The application of deep learning for partial differential equation (PDE)-constrained control is gaining increasing attention. However, existing methods rarely consider safety requirements crucial in real-world applications. To address this limitation, we propose Safe Diffusion Models for PDE Control (SafeDiffCon), which introduce the uncertainty quantile as model uncertainty quantification to achieve optimal control under safety constraints through both post-training and inference phases. Firstly, our approach post-trains a pre-trained diffusion model to generate control sequences that better satisfy safety constraints while achieving improved control objectives via a reweighted diffusion loss, which incorporates the uncertainty quantile estimated using conformal prediction. Secondly, during inference, the diffusion model dynamically adjusts both its generation process and parameters through iterative guidance and fine-tuning, conditioned on control targets while simultaneously integrating the estimated uncertainty quantile. We evaluate SafeDiffCon on three control tasks: 1D Burgers’ equation, 2D incompressible fluid, and controlled nuclear fusion problem. Results demonstrate that SafeDiffCon is the only method that satisfies all safety constraints, whereas other classical and deep learning baselines fail. Furthermore, while adhering to safety constraints, SafeDiffCon achieves the best control performance. The code can be found at https://github.com/AI4Science-WestlakeU/safediffcon.

Multi-relational clustering is a challenging task due to the fact that diverse semantic information conveyed in multi-layer graphs is difficult to extract and fuse. Recent methods integrate topology structure and node attribute information through graph filtering. However, they often use a low-pass filter without fully considering the correlation among multiple graphs. To overcome this drawback, we propose to learn a graph filter motivated by the theoretical analysis of Barlow Twins. We find that input with a negative semi-definite inner product provides a lower bound for Barlow Twins loss, which prevents it from reaching a better solution. We thus learn a filter that yields an upper bound for Barlow Twins. Afterward, we design a simple clustering architecture and demonstrate its state-of-the-art performance on four benchmark datasets. The source code is available at https://github.com/XweiQ/BTGF.

Fair graph learning plays a pivotal role in numerous practical applications. Recently, many fair graph learning methods have been proposed; however, their evaluation often relies on poorly constructed semi-synthetic datasets or substandard real-world datasets. In such cases, even a basic Multilayer Perceptron (MLP) can outperform Graph Neural Networks (GNNs) in both utility and fairness. In this work, we illustrate that many datasets fail to provide meaningful information in the edges, which may challenge the necessity of using graph structures in these problems. To address these issues, we develop and introduce a collection of synthetic, semi-synthetic, and real-world datasets that fulfill a broad spectrum of requirements. These datasets are thoughtfully designed to include relevant graph structures and bias information crucial for the fair evaluation of models. The proposed synthetic and semi-synthetic datasets offer the flexibility to create data with controllable bias parameters, thereby enabling the generation of desired datasets with user-defined bias values with ease. Moreover, we conduct systematic evaluations of these proposed datasets and establish a unified evaluation approach for fair graph learning models. Our extensive experimental results with fair graph learning methods across our datasets demonstrate their effectiveness in benchmarking the performance of these methods. Our datasets and the code for reproducing our experiments are available at https://github.com/XweiQ/Benchmark-GraphFairness.