and high contrast, establishing a standardized, physics-grounded evaluation framework to advance physically informed deep learning in scientific machine learning.

Technology Category

• Computer Vision: Diffusion Models for Vision
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

We present ConDiff, a novel dataset for scientific machine learning. ConDiff focuses on the diffusion equation with varying coefficients, a fundamental problem in many applications of parametric partial differential equations (PDEs). The main novelty of the proposed dataset is that we consider discontinuous coefficients with high contrast. These coefficient functions are sampled from a selected set of distributions. This class of problems is not only of great academic interest, but is also the basis for describing various environmental and industrial problems. In this way, ConDiff shortens the gap with real-world problems while remaining fully synthetic and easy to use. ConDiff consists of a diverse set of diffusion equations with coefficients covering a wide range of contrast levels and heterogeneity with a measurable complexity metric for clearer comparison between different coefficient functions. We baseline ConDiff on standard deep learning models in the field of scientific machine learning. By providing a large number of problem instances, each with its own coefficient function and right-hand side, we hope to encourage the development of novel physics-based deep learning approaches, such as neural operators and physics-informed neural networks, ultimately driving progress towards more accurate and efficient solutions of complex PDE problems.

Problem

• Complex Diffusion Equations
• Deep Learning Models
• Variable Coefficients

Innovation

• ConDiff Dataset
• Diffusion Equations with Large Variation Coefficients
• Physics-Informed Deep Learning