ConDiff: A Challenging Dataset for Neural Solvers of Partial Differential Equations
This dataset addresses the need for more realistic benchmarks in scientific machine learning, particularly for researchers working on physics-based deep learning methods, though it is incremental as it builds on existing synthetic datasets.
The authors tackled the challenge of creating a realistic dataset for neural PDE solvers by introducing ConDiff, a synthetic dataset featuring parametric diffusion equations with discontinuous, high-contrast coefficients, and they provided baseline results using standard deep learning models to encourage development of novel approaches.
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.