While significant progress has been made on Physics-Informed Neural Networks (PINNs), a comprehensive comparison of these methods across a wide range of Partial Differential Equations (PDEs) is still lacking. This study introduces PINNacle, a benchmarking tool designed to fill this gap. PINNacle provides a diverse dataset, comprising over 20 distinct PDEs from various domains, including heat conduction, fluid dynamics, biology, and electromagnetics. These PDEs encapsulate key challenges inherent to real-world problems, such as complex geometry, multi-scale phenomena, nonlinearity, and high dimensionality. PINNacle also offers a user-friendly toolbox, incorporating about 10 state-of-the-art PINN methods for systematic evaluation and comparison. We have conducted extensive experiments with these methods, offering insights into their strengths and weaknesses. In addition to providing a standardized means of assessing performance, PINNacle also offers an in-depth analysis to guide future research, particularly in areas such as domain decomposition methods and loss reweighting for handling multi-scale problems and complex geometry. To the best of our knowledge, it is the largest benchmark with a diverse and comprehensive evaluation that will undoubtedly foster further research in PINNs.

The development, integration, and maintenance of geospatial databases rely heavily on efficient and accurate matching procedures of Geospatial Entity Resolution (ER). While resolution of points-of-interest (POIs) has been widely addressed, resolution of entities with diverse geometries has been largely overlooked. This is partly due to the lack of a uniform technique for embedding heterogeneous geometries seamlessly into a neural network framework. Existing neural approaches simplify complex geometries to a single point, resulting in significant loss of spatial information. To address this limitation, we propose Omni, a geospatial ER model featuring an omni-geometry encoder. This encoder is capable of embedding point, line, polyline, polygon, and multi-polygon geometries, enabling the model to capture the complex geospatial intricacies of the places being compared. Furthermore, Omni leverages transformer-based pre-trained language models over individual textual attributes of place records in an Attribute Affinity mechanism. The model is rigorously tested on existing point-only datasets and a new diverse-geometry geospatial ER dataset. Omni produces up to 12% (F1) improvement over existing methods. Furthermore, we test the potential of Large Language Models (LLMs) to conduct geospatial ER, experimenting with prompting strategies and learning scenarios, comparing the results of pre-trained language model-based methods with LLMs. Results indicate that LLMs show competitive results.

-- Robotic assembly of complex, non-convex geometries with tight clearances remains a challenging problem, demanding precise state estimation for successful insertion. In this work, we propose a novel framework that relies solely on contact states to estimate the full SE (3) pose of a peg relative to a hole. Our method constructs an online submanifold of contact states through primitive motions with just 6 seconds of online execution, subsequently mapping it to an offline contact manifold for precise pose estimation. We demonstrate that without such state estimation, robots risk jamming and excessive force application, potentially causing damage. We evaluate our approach on five industrially relevant, complex geometries with 0.1 to 1.0 mm clearances, achieving a 96.7% success rate-a 6 improvement over primitive-based insertion without state estimation. Additionally, we analyze insertion forces, and overall insertion times, showing our method significantly reduces the average wrench, enabling safer and more efficient assembly. I. INTRODUCTION The peg-in-hole insertion task is one of the most fundamental problems in robotics. In the realm of contact-rich manipulation and assembly, insertion-based tasks are often framed as peg-in-hole problems.

While significant progress has been made on Physics-Informed Neural Networks (PINNs), a comprehensive comparison of these methods across a wide range of Partial Differential Equations (PDEs) is still lacking. This study introduces PINNacle, a benchmarking tool designed to fill this gap. PINNacle provides a diverse dataset, comprising over 20 distinct PDEs from various domains, including heat conduction, fluid dynamics, biology, and electromagnetics. These PDEs encapsulate key challenges inherent to real-world problems, such as complex geometry, multi-scale phenomena, nonlinearity, and high dimensionality. PINNacle also offers a user-friendly toolbox, incorporating about 10 state-of-the-art PINN methods for systematic evaluation and comparison.

Rapid yet accurate simulations of fluid dynamics around complex geometries is critical in a variety of engineering and scientific applications, including aerodynamics and biomedical flows. However, while scientific machine learning (SciML) has shown promise, most studies are constrained to simple geometries, leaving complex, real-world scenarios underexplored. This study addresses this gap by benchmarking diverse SciML models, including neural operators and vision transformer-based foundation models, for fluid flow prediction over intricate geometries. Using a high-fidelity dataset of steady-state flows across various geometries, we evaluate the impact of geometric representations -- Signed Distance Fields (SDF) and binary masks -- on model accuracy, scalability, and generalization. Central to this effort is the introduction of a novel, unified scoring framework that integrates metrics for global accuracy, boundary layer fidelity, and physical consistency to enable a robust, comparative evaluation of model performance. Our findings demonstrate that foundation models significantly outperform neural operators, particularly in data-limited scenarios, and that SDF representations yield superior results with sufficient training data. Despite these advancements, all models struggle with out-of-distribution generalization, highlighting a critical challenge for future SciML applications. By advancing both evaluation methodologies and modeling capabilities, this work paves the way for robust and scalable ML solutions for fluid dynamics across complex geometries.

Simulating fluid flow around arbitrary shapes is key to solving various engineering problems. However, simulating flow physics across complex geometries remains numerically challenging and computationally resource-intensive, particularly when using conventional PDE solvers. Machine learning methods offer attractive opportunities to create fast and adaptable PDE solvers. However, benchmark datasets to measure the performance of such methods are scarce, especially for flow physics across complex geometries. We introduce FlowBench, a dataset for neural simulators with over 10K samples, which is currently larger than any publicly available flow physics dataset. FlowBench contains flow simulation data across complex geometries (\textit{parametric vs. non-parametric}), spanning a range of flow conditions (\textit{Reynolds number and Grashoff number}), capturing a diverse array of flow phenomena (\textit{steady vs. transient; forced vs. free convection}), and for both 2D and 3D. FlowBench contains over 10K data samples, with each sample the outcome of a fully resolved, direct numerical simulation using a well-validated simulator framework designed for modeling transport phenomena in complex geometries. For each sample, we include velocity, pressure, and temperature field data at 3 different resolutions and several summary statistics features of engineering relevance (such as coefficients of lift and drag, and Nusselt numbers). %Additionally, we include masks and signed distance fields for each shape. We envision that FlowBench will enable evaluating the interplay between complex geometry, coupled flow phenomena, and data sufficiency on the performance of current, and future, neural PDE solvers. We enumerate several evaluation metrics to help rank order the performance of neural PDE solvers. We benchmark the performance of several baseline methods including FNO, CNO, WNO, and DeepONet.

We propose a geometry-to-flow diffusion model that utilizes the input of obstacle shape to predict a flow field past the obstacle. The model is based on a learnable Markov transition kernel to recover the data distribution from the Gaussian distribution. The Markov process is conditioned on the obstacle geometry, estimating the noise to be removed at each step, implemented via a U-Net. A cross-attention mechanism incorporates the geometry as a prompt. We train the geometry-to-flow diffusion model using a dataset of flows past simple obstacles, including the circle, ellipse, rectangle, and triangle. For comparison, the CNN model is trained using the same dataset. Tests are carried out on flows past obstacles with simple and complex geometries, representing interpolation and extrapolation on the geometry condition, respectively. In the test set, challenging scenarios include a cross and characters `PKU'. Generated flow fields show that the geometry-to-flow diffusion model is superior to the CNN model in predicting instantaneous flow fields and handling complex geometries. Quantitative analysis of the model accuracy and divergence in the fields demonstrate the high robustness of the diffusion model, indicating that the diffusion model learns physical laws implicitly.

Controlling hand exoskeletons to assist individuals with grasping tasks poses a challenge due to the difficulty in understanding user intentions. We propose that most daily grasping tasks during activities of daily living (ADL) can be deduced by analyzing object geometries (simple and complex) from 3D point clouds. The study introduces PointGrasp, a real-time system designed for identifying household scenes semantically, aiming to support and enhance assistance during ADL for tailored end-to-end grasping tasks. The system comprises an RGB-D camera with an inertial measurement unit and a microprocessor integrated into a tendon-driven soft robotic glove. The RGB-D camera processes 3D scenes at a rate exceeding 30 frames per second. The proposed pipeline demonstrates an average RMSE of 0.8 $\pm$ 0.39 cm for simple and 0.11 $\pm$ 0.06 cm for complex geometries. Within each mode, it identifies and pinpoints reachable objects. This system shows promise in end-to-end vision-driven robotic-assisted rehabilitation manual tasks.

This article introduces GIT-Net, a deep neural network architecture for approximating Partial Differential Equation (PDE) operators, inspired by integral transform operators. GIT-NET harnesses the fact that differential operators commonly used for defining PDEs can often be represented parsimoniously when expressed in specialized functional bases (e.g., Fourier basis). Unlike rigid integral transforms, GIT-Net parametrizes adaptive generalized integral transforms with deep neural networks. When compared to several recently proposed alternatives, GIT-Net's computational and memory requirements scale gracefully with mesh discretizations, facilitating its application to PDE problems on complex geometries. Numerical experiments demonstrate that GIT-Net is a competitive neural network operator, exhibiting small test errors and low evaluations across a range of PDE problems. This stands in contrast to existing neural network operators, which typically excel in just one of these areas.

Physics-informed neural networks (PINNs) effectively embed physical principles into machine learning, but often struggle with complex or alternating geometries. We propose a novel method for integrating geometric transformations within PINNs to robustly accommodate geometric variations. Our method incorporates a diffeomorphism as a mapping of a reference domain and adapts the derivative computation of the physics-informed loss function. This generalizes the applicability of PINNs not only to smoothly deformed domains, but also to lower-dimensional manifolds and allows for direct shape optimization while training the network. We demonstrate the effectivity of our approach on several problems: (i) Eikonal equation on Archimedean spiral, (ii) Poisson problem on surface manifold, (iii) Incompressible Stokes flow in deformed tube, and (iv) Shape optimization with Laplace operator. Through these examples, we demonstrate the enhanced flexibility over traditional PINNs, especially under geometric variations. The proposed framework presents an outlook for training deep neural operators over parametrized geometries, paving the way for advanced modeling with PDEs on complex geometries in science and engineering.