SPDEBench: An Extensive Benchmark for Learning Regular and Singular Stochastic PDEs

Stochastic Partial Differential Equations (SPDEs) driven by random noise play a central role in modelling physical processes whose spatio-temporal dynamics can be rough, such as turbulence flows, superconductors, and quantum dynamics. To efficiently model these processes and make predictions, machine learning (ML)-based surrogate models are proposed, with their network architectures incorporating the spatio-temporal roughness in their design. However, it lacks an extensive and unified datasets for SPDE learning; especially, existing datasets do not account for the computational error introduced by noise sampling and the necessary renormalization required for handling singular SPDEs. We thus introduce SPDEBench, which is designed to solve typical SPDEs of physical significance (e.g., the $\Phi^4_d$, wave, incompressible Navier--Stokes, and KdV equations) on 1D or 2D tori driven by white noise via ML methods. New datasets for singular SPDEs based on the renormalization process have been constructed, and novel ML models achieving the best results to date have been proposed. In particular, we investigate the impact of computational error introduced by noise sampling and renormalization on the performance comparison of ML models and highlight the importance of selecting high-quality test data for accurate evaluation. Results are benchmarked with traditional numerical solvers and ML-based models, including FNO, NSPDE and DLR-Net, etc. It is shown that, for singular SPDEs, naively applying ML models on data without specifying the numerical schemes can lead to significant errors and misleading conclusions. Our SPDEBench provides an open-source codebase that ensures full reproducibility of benchmarking across a variety of SPDE datasets while offering the flexibility to incorporate new datasets and machine learning baselines, making it a valuable resource for the community.

Training-Free Hierarchical Scene Understanding for Gaussian Splatting with Superpoint Graphs

Bridging natural language and 3D geometry is a crucial step toward flexible, language-driven scene understanding. While recent advances in 3D Gaussian Splatting (3DGS) have enabled fast and high-quality scene reconstruction, research has also explored incorporating open-vocabulary understanding into 3DGS. However, most existing methods require iterative optimization over per-view 2D semantic feature maps, which not only results in inefficiencies but also leads to inconsistent 3D semantics across views. To address these limitations, we introduce a training-free framework that constructs a superpoint graph directly from Gaussian primitives. The superpoint graph partitions the scene into spatially compact and semantically coherent regions, forming view-consistent 3D entities and providing a structured foundation for open-vocabulary understanding. Based on the graph structure, we design an efficient reprojection strategy that lifts 2D semantic features onto the superpoints, avoiding costly multi-view iterative training. The resulting representation ensures strong 3D semantic coherence and naturally supports hierarchical understanding, enabling both coarse- and fine-grained open-vocabulary perception within a unified semantic field. Extensive experiments demonstrate that our method achieves state-of-the-art open-vocabulary segmentation performance, with semantic field reconstruction completed over $30\times$ faster. Our code will be available at https://github.com/Atrovast/THGS.