This paper presents an analytical solution to the inverse kinematic problem(IKP) for the seven degree-of-freedom (7-DOF) Moz1 Robot Arm with offsets on wrist. We provide closed-form solutions with the novel arm angle . It also provides information on how the redundancy is resolved in a new arm angle representation where traditional SEW angle faied to be defined and how singularities are handled. The solution is simple, fast and exact, providing full solution space (i.e. Research on light-weight redundant manipulators, has grown in various directions like human robot interaction [1] or machine learning [2].

We present HyPINO, a multi-physics neural operator designed for zero-shot generalization across a broad class of PDEs without requiring task-specific fine-tuning. Our approach combines a Swin Transformer-based hypernetwork with mixed supervision: (i) labeled data from analytical solutions generated via the Method of Manufactured Solutions (MMS), and (ii) unlabeled samples optimized using physics-informed objectives. The model maps PDE parameterizations to target Physics-Informed Neural Networks (PINNs) and can handle linear elliptic, hyperbolic, and parabolic equations in two dimensions with varying source terms, geometries, and mixed Dirichlet/Neumann boundary conditions, including interior boundaries. HyPINO achieves strong zero-shot accuracy on seven benchmark problems from PINN literature, outperforming U-Nets, Poseidon, and Physics-Informed Neural Operators (PINO). Further, we introduce an iterative refinement procedure that treats the residual of the generated PINN as "delta PDE" and performs another forward pass to generate a corrective PINN. Summing their contributions and repeating this process forms an ensemble whose combined solution progressively reduces the error on six benchmarks and achieves a >100x lower $L_2$ loss in the best case, while retaining forward-only inference. Additionally, we evaluate the fine-tuning behavior of PINNs initialized by HyPINO and show that they converge faster and to lower final error than both randomly initialized and Reptile-meta-learned PINNs on five benchmarks, performing on par on the remaining two. Our results highlight the potential of this scalable approach as a foundation for extending neural operators toward solving increasingly complex, nonlinear, and high-dimensional PDE problems. The code and model weights are publicly available at https://github.com/rbischof/hypino.

Engineering risk is concerned with the likelihood of failure and the scenarios when it occurs. The sensitivity of failure probability to change in system parameters is relevant to risk-informed decision making. Computing sensitivity is at least one level more difficult than the probability itself, which is already challenged by a large number of input random variables, rare events and implicit nonlinear `black-box' response. Finite difference with Monte Carlo probability estimates is spurious, requiring the number of samples to grow with the reciprocal of step size to suppress estimation variance. Many existing works gain efficiency by exploiting a specific class of input variables, sensitivity parameters, or response in its exact or surrogate form. For general systems, this work presents a theory and associated Monte Carlo strategy for computing sensitivity using response values and gradients with respect to sensitivity parameters. It is shown that the sensitivity at a given response threshold can be expressed via the expectation of response gradient conditional on the threshold. Determining the expectation requires conditioning on the threshold that is a zero-probability event, but it can be resolved by the concept of kernel smoothing. The proposed method offers sensitivity estimates for all response thresholds generated in a single Monte Carlo run. It is investigated in a number of examples featuring sensitivity parameters of different nature. As response gradient becomes increasingly available, it is hoped that this work can provide the basis for embedding sensitivity calculations with reliability in the same Monte Carlo run.

Reinforcement learning (RL) is central to improving reasoning in large language models (LLMs) but typically requires ground-truth rewards. Test-Time Reinforcement Learning (TTRL) removes this need by using majority-vote rewards, but relies on heavy online RL and incurs substantial computational cost. We propose RoiRL: Reasoning with offline iterative Reinforcement Learning, a family of lightweight offline learning alternatives that can target the same regularized optimal policies. Unlike TTRL, RoiRL eliminates the need to maintain a reference model and instead optimizes weighted log-likelihood objectives, enabling stable training with significantly lower memory and compute requirements. Experimental results show that RoiRL trains to 2.5x faster and consistently outperforms TTRL on reasoning benchmarks, establishing a scalable path to self-improving LLMs without labels.

To all reviewers, thank you very much for your thoughtful comments and suggestions. R#1: "...importance of similarity among the selected tasks... " R#1: "...domain randomization, when enough samples are used, is a better alternative to meta-learning... " R#2: "...Theorems 1 and 2 are asymptotic... " Hence, the theorems are NOT asymptotic. We will remove the asymptotic parts for clarity. R#2: 'Assumption 2 ... the per-task optimal models are centered around the corresponding optimal solutions. This assumption can easily be dropped with the cost of including the distance as a term.

This work focuses on visualizing uncertainty of local divergence of two-dimensional vector fields. Divergence is one of the fundamental attributes of fluid flows, as it can help domain scientists analyze potential positions of sources (positive divergence) and sinks (negative divergence) in the flow. However, uncertainty inherent in vector field data can lead to erroneous divergence computations, adversely impacting downstream analysis. While Monte Carlo (MC) sampling is a classical approach for estimating divergence uncertainty, it suffers from slow convergence and poor scalability with increasing data size and sample counts. Thus, we present a two-fold contribution that tackles the challenges of slow convergence and limited scalability of the MC approach. (1) We derive a closed-form approach for highly efficient and accurate uncertainty visualization of local divergence, assuming independently Gaussian-distributed vector uncertainties. (2) We further integrate our approach into Viskores, a platform-portable parallel library, to accelerate uncertainty visualization. In our results, we demonstrate significantly enhanced efficiency and accuracy of our serial analytical (speed-up up to 1946X) and parallel Viskores (speed-up up to 19698X) algorithms over the classical serial MC approach. We also demonstrate qualitative improvements of our probabilistic divergence visualizations over traditional mean-field visualization, which disregards uncertainty. We validate the accuracy and efficiency of our methods on wind forecast and ocean simulation datasets.

In this study, we firstly propose an auxiliary equation neural networks method (AENNM), an innovative analytical method that integrates neural networks (NNs) models with the auxiliary equation method to obtain exact solutions of nonlinear partial differential equations (NLPDEs). A key novelty of this method is the introduction of a novel activation function derived from the solutions of the Riccati equation, establishing a new mathematical link between differential equations theory and deep learning. By combining the strong approximation capability of NNs with the high precision of symbolic computation, AENNM significantly enhances computational efficiency and accuracy. To demonstrate the effectiveness of the AENNM in solving NLPDEs, three numerical examples are investigated, including the nonlinear evolution equation, the Korteweg-de Vries-Burgers equation, and the (2+1)-dimensional Boussinesq equation. Furthermore, some new trial functions are constructed by setting specific activation functions within the "2-2-2-1" and "3-2-2-1" NNs models. By embedding the auxiliary equation method into the NNs framework, we derive previously unreported solutions. The exact analytical solutions are expressed in terms of hyperbolic functions, trigonometric functions, and rational functions. Finally, three-dimensional plots, contour plots, and density plots are presented to illustrate the dynamic characteristics of the obtained solutions. This research provides a novel methodological framework for addressing NLPDEs, with broad applicability across scientific and engineering fields.

Grounded ice thickness plays a critical role in understanding the behavior and stability of ice sheets, particularly in polar regions such as Greenland, Antarctica, and the Canadian Arctic. Ice sheet dynamics are governed by complex interactions between ice flow, surface accumulation, and bedrock topography, making the accurate modeling of these processes essential for predicting long-term ice sheet behavior and their contributions to global sea level rise [14, 18]. In particular, the Shallow Ice Approximation (SIA) provides a framework for modeling grounded ice, where ice flow is driven by internal deformation and the base is often assumed to be frozen, constraining the ice thickness by bedrock topography [12, 15]. A key challenge in modeling grounded ice involves solving the partial differential equations (PDEs) that govern ice thickness evolution, while incorporating these constraints. This leads to a free boundary problem, where the ice thickness must remain non-negative and above the bedrock, giving rise to an obstacle problem [21, 3].

In contrast to the assumptions of most existing Electromigration (EM) analysis tools, the evolution of EM-induced stress is inherently non-deterministic, influenced by factors such as input current fluctuations and manufacturing non-idealities. Traditional approaches for estimating stress variations typically involve computationally expensive and inefficient Monte Carlo simulations with industrial solvers, which quantify variations using mean and variance metrics. In this work, we introduce a novel machine learning-based framework, termed BPINNEM- Post, for efficient stochastic analysis of EM-induced postvoiding aging processes. This new approach integrates closedform analytical solutions with a Bayesian Physics-Informed Neural Network (BPINN) framework to accelerate the analysis for the first time. The closed-form solutions enforce physical laws at the individual wire segment level, while the BPINN ensures that physics constraints at inter-segment junctions are satisfied and stochastic behaviors are accurately modeled. By reducing the number of variables in the loss functions through the use of analytical solutions, our method significantly improves training efficiency without accuracy loss and naturally incorporates variational effects. Additionally, the analytical solutions effectively address the challenge of incorporating initial stress distributions in interconnect structures during post-void stress calculations. Numerical results demonstrate that BPINN-EM-Post achieves over 240x speedup compared to Monte Carlo simulations using the FEM-based COMSOL solver and more than 65x speedup compared to Monte Carlo simulations using the FDM-based EMSpice method.

Physics-Informed Neural Networks (PINNs) have emerged as a powerful framework for solving partial differential equations (PDEs) by embedding physical laws into neural network training. However, traditional PINN models are typically designed for single PDEs, limiting their generalizability across different physical systems. In this work, we explore the potential of a foundation PINN model capable of solving multiple PDEs within a unified architecture. We investigate the efficacy of a single PINN framework trained on four distinct PDEs--the Simple Harmonic Oscillator (SHO), the 1D Heat Equation, the 1D Wave Equation, and the 2D Laplace Equation--demonstrating its ability to learn diverse physical dynamics. To enhance sample efficiency, we incorporate Active Learning (AL) using Monte Carlo (MC) Dropout-based uncertainty estimation, selecting the most informative training samples iteratively. We evaluate different active learning strategies, comparing models trained on 10%, 20%, 30%, 40%, and 50% of the full dataset, and analyze their impact on solution accuracy. Our results indicate that targeted uncertainty sampling significantly improves performance with fewer training samples, leading to efficient learning across multiple PDEs. This work highlights the feasibility of a generalizable PINN-based foundation model, capable of adapting to different physics-based problems without redesigning network architectures. Our findings suggest that multi-PDE PINNs with active learning can serve as an effective approach for reducing computational costs while maintaining high accuracy in physics-based deep learning applications.