Explainable deep learning reveals the physical mechanisms behind the turbulent kinetic energy equation

In this work, we investigate the physical mechanisms governing turbulent kinetic energy transport using explainable deep learning (XDL). An XDL model based on SHapley Additive exPlanations (SHAP) is used to identify and percolate high-importance structures for the evolution of the turbulent kinetic energy budget terms of a turbulent channel flow at a friction Reynolds number of $Re_τ= 125$. The results show that the important structures are predominantly located in the near-wall region and are more frequently associated with sweep-type events. In the viscous layer, the SHAP structures relevant for production and viscous diffusion are almost entirely contained within those relevant for dissipation, revealing a clear hierarchical organization of near-wall turbulence. In the outer layer, this hierarchical organization breaks down and only velocity-pressure-gradient correlation and turbulent transport SHAP structures remain, with a moderate spatial coincidence of approximately $60\%$. Finally, we show that none of the coherent structures classically studied in turbulence are capable of representing the mechanisms behind the various terms of the turbulent kinetic energy budget throughout the channel. These results reveal dissipation as the dominant organizing mechanism of near-wall turbulence, constraining production and viscous diffusion within a single structural hierarchy that breaks down in the outer layer.

Evaluating Visual Mathematics in Multimodal LLMs: A Multilingual Benchmark Based on the Kangaroo Tests

Multimodal Large Language Models (MLLMs) promise advanced vision language capabilities, yet their effectiveness in visually presented mathematics remains underexplored. This paper analyzes the development and evaluation of MLLMs for mathematical problem solving, focusing on diagrams, multilingual text, and symbolic notation. We then assess several models, including GPT 4o, Pixtral, Qwen VL, Llama 3.2 Vision variants, and Gemini 2.0 Flash in a multilingual Kangaroo style benchmark spanning English, French, Spanish, and Catalan. Our experiments reveal four key findings. First, overall precision remains moderate across geometry, visual algebra, logic, patterns, and combinatorics: no single model excels in every topic. Second, while most models see improved accuracy with questions that do not have images, the gain is often limited; performance for some remains nearly unchanged without visual input, indicating underutilization of diagrammatic information. Third, substantial variation exists across languages and difficulty levels: models frequently handle easier items but struggle with advanced geometry and combinatorial reasoning. Notably, Gemini 2.0 Flash achieves the highest precision on image based tasks, followed by Qwen VL 2.5 72B and GPT 4o, though none approach human level performance. Fourth, a complementary analysis aimed at distinguishing whether models reason or simply recite reveals that Gemini and GPT 4o stand out for their structured reasoning and consistent accuracy. In contrast, Pixtral and Llama exhibit less consistent reasoning, often defaulting to heuristics or randomness when unable to align their outputs with the given answer options.

Additive-feature-attribution methods: a review on explainable artificial intelligence for fluid dynamics and heat transfer

The use of data-driven methods in fluid mechanics has surged dramatically in recent years due to their capacity to adapt to the complex and multi-scale nature of turbulent flows, as well as to detect patterns in large-scale simulations or experimental tests. In order to interpret the relationships generated in the models during the training process, numerical attributions need to be assigned to the input features. One important example are the additive-feature-attribution methods. These explainability methods link the input features with the model prediction, providing an interpretation based on a linear formulation of the models. The SHapley Additive exPlanations (SHAP values) are formulated as the only possible interpretation that offers a unique solution for understanding the model. In this manuscript, the additive-feature-attribution methods are presented, showing four common implementations in the literature: kernel SHAP, tree SHAP, gradient SHAP, and deep SHAP. Then, the main applications of the additive-feature-attribution methods are introduced, dividing them into three main groups: turbulence modeling, fluid-mechanics fundamentals, and applied problems in fluid dynamics and heat transfer. This review shows thatexplainability techniques, and in particular additive-feature-attribution methods, are crucial for implementing interpretable and physics-compliant deep-learning models in the fluid-mechanics field.

Explaining wall-bounded turbulence through deep learning

Despite its great scientific and technological importance, wall-bounded turbulence is an unresolved problem that requires new perspectives to be tackled. One of the key strategies has been to study interactions among the coherent structures in the flow. Such interactions are explored in this study for the first time using an explainable deep-learning method. The instantaneous velocity field in a turbulent channel is used to predict the velocity field in time through a convolutional neural network. The predicted flow is used to assess the importance of each structure for this prediction using a game-theoretic algorithm (SHapley Additive exPlanations). This work provides results in agreement with previous observations in the literature and extends them by quantifying the importance of the Reynolds-stress structures, finding a causal connection between these structures and the dynamics of the flow. The process, based on deep-learning explainability, has the potential to shed light on numerous fundamental phenomena of wall-bounded turbulence, including the objective definition of new types of flow structures.

Aim in Climate Change and City Pollution

The sustainability of urban environments is an increasingly relevant problem. Air pollution plays a key role in the degradation of the environment as well as the health of the citizens exposed to it. In this chapter we provide a review of the methods available to model air pollution, focusing on the application of machine-learning methods. In fact, machine-learning methods have proved to importantly increase the accuracy of traditional air-pollution approaches while limiting the development cost of the models. Machine-learning tools have opened new approaches to study air pollution, such as flow-dynamics modelling or remote-sensing methodologies.

Towards extraction of orthogonal and parsimonious non-linear modes from turbulent flows

We propose a deep probabilistic-neural-network architecture for learning a minimal and near-orthogonal set of non-linear modes from high-fidelity turbulent-flow-field data useful for flow analysis, reduced-order modeling, and flow control. Our approach is based on $\beta$-variational autoencoders ($\beta$-VAEs) and convolutional neural networks (CNNs), which allow us to extract non-linear modes from multi-scale turbulent flows while encouraging the learning of independent latent variables and penalizing the size of the latent vector. Moreover, we introduce an algorithm for ordering VAE-based modes with respect to their contribution to the reconstruction. We apply this method for non-linear mode decomposition of the turbulent flow through a simplified urban environment, where the flow-field data is obtained based on well-resolved large-eddy simulations (LESs). We demonstrate that by constraining the shape of the latent space, it is possible to motivate the orthogonality and extract a set of parsimonious modes sufficient for high-quality reconstruction. Our results show the excellent performance of the method in the reconstruction against linear-theory-based decompositions. Moreover, we compare our method with available AE-based models. We show the ability of our approach in the extraction of near-orthogonal modes that may lead to interpretability.