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.

我们提出了 ConDiff，这是一个新颖的科学机器学习数据集，该数据集的主题是具有可变系数的扩散方程，这是参数化偏微分方程的许多应用中的一个基本问题。

ConDiff: 一个用于偏微分方程的神经求解器的挑战数据集

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

Inferring a diffusion equation from discretely-observed measurements is a
statistical challenge of significant importance in a variety of fields, from
single-molecule tracking in biophysical systems to modeling financial
instruments. Assuming that the underlying dynamical process obeys a
$d$-dimensional stochastic differential equation of the form
$$\mathrm{d}\boldsymbol{x}_t=\boldsymbol{b}(\boldsymbol{x}_t)\mathrm{d}
t+\Sigma(\boldsymbol{x}_t)\mathrm{d}\boldsymbol{w}_t,$$ we propose neural
network-based estimators of both the drift $\boldsymbol{b}$ and the
spatially-inhomogeneous diffusion tensor $D = \Sigma\Sigma^{T}$ and provide
statistical convergence guarantees when $\boldsymbol{b}$ and $D$ are
$s$-H\"older continuous. Notably, our bound aligns with the minimax optimal
rate $N^{-\frac{2s}{2s+d}}$ for nonparametric function estimation even in the
presence of correlation within observational data, which necessitates careful
handling when establishing fast-rate generalization bounds. Our theoretical
results are bolstered by numerical experiments demonstrating accurate inference
of spatially-inhomogeneous diffusion tensors.

从离散观测数据中推算出扩散方程是一个在各个领域中具有重要统计意义的挑战，我们提出了使用基于神经网络的估计方法，对漂移项和空间非均匀扩散张量进行估计，并在漂移项和扩散张量连续可微的情况下提供了统计收敛保证，我们的界限与非参数函数估计的极小极限速率 $N^{-rac {2s}{2s+d}}$ 一致，即使存在观测数据内的相关性，这需要在建立快速泛化界限时进行仔细处理，我们的理论结果得到通过数值实验证明了对空间非均匀扩散张量的准确推断。