Partial differential equations (PDEs) underpin quantitative descriptions across the physical sciences and engineering, yet high-fidelity simulation remains a major computational bottleneck for many-query, real-time, and design tasks. Data-driven surrogates can be strikingly fast but are often unreliable when applied outside their training distribution. Here we introduce Neural Operator Warm Starts (NOWS), a hybrid strategy that harnesses learned solution operators to accelerate classical iterative solvers by producing high-quality initial guesses for Krylov methods such as conjugate gradient and GMRES. NOWS leaves existing discretizations and solver infrastructures intact, integrating seamlessly with finite-difference, finite-element, isogeometric analysis, finite volume method, etc. Across our benchmarks, the learned initialization consistently reduces iteration counts and end-to-end runtime, resulting in a reduction of the computational time of up to 90 %, while preserving the stability and convergence guarantees of the underlying numerical algorithms. By combining the rapid inference of neural operators with the rigor of traditional solvers, NOWS provides a practical and trustworthy approach to accelerate high-fidelity PDE simulations.

Solving partial differential equations (PDEs) is a required step in the simulation of natural and engineering systems. The associated computational costs significantly increase when exploring various scenarios, such as changes in initial or boundary conditions or different input configurations. This study proposes the Variational Physics-Informed Neural Operator (VINO), a deep learning method designed for solving PDEs by minimizing the energy formulation of PDEs. This framework can be trained without any labeled data, resulting in improved performance and accuracy compared to existing deep learning methods and conventional PDE solvers. By discretizing the domain into elements, the variational format allows VINO to overcome the key challenge in physics-informed neural operators, namely the efficient evaluation of the governing equations for computing the loss. Comparative results demonstrate VINO's superior performance, especially as the mesh resolution increases. As a result, this study suggests a better way to incorporate physical laws into neural operators, opening a new approach for modeling and simulating nonlinear and complex processes in science and engineering.

Reasoning is a crucial part of natural language argumentation. To comprehend an argument, one must analyze its warrant, which explains why its claim follows from its premises. As arguments are highly contextualized, warrants are usually presupposed and left implicit. Thus, the comprehension does not only require language understanding and logic skills, but also depends on common sense. In this paper we develop a methodology for reconstructing warrants systematically. We operationalize it in a scalable crowdsourcing process, resulting in a freely licensed dataset with warrants for 2k authentic arguments from news comments. On this basis, we present a new challenging task, the argument reasoning comprehension task. Given an argument with a claim and a premise, the goal is to choose the correct implicit warrant from two options. Both warrants are plausible and lexically close, but lead to contradicting claims. A solution to this task will define a substantial step towards automatic warrant reconstruction. However, experiments with several neural attention and language models reveal that current approaches do not suffice.

This paper presents a 55-line code written in python for 2D and 3D topology optimization (TO) based on the open-source finite element computing software (FEniCS), equipped with various finite element tools and solvers. PETSc is used as the linear algebra back-end, which results in significantly less computational time than standard python libraries. The code is designed based on the popular solid isotropic material with penalization (SIMP) methodology. Extensions to multiple load cases, different boundary conditions, and incorporation of passive elements are also presented. Thus, this implementation is the most compact implementation of SIMP based topology optimization for 3D as well as 2D problems. Utilizing the concept of Euclidean distance matrix to vectorize the computation of the weight matrix for the filter, we have achieved a substantial reduction in the computational time and have also made it possible for the code to work with complex ground structure configurations. We have also presented the code's extension to large-scale topology optimization problems with support for parallel computations on complex structural configuration, which could help students and researchers explore novel insights into the TO problem with dense meshes. Appendix-A contains the complete code, and the website: \url{this https URL} also contains the complete code.