Solving partial differential equations (PDEs) is a fundamental problem in engineering and science. While neural PDE solvers can be more efficient than established numerical solvers, they often require large amounts of training data that is costly to obtain. Active learning (AL) could help surrogate models reach the same accuracy with smaller training sets by querying classical solvers with more informative initial conditions and PDE parameters. While AL is more common in other domains, it has yet to be studied extensively for neural PDE solvers. To bridge this gap, we introduce AL4PDE, a modular and extensible active learning benchmark. It provides multiple parametric PDEs and state-of-the-art surrogate models for the solver-in-the-loop setting, enabling the evaluation of existing and the development of new AL methods for PDE solving. We use the benchmark to evaluate batch active learning algorithms such as uncertainty- and feature-based methods. We show that AL reduces the average error by up to 71\% compared to random sampling and significantly reduces worst-case errors. Moreover, AL generates similar datasets across repeated runs, with consistent distributions over the PDE parameters and initial conditions. The acquired datasets are reusable, providing benefits for surrogate models not involved in the data generation.

Transformer models are increasingly used for solving Partial Differential Equations (PDEs). Several adaptations have been proposed, all of which suffer from the typical problems of Transformers, such as quadratic memory and time complexity. Furthermore, all prevalent architectures for PDE solving lack at least one of several desirable properties of an ideal surrogate model, such as (i) generalization to PDE parameters not seen during training, (ii) spatial and temporal zero-shot super-resolution, (iii) continuous temporal extrapolation, (iv) support for 1D, 2D, and 3D PDEs, and (v) efficient inference for longer temporal rollouts. To address these limitations, we propose *Vectorized Conditional Neural Fields* (VCNeFs), which represent the solution of time-dependent PDEs as neural fields. Contrary to prior methods, however, VCNeFs compute, for a set of multiple spatio-temporal query points, their solutions in parallel and model their dependencies through attention mechanisms. Moreover, VCNeF can condition the neural field on both the initial conditions and the parameters of the PDEs. An extensive set of experiments demonstrates that VCNeFs are competitive with and often outperform existing ML-based surrogate models.

Interest in Artificial Intelligence (AI) and its applications has seen unprecedented growth in the last few years. This success can be partly attributed to the advancements made in the sub-fields of AI such as machine learning, computer vision, and natural language processing. Much of the growth in these fields has been made possible with deep learning, a sub-area of machine learning that uses artificial neural networks. This has created significant interest in the integration of vision and language. In this survey, we focus on ten prominent tasks that integrate language and vision by discussing their problem formulation, methods, existing datasets, evaluation measures, and compare the results obtained with corresponding state-of-the-art methods. Our efforts go beyond earlier surveys which are either task-specific or concentrate only on one type of visual content, i.e., image or video. Furthermore, we also provide some potential future directions in this field of research with an anticipation that this survey stimulates innovative thoughts and ideas to address the existing challenges and build new applications.