ConDiff: A Challenging Dataset for Neural Solvers of Partial Differential Equations
Vladislav Trifonov, Alexander Rudikov, Oleg Iliev, Yuri M. Laevsky, Ivan Oseledets, Ekaterina Muravleva
TL;DR
ConDiff introduces a synthetic, high-contrast benchmark for parametric PDEs, focusing on a 2D steady diffusion equation with space-dependent coefficients $k(x)=\exp(\phi(x))$ drawn from Gaussian random fields under cubic, exponential, and Gaussian covariances. By generating $24$ PDE realizations with $1000$ training and $200$ test samples each across $64\times64$ and $128\times128$ grids, the authors create $28,800$ total samples to stress neural operators and related surrogates, with complexity controlled by the GRF variance through the contrast $\text{contrast}=\exp(\max\phi-\min\phi)$. They benchmark several architectures (SNO, F-FNO, DilResNet, U-Net) using a relative $L_2$ loss, showing that higher variance and rougher coefficients increase difficulty and affect generalization, while Gaussian GRFs pose the greatest challenge due to ill-conditioning. The dataset is publicly available with accompanying code, enabling reproducible evaluation and encouraging the development of physics-informed neural solvers and neural operators for complex, high-contrast PDEs. Overall, ConDiff addresses the gap between fully synthetic benchmarks and real-world heterogeneous media problems, offering a structured platform for method development and comparison.
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.