A.1 Initial Value Problem518 We use the datasets from Pfaff et al. (2021), and take the first and last frames of each trajectory as the519 input and output data for the initial value problem.520 Cylinder The dataset includes computational fluid dynamics (CFD) simulations of the flow around521 a cylinder, governed by the incompressible Navier-Stokes equation. These simulations were generated522 using COMSOL software, employing an irregular 2D-triangular mesh. The trajectory consists of 600523 timestamps, with a time interval of t =0 .01s between each timestamp.524 Airfoil The dataset contains CFD simulations of the flow around an airfoil, following the com-525 pressible Navier-Stokes equation. These simulations were conducted using SU2 software, using an526 irregular 2D-triangular mesh. The trajectory encompasses 600 timestamps, with a time interval of527 t =0 .008s between each timestamp.528 A.2 Dynamics Modeling529 2D-Navier-Stokes (Navier-Stokes) We consider the 2DNavier-Stokes equation as presented in Li530 et al. (2021); Yin et al. (2022).

Differentially private $K$-means clustering enables releasing cluster centers derived from a dataset while protecting the privacy of the individuals. Non-interactive clustering techniques based on privatized histograms are attractive because the released data synopsis can be reused for other downstream tasks without additional privacy loss. The choice of the number of grids for discretizing the data points is crucial, as it directly controls the quantization bias and the amount of noise injected to preserve privacy. The widely adopted strategy selects a grid size that is independent of the number of clusters and also relies on empirical tuning. In this work, we revisit this choice and propose a refined grid-size selection rule derived by minimizing an upper bound on the expected deviation in the K-means objective function, leading to a more principled discretization strategy for non-interactive private clustering. Compared to prior work, our grid resolution differs both in its dependence on the number of clusters and in the scaling with dataset size and privacy budget. Extensive numerical results elucidate that the proposed strategy results in accurate clustering compared to the state-of-the-art techniques, even under tight privacy budgets.

Kernel density estimation is a key component of a wide variety of algorithms in machine learning, Bayesian inference, stochastic dynamics and signal processing. However, the unsupervised density estimation technique requires tuning a crucial hyperparameter: the kernel bandwidth. The choice of bandwidth is critical as it controls the bias-variance trade-off by over- or under-smoothing the topological features. Topological data analysis provides methods to mathematically quantify topological characteristics, such as connected components, loops, voids et cetera, even in high dimensions where visualization of density estimates is impossible. In this paper, we propose an unsupervised learning approach using a topology-based loss function for the automated and unsupervised selection of the optimal bandwidth and benchmark it against classical techniques -- demonstrating its potential across different dimensions.

This paper explores the spatial reasoning capability of large language models (LLMs) over textual input through a suite of five tasks aimed at probing their spatial understanding and computational abilities. The models were tested on both fundamental spatial reasoning and multi-step problem-solving within structured grid-based environments using tasks such as quadrant identification, geometric transformations, distance evaluation, word searches, and tile sliding. Each task was scaled in complexity through increasing grid dimensions, requiring models to extend beyond simple pattern recognition into abstract spatial reasoning. Our results reveal that while LLMs demonstrate moderate success in all tasks with small complexity and size, performance drops off rapidly as scale increases, with an average loss in accuracy of 42.7%, and reaching as high as 84%. Every test that began with over 50% accuracy showed a loss of at least 48%, illustrating the consistent nature of the deterioration. Furthermore, their struggles with scaling complexity hint at a lack of robust spatial representations in their underlying architectures. This paper underscores the gap between linguistic and spatial reasoning in LLMs, offering insights into their current limitations, and laying the groundwork for future integrative benchmarks at the intersection of language and geometry.

AC Optimal Power Flow (ACOPF) is computationally expensive for large-scale power systems, with conventional solvers requiring prohibitive solution times. Machine learning approaches offer computational speedups but struggle with scalability and topology adaptability without expensive retraining. To enable scalability across grid sizes and adaptability to topology changes, we propose a Hybrid Heterogeneous Message Passing Neural Network (HH-MPNN). HH-MPNN models buses, generators, loads, shunts, transmission lines and transformers as distinct node or edge types, combined with a scalable transformer model for handling long-range dependencies. On grids from 14 to 2,000 buses, HH-MPNN achieves less than 1% optimality gap on default topologies. Applied zero-shot to thousands of unseen topologies, HH-MPNN achieves less than 3% optimality gap despite training only on default topologies. Pre-training on smaller grids also improves results on a larger grid. Computational speedups reach 1,000x to 10,000x compared to interior point solvers. These results advance practical, generalizable machine learning for real-time power system operations.