Zhi-Feng Wei

Zhi-Feng Wei, Saad Qadeer, Panos Stinis

Reduced-order modeling of high-dimensional dynamical systems is often hindered by the non-Markovian closure term that represents the effect of unresolved variables on the resolved dynamics. Inspired by the Mori--Zwanzig formalism, in which the closure takes the form of a memory functional of the resolved trajectory, we recast closure modeling as a sequence modeling problem and propose the Mamba-Assisted Closure (MAC) framework: a Mamba-based sequence model, trained to predict the closure from the resolved trajectory, is coupled with the reduced-order governing equations through a numerical integrator to advance the resolved variables in time. A key feature of the framework is its exploitation of the dual representation of state-space models -- the model is trained in a sequence-to-sequence fashion via the convolutional form, and deployed for step-by-step autoregressive rollout via the recurrent form, yielding both efficient long-trajectory training and constant per-step inference cost. On the viscous Burgers' equation and the chaotic two-scale Lorenz '96 system, the MAC model substantially outperforms the Markovian reduced-order model, the GRU-based sequence model, and the Wilks method in predictive accuracy and long-time rollout stability.

Zhi-Feng Wei, Pablo Moriano, Ramakrishnan Kannan

This study investigates the robustness of graph embedding methods for community detection in the face of network perturbations, specifically edge deletions. Graph embedding techniques, which represent nodes as low-dimensional vectors, are widely used for various graph machine learning tasks due to their ability to capture structural properties of networks effectively. However, the impact of perturbations on the performance of these methods remains relatively understudied. The research considers state-of-the-art graph embedding methods from two families: matrix factorization (e.g., LE, LLE, HOPE, M-NMF) and random walk-based (e.g., DeepWalk, LINE, node2vec). Through experiments conducted on both synthetic and real-world networks, the study reveals varying degrees of robustness within each family of graph embedding methods. The robustness is found to be influenced by factors such as network size, initial community partition strength, and the type of perturbation. Notably, node2vec and LLE consistently demonstrate higher robustness for community detection across different scenarios, including networks with degree and community size heterogeneity. These findings highlight the importance of selecting an appropriate graph embedding method based on the specific characteristics of the network and the task at hand, particularly in scenarios where robustness to perturbations is crucial.

Zhi-Feng Wei, Wenqian Chen, Panos Stinis

Operator learning has emerged as a promising tool for accelerating the solution of partial differential equations (PDEs). The Deep Operator Networks (DeepONets) represent a pioneering framework in this area: the "vanilla" DeepONet is valued for its simplicity and efficiency, while the modified DeepONet achieves higher accuracy at the cost of increased training time. In this work, we propose a series of Transformer-inspired DeepONet variants that introduce bidirectional cross-conditioning between the branch and trunk networks in DeepONet. Query-point information is injected into the branch network and input-function information into the trunk network, enabling dynamic dependencies while preserving the simplicity and efficiency of the "vanilla" DeepONet in a non-intrusive manner. Experiments on four PDE benchmarks -- advection, diffusion-reaction, Burgers', and Korteweg-de Vries equations -- show that for each case, there exists a variant that matches or surpasses the accuracy of the modified DeepONet while offering improved training efficiency. Moreover, the best-performing variant for each equation aligns naturally with the equation's underlying characteristics, suggesting that the effectiveness of cross-conditioning depends on the characteristics of the equation and its underlying physics. To ensure robustness, we validate the effectiveness of our variants through a range of rigorous statistical analyses, among them the Wilcoxon Two One-Sided Test, Glass's Delta, and Spearman's rank correlation.