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Despite the recent popularity of attention-based neural architectures in core AI fields like natural language processing (NLP) and computer vision (CV), their potential in modeling complex physical systems remains underexplored. Learning problems in physical systems are often characterized as discovering operators that map between function spaces based on a few instances of function pairs. This task frequently presents a severely ill-posed PDE inverse problem. In this work, we propose a novel neural operator architecture based on the attention mechanism, which we refer to as the Nonlocal Attention Operator (NAO), and explore its capability in developing a foundation physical model. In particular, we show that the attention mechanism is equivalent to a double integral operator that enables nonlocal interactions among spatial tokens, with a data-dependent kernel characterizing the inverse mapping from data to the hidden parameter field of the underlying operator. As such, the attention mechanism extracts global prior information from training data generated by multiple systems, and suggests the exploratory space in the form of a nonlinear kernel map. Consequently, NAO can address ill-posedness and rank deficiency in inverse PDE problems by encoding regularization and achieving generalizability. We empirically demonstrate the advantages of NAO over baseline neural models in terms of generalizability to unseen data resolutions and system states. Our work not only suggests a novel neural operator architecture for learning interpretable foundation models of physical systems, but also offers a new perspective towards understanding the attention mechanism. Our code and data accompanying this paper are available at https://github.com/fishmoon1234/NAO. 

Abstract Systems thinking (ST) includes a set of critical skills and approaches for addressing today's complex societal problems. Therefore, it has been introduced into the curricula of many educational programmes around the world. Despite all the attention to ST, there is less consensus when it comes to evaluating and assessing ST skills. Particularly, a quantitative assessment approach that captures ST's multi‐dimensionality is crucial when evaluating the degree to which one has learned and utilizes ST. This paper proposes a systematic approach to create such a multi‐dimensional Index of ST from textual data. Initially, we provide an overview of the theoretical background as it pertains to different measurement approaches of ST skills. Then we provide a conceptual framework based on ST skill measures and transform this framework into a quantifiable model. Finally, using student data, we provide an illustration of an integrated index of ST skills. We compute this index by using a mixed methods approach, including robust principal component analysis, data envelopment analysis and two‐staged bootstrapping approach. The results show that (i) our model serves as a systematic multi‐dimensional ST approach by including multiple measures of ST skills and (ii) international students and self‐reported math skills are found as significant predictors of one's level of ST in the graduate student dataset (N = 30), however no significant factors are found in the first‐year engineering student dataset (N = 144). 

In plasma discharges, the acceleration of electrons by a fast varying electric field and the subsequent collisional electron energy transfer determines the plasma dynamics, chemical reactivity, and breakdown. Current in situ electric field measurements require reconstruction of the temporal profile over many observations. However, such methods are unsuitable for non-repetitive and unstable plasmas. Here, we present a method for creating “movies” of dynamic electric fields in a single acquisition at sample rates of 500 × 106 fps. This ultrafast diagnostic was demonstrated in radio frequency electric fields between two parallel plates in air, as well as in Ar nanosecond-pulsed single-sided dielectric barrier discharges.