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The advent of artificial intelligence and machine learning has led to significant technological and scientific progress, but also to new challenges. Partial differential equations, usually used to model systems in the sciences, have shown to be useful tools in a variety of tasks in the data sciences, be it just as physical models to describe physical data, as more general models to replace or construct artificial neural networks, or as analytical tools to analyse stochastic processes appearing in the training of machine-learning models. This article acts as an introduction of a theme issue covering synergies and intersections of partial differential equations and data science. We briefly review some aspects of these synergies and intersections in this article and end with an editorial foreword to the issue. This article is part of the theme issue ‘Partial differential equations in data science’. 

Spiral gravity separators are designed to separate multi-species slurry components based on differences in density and size. Previous studies [S. Lee et al., Phys. Fluids 26, 043302 (2014); D. Arnold et al., Phys. Fluids 31, 073305 (2019)] have investigated steady-state solutions for mixtures of liquids and single particle species in thin-film flows. However, these models are constrained to single-species systems and cannot describe the dynamics of multi-species separation. In contrast, our analysis extends to mixtures containing two particle species of differing densities, revealing that they undergo radial separation—an essential mechanism for practical applications in separating particles of varying densities. This work models gravity-driven bidensity slurries in a spiral trough by incorporating particle interactions, using empirically derived formulas for particle fluxes from previous bidensity studies on inclined planes [J. T. Wong and A. L. Bertozzi, Phys. D 330, 47–57 (2016)]. Specifically, we study a thin-film bidensity slurry flowing down a rectangular channel helically wound around a vertical axis. Through a thin-film approximation, we derive equilibrium profiles for the concentration of each particle species and the fluid depth. Additionally, we analyze the influence of key design parameters, such as spiral radius and channel width, on particle concentration profiles. Our findings provide valuable insights into optimizing spiral separator designs for enhanced applicability and adaptability. 

Narrative data spans all disciplines and provides a coherent model of the world to the reader or viewer. Recent advancement in machine learning and Large Language Models (LLMs) have enable great strides in analyzing natural language. However, Large language models (LLMs) still struggle with complex narrative arcs as well as narratives containing conflicting information. Recent work indicates LLMs augmented with external knowledge bases can improve the accuracy and interpretability of the resulting models. In this work, we analyze the effectiveness of applying knowledge graphs (KGs) in understanding true-crime podcast data from both classical Natural Language Processing (NLP) and LLM approaches. We directly compare KG-augmented LLMs (KGLLMs) with classical methods for KG construction, topic modeling, and sentiment analysis. Additionally, the KGLLM allows us to query the knowledge base in natural language and test its ability to factually answer questions. We examine the robustness of the model to adversarial prompting in order to test the model's ability to deal with conflicting information. Finally, we apply classical methods to understand more subtle aspects of the text such as the use of hearsay and sentiment in narrative construction and propose future directions. Our results indicate that KGLLMs outperform LLMs on a variety of metrics, are more robust to adversarial prompts, and are more capable of summarizing the text into topics. 

Theoretical and empirical comparisons have been made to assess the expressive power and performance of invariant and equivariant GNNs. However, there is currently no theoretical result comparing the expressive power of k-hop invariant GNNs and equivariant GNNs. Additionally, little is understood about whether the performance of equivariant GNNs, employing steerable features up to type-L, increases as L grows – especially when the feature dimension is held constant. In this study, we introduce a key lemma that allows us to analyze steerable features by examining their corresponding invariant features. The lemma facilitates us in understanding the limitations of k-hop invariant GNNs, which fail to capture the global geometric structure due to the loss of geometric information between local structures. Furthermore, we analyze the ability of steerable features to carry information by studying their corresponding invariant features. In particular, we establish that when the input spatial embedding has full rank, the informationcarrying ability of steerable features is characterized by their dimension and remains independent of the feature types. This suggests that when the feature dimension is constant, increasing L does not lead to essentially improved performance in equivariant GNNs employing steerable features up to type-L. We substantiate our theoretical insights with numerical evidence.