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

TL;DR

ConDiff is a new synthetic dataset designed to challenge neural PDE solvers by including complex, high-contrast, discontinuous coefficients in parametric diffusion equations, aiming to advance scientific machine learning methods.

Contribution

The paper introduces ConDiff, a diverse, synthetic dataset with high-contrast, discontinuous coefficients for PDEs, facilitating benchmarking and development of neural solvers.

Findings

Baseline deep learning models perform variably on ConDiff.

The dataset covers a wide range of contrast levels and heterogeneity.

ConDiff encourages development of advanced physics-informed neural methods.

Abstract

We present ConDiff, a novel dataset for scientific machine learning. ConDiff focuses on the parametric diffusion equation with space dependent coefficients, a fundamental problem in many applications of 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, ultimately driving progress towards more accurate and efficient solutions of complex PDE problems.