arXiv subjects

Lujun Zhang

Publications and source records attributed to Lujun Zhang.

5 recordsLinked to original sources

arXiv2026

Stability of Type A Mirkovi\'c-Vilonen Polytopes under Minkowski Sum via Weak Separation

Mirkovi\'c--Vilonen (MV) polytopes play a key role in the representation theory of reductive algebraic groups, while the geometric behavior of prime MV polytopes under Minkowski addition remains a subtle open problem. This paper focuses on type A and regards Schubert matroid polytopes as fundamental prime MV building blocks. Using the crystal structure on MV polytopes, we strengthen Sanchez's compatibility condition and establish a necessary and sufficient condition: the positive Minkowski sum of such polytopes is again an MV polytope precisely when the indexing family is weakly separated. Working within discrete convex analysis, we relate discrete concave tropical Pl\"ucker functions to concave extensions on the hypercube and the resulting generalized matroid subdivisions, showing that weak separation is equivalent to the stability of these subdivisions under common refinement. We further clarify the intrinsic connection between our subdivision constructions and the hypersimplex matroid subdivisions developed by Early, providing a natural flag-type generalization of his classical results. We briefly discuss generalized positroids and generalized polypositroids, and identify the MV fan $\mathcal{MV}$ as the secondary fan of hypercube generalized positroid subdivisions. Accordingly, maximal weakly separated sets correspond to maximal cones in $\mathcal{MV}$ and produce the finest such subdivisions. This work unifies MV polytope theory with tropical matroid geometry, advances the understanding of compatibility phenomena in MV combinatorics, and offers new perspectives at the interface of representation theory and combinatorics.

math.RT↗

arXiv2025

HydroDiffusion: Diffusion-Based Probabilistic Streamflow Forecasting with a State Space Backbone

Recent advances have introduced diffusion models for probabilistic streamflow forecasting, demonstrating strong early flood-warning skill. However, current implementations rely on recurrent Long Short-Term Memory (LSTM) backbones and single-step training objectives, which limit their ability to capture long-range dependencies and produce coherent forecast trajectories across lead times. To address these limitations, we developed HydroDiffusion, a diffusion-based probabilistic forecasting framework with a decoder-only state space model backbone. The proposed framework jointly denoises full multi-day trajectories in a single pass, ensuring temporal coherence and mitigating error accumulation common in autoregressive prediction. HydroDiffusion is evaluated across 531 watersheds in the contiguous United States (CONUS) in the CAMELS dataset. We benchmark HydroDiffusion against two diffusion baselines with LSTM backbones, as well as the recently proposed Diffusion-based Runoff Model (DRUM). Results show that HydroDiffusion achieves strong nowcast accuracy when driven by observed meteorological forcings, and maintains consistent performance across the full simulation horizon. Moreover, HydroDiffusion delivers stronger deterministic and probabilistic forecast skill than DRUM in operational forecasting. These results establish HydroDiffusion as a robust generative modeling framework for medium-range streamflow forecasting, providing both a new modeling benchmark and a foundation for future research on probabilistic hydrologic prediction at continental scales.

cs.LG↗

arXiv2025

Finest positroid subdivisions from maximal weakly separated collections

We adopt a formal and algebraic approach of Early \cite{E2} to study the positive tropical Grassmannian $\operatorname{Trop}^+ Gr_{k,n}$. Specifically, we deal with positroid subdivision of hypersimplex induced by translated blades from any maximal weakly separated collection. One of our main results gives a necessary and sufficient condition on a maximal weakly separated collection to form a positroid subdivision of a hypersimplex corresponding to a simplicial cone in $\rm Trop^+Gr_{k,n}$. For k = 2 our condition says that any weakly separated collection of two-elements sets gives such a simplicial cone, and all cones are of such a form. We also show that the maximality of any weakly separated collection is preserved under the boundary map, which armatively answers a question by Early in \cite{E1}. Plabic graphs, invented by Postnikov \cite{P}, are of use in proving this result. As a corollary, we get that all those positroid subdivisions are the finest. Thus, the flip of two maximal weakly separatedcollections corresponds to a pair of adjacent maximal cones in positive tropical Grassmannian.

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arXiv2025

A Deep State Space Model for Rainfall-Runoff Simulations

The classical way of studying the rainfall-runoff processes in the water cycle relies on conceptual or physically-based hydrologic models. Deep learning (DL) has recently emerged as an alternative and blossomed in hydrology community for rainfall-runoff simulations. However, the decades-old Long Short-Term Memory (LSTM) network remains the benchmark for this task, outperforming newer architectures like Transformers. In this work, we propose a State Space Model (SSM), specifically the Frequency Tuned Diagonal State Space Sequence (S4D-FT) model, for rainfall-runoff simulations. The proposed S4D-FT is benchmarked against the established LSTM and a physically-based Sacramento Soil Moisture Accounting model across 531 watersheds in the contiguous United States (CONUS). Results show that S4D-FT is able to outperform the LSTM model across diverse regions. Our pioneering introduction of the S4D-FT for rainfall-runoff simulations challenges the dominance of LSTM in the hydrology community and expands the arsenal of DL tools available for hydrological modeling.

cs.LG↗

arXiv2024

Using Long Short-term Memory (LSTM) to merge precipitation data over mountainous area in Sierra Nevada

Obtaining reliable precipitation estimation with high resolutions in time and space is of great importance to hydrological studies. However, accurately estimating precipitation is a challenging task over high mountainous complex terrain. The three widely used precipitation measurement approaches, namely rainfall gauge, precipitation radars, and satellite-based precipitation sensors, have their own pros and cons in producing reliable precipitation products over complex areas. One way to decrease the detection error probability and improve data reliability is precipitation data merging. With the rapid advancements in computational capabilities and the escalating volume and diversity of earth observational data, Deep Learning (DL) models have gained considerable attention in geoscience. In this study, a deep learning technique, namely Long Short-term Memory (LSTM), was employed to merge a radar-based and a satellite-based Global Precipitation Measurement (GPM) precipitation product Integrated Multi-Satellite Retrievals for GPM (IMERG) precipitation product at hourly scale. The merged results are compared with the widely used reanalysis precipitation product, Multi-Radar Multi-Sensor (MRMS), and assessed against gauge observational data from the California Data Exchange Center (CDEC). The findings indicated that the LSTM-based merged precipitation notably underestimated gauge observations and, at times, failed to provide meaningful estimates, showing predominantly near-zero values. Relying solely on individual Quantitative Precipitation Estimates (QPEs) without additional meteorological input proved insufficient for generating reliable merged QPE. However, the merged results effectively captured the temporal trends of the observations, outperforming MRMS in this aspect. This suggested that incorporating bias correction techniques could potentially enhance the accuracy of the merged product.

cs.LG↗