diff --git a/_research/ai-models/air-pollution/2026-low-cost-sensor-calibration.md b/_research/ai-models/air-pollution/2026-low-cost-sensor-calibration.md
index 8ea95787995e..7745de430ce0 100644
--- a/_research/ai-models/air-pollution/2026-low-cost-sensor-calibration.md
+++ b/_research/ai-models/air-pollution/2026-low-cost-sensor-calibration.md
@@ -276,6 +276,7 @@ This calibration framework can support:
- **Tutorial:** Low-cost sensor calibration tutorial
- **Data:** OxAria low-cost air-quality sensor dataset
+
---
## References
diff --git a/_research/ai-models/air-pollution/2026_air_pollution.md b/_research/ai-models/air-pollution/2026_air_pollution.md
index c66d618f71de..a906e783e9fd 100644
--- a/_research/ai-models/air-pollution/2026_air_pollution.md
+++ b/_research/ai-models/air-pollution/2026_air_pollution.md
@@ -2,9 +2,9 @@
layout: post
category: "Data-driven & Urban Sustainability"
topic: "Air Pollution"
-thumbnail: "assets/img/Gappy.png"
-tldr: "A Robust Cross-Efficiency DEA Model with Undesirable Outputs for Urban Energy Efficiency Evaluation."
-title: Data-driven
+thumbnail: "assets/img/2026-workshop-li-urban-1.jpg"
+tldr: "Urban Energy Efficiency Assessment under Uncertainty."
+title: Sustainable Cities & Energy Systems
subtitle: Our studies focused on urban sustainability
---
@@ -12,13 +12,15 @@ subtitle: Our studies focused on urban sustainability
## Introduction
-As industrialisation and urbanisation continue to advance, energy consumption—whilst driving economic growth and social development—has also given rise to severe resource constraints and environmental pollution. For a long time, fossil fuels, primarily coal, oil and natural gas, have underpinned the rapid expansion of China’s urban economies; however, issues such as carbon dioxide emissions, smog pollution and ecological degradation caused by high levels of energy consumption have become increasingly prominent. Particularly against the backdrop of the ‘dual carbon’ goals, achieving a balanced development between economic growth, energy utilisation and environmental protection has become a critical issue for the sustainable development of Chinese cities. As the primary vehicles for energy consumption and pollution emissions, the energy efficiency levels of cities not only relate to high-quality regional economic development but also directly impact the achievement of national energy conservation and emission reduction targets. Therefore, scientifically evaluating urban energy efficiency and revealing its patterns of change under conditions of uncertainty holds significant practical importance for promoting sustainable development.
+Rapid urbanization and economic growth have significantly increased energy demand across Chinese cities, creating new challenges related to sustainability, resource management, and environmental performance. Under the framework of China's carbon neutrality objectives, understanding how efficiently cities utilize energy resources has become increasingly important for supporting sustainable development policies.
+This study evaluates the energy efficiency of 280 Chinese prefecture-level cities under uncertainty conditions, providing a robust framework to assess urban energy performance while accounting for data variability and measurement uncertainty.
## Study Area
-The selection of 280 prefecture-level cities in China as the research subjects was primarily based on the following considerations.
+The analysis considers 280 prefecture-level cities distributed across China's seven major regions: North China, Northeast China, Central China, East China, South China, Southwest China, and Northwest China.
+This large-scale dataset enables the investigation of regional disparities in energy efficiency and the identification of long-term development patterns across different economic and geographical contexts.
-
- 
+ 
@@ -26,20 +28,67 @@ The selection of 280 prefecture-level cities in China as the research subjects w
## Research Methodology
-Data Envelopment Analysis (DEA) is a widely recognized non-parametric method for evaluating the relative efficiency of Decision-Making Units (DMUs). However, traditional DEA models rely heavily on deterministic input and output data, rendering their efficiency scores highly sensitive to measurement errors or data fluctuations. To address this limitation, this study integrates Robust Optimization (RO) principles into the conventional DEA framework. By constructing a bounded uncertainty set (such as a polyhedral or budget uncertainty set), the proposed Robust DEA model accounts for the worst-case scenario within a predefined protection level.
+Research Methodology
+Robust Data Envelopment Analysis (RDEA)
+Data Envelopment Analysis (DEA) is a widely used non-parametric approach for evaluating the relative efficiency of Decision-Making Units (DMUs). However, conventional DEA models assume deterministic input and output variables, making their results sensitive to measurement errors and data uncertainty.
+To overcome this limitation, this work integrates Robust Optimization (RO) into the DEA framework through the Robust Data Envelopment Analysis (RDEA) methodology.
+By introducing uncertainty sets and protection levels, RDEA evaluates efficiency under worst-case scenarios, producing more reliable and stable efficiency estimates.
+Two robust formulations are considered:
-## Results
+- CU-RDEA (Conservative Uncertainty Robust DEA)
+- CEU-RDEA (Conservative Extended Uncertainty Robust DEA)
-The energy efficiency results of CU-RDEA and CEU-RDEA. A comparison of city maps across China's seven major regions reveals that the energy efficiency performance of Chinese cities is highly sensitive to data uncertainty. Empirical results show that the efficiency scores derived from both the CU-RDEA and CEU-RDEA models exhibit a marked downward trend as the perturbation level T increases from 0 to 0.10. This indicates that robust DEA methods can effectively mitigate efficiency bias issues caused by data fluctuations, changes in energy demand, and policy uncertainty.
-From a spatial perspective, there are significant differences in energy utilization efficiency among various regions in China. The East China region exhibits a distinct gradient of competition, while the Northwest and Southwest regions have a lower overall efficiency level. In terms of regional distribution, the 2010 CU-RDEA and CEU-RDEA scores reveal that early high-efficiency cities were predominantly concentrated in specific resource- or policy-oriented regions such as Ordos and Suihua; by 2024, however, core central cities such as Beijing, Chongqing, Wuhan, and Jiangmen have emerged as leaders under the CEU-RDEA model, occupying leading positions across their respective regions. This indicates that China’s urban energy development model is gradually shifting from a traditional resource-driven approach to one driven by the synergy of technological innovation, industrial upgrading, and green, low-carbon initiatives.
-
-
-
-
+This approach allows the assessment of how uncertainty affects urban energy efficiency rankings and regional performance.
+
+
+ 
+
+
+
+The diagram should show how urban energy data are processed through DEA and Robust Optimization to build the CU-RDEA and CEU-RDEA models. The goal is to communicate the novelty of the methodology rather than reproduce the mathematical formulation presented in the article.
+
+## Main Findings
+
+The results reveal that urban energy efficiency is highly sensitive to data uncertainty.
+As the perturbation level (Γ) increases, both CU-RDEA and CEU-RDEA efficiency scores show a systematic decline, demonstrating the importance of incorporating uncertainty into efficiency assessments.
+The analysis also highlights substantial regional disparities across China:
+
+- East China exhibits the highest overall efficiency levels.
+- Northwest and Southwest China show comparatively lower efficiency performance.
+- Significant spatial heterogeneity exists among cities within the same region.
+
+Furthermore, the results indicate a progressive transition from traditional resource-driven development models toward growth supported by technological innovation, industrial upgrading, and green low-carbon strategies.
+
+
+ 
+
+
+
+The figure should clearly illustrate (i) regional differences in urban energy efficiency and (ii) the impact of uncertainty levels (Γ) on efficiency scores. A China efficiency map combined with a regional comparison chart would communicate the results more effectively for a web audience than the full set of detailed regional plots
+
+## Scientific Contribution
+This work demonstrates that Robust Data Envelopment Analysis provides a reliable framework for evaluating urban energy efficiency under uncertainty.
+By combining DEA and Robust Optimization, the proposed methodology reduces sensitivity to data fluctuations and offers a more realistic assessment of urban sustainability performance.
+The framework can support policy evaluation, urban planning, energy management, and sustainable development strategies in complex socio-economic systems.
+
+## Scientific Contribution
+
+This work demonstrates that Robust Data Envelopment Analysis provides a reliable framework for evaluating urban energy efficiency under uncertainty.
+By combining DEA and Robust Optimization, the proposed methodology reduces sensitivity to data fluctuations and offers a more realistic assessment of urban sustainability performance.
+The framework can support policy evaluation, urban planning, energy management, and sustainable development strategies in complex socio-economic systems.
+
+## Relevance for ModelFLOWs
-### Other Relevant References
+This research illustrates the application of advanced data-driven methodologies to large-scale urban systems.
+Beyond traditional engineering applications, ModelFLOWs methodologies can be used to analyze complex socio-economic and environmental datasets, enabling robust decision-support tools for sustainability, smart cities, and energy planning.
-[*Yang, Z., & Wei, X., 2019. The measurement and influences of China's urban total factor energy efficiency under environmental pollution: Based on the game cross-efficiency DEA. Journal of cleaner production, 209, 439-450.*](https://doi.org/10.1016/j.jclepro.2018.10.271)
+## References
-[*Wang, R., Wang, Q., & Yao, S., 2021. Evaluation and difference analysis of regional energy efficiency in China under the carbon neutrality targets: Insights from DEA and Theil models. Journal of environmental management, 293, 112958.*](https://doi.org/10.1016/j.jenvman.2021.112958)
+Jiannan Li, Soledad Le Clainche, and collaborators.
+A Robust Cross-Efficiency DEA Model with Undesirable Outputs for Urban Energy Efficiency Evaluation.
+Manuscript in preparation.
diff --git a/_research/ai-models/flow-patterns-complex-flows/2026-viscoelastic-jets.md.md b/_research/ai-models/flow-patterns-complex-flows/2026-viscoelastic-jets.md.md
new file mode 100644
index 000000000000..638409268e5e
--- /dev/null
+++ b/_research/ai-models/flow-patterns-complex-flows/2026-viscoelastic-jets.md.md
@@ -0,0 +1,134 @@
+---
+layout: post
+title: "Reduced-order modeling of viscoelastic flows"
+category: "AI & Data-Driven Models"
+topic: "Flow Patterns in Complex Flows"
+thumbnail: "/assets/img/flow-patterns/thumbnail_viscoelastic-jet.png"
+tldr: "Modal decompositions for studying and modeling viscoelastic turbulent planar jets"
+---
+
+Newtonian fluids, like air and water, follow Newton's Law of Viscosity, which states that the shear stress is directly proportional to the velocity gradient. The constant of proportionality is commonly known as the dynamic viscosity of the fluid. On the contrary, non-Newtonian fluids do not follow this law. This is the case of mayonnaise, topical creams or paints, which showcase more complex behaviors such as fluid elasticity, yield-stress or shear-dependent viscosity.
+
+In this post, we focus on polymeric solutions that have viscous and elastic properties, or viscoelastic flows. In turbulent flows, polymers draw kinetic energy, they turn into elastic energy by polymer stretching, and either return to the flow or dissipate. This mechanism can change drastically the flow, and it translates into macroscopic effects such as drag reduction, enhanced heat transfer, and elastic instabilities.
+
+These effects are commonly studied through experiments and numerical simulations. Numerical simulations provide a complete characterization of the flow and the polymer dynamics. However, the hyperbolic nature of the polymer equation supposes a great challenge, where numerical instabilities appear as polymer elasticity increases. To solve this issue, numerical solvers implement sophisticated methods, that in turn increase significantly the cost of numerical simulations.
+
+Here, we propose reduced-order models for accelerating numerical simulations. We split this contribution in two parts:
+
+1. **Dimensionality reduction through modal decompositions**
+We demonstrate that the complex dynamics can be interpreted as a superposition of a few coherent structures.
+
+2. **Reduced-order modeling with hybrid machine learning**
+We describe a surrogate model that combines modal decompositions with a deep neural network for temporal forecasting.
+
+---
+
+## Coherent structures
+We first show the application of modal decompositions for identifying the dominant coherent structures in the viscoelastic turbulent planar jet.
+
+Modal decompositions decompose the data into a superposition of modes, each accompanied by characteristic values that represent either their energy content or dynamical traits. Modal decompositions require large datasets, and they can be used, for example, for [reconstructing flow fields from sparse measurements](https://modelflows.github.io/modelflowsapp/research/2026-from-sensors-to-3d-reconstruction/). Here, we use modal decompositions for compressing high-dimensional data into a more interpretable low-dimensional form by means of the higher order dynamic mode decomposition (HODMD); more information about the method and applications is available [here](https://modelflows.github.io/modelflowsapp/software/notebooks/2026-modaldecomposition/).
+
+HODMD yields a decomposition based on DMD modes, each associated to a frequency and temporal growth/decay rate; the weighted superposition of these modes reconstructs the original data based on their most dominant coherent dynamics. The method can be tuned to consider small-amplitude dynamics, though it must be more carefully calibrated so the reconstruction does not overfit noisy or spurious dynamics.
+
+
+ 
+
+
+
+
+ HODMD spectra of robust DMD modes in the viscoelastic jet. Modes are compared with those from a Newtonian turbulent planar jet.
+
+
+
+We reconstruct the viscoelastic turbulent planar jet using sixteen robust modes. We also performed HODMD to a Newtonian turbulent planar jet, which yielded twenty-one robust modes. We find that the turbulent dynamics in the Newtonian jet are more complex than those from the viscoelastic jet, since it is required a larger number of DMD modes to reconstruct the flow field with similar accuracy. The DMD modes can also be represented in space, where low-frequency modes are associated with global, streamwise-coherent streaks, and high-frequency modes with localized, spanwise-coherent wave packets. Remarkably, the method is also able to reconstruct the spanwise-homogeneous or roller-like structures in the near-field of the Newtonian jet, whose shape and frequency range matches with the symmetric or varicose mode of instability.
+
+
+ 
+
+
+
+
+ Spatial structures of representative robust DMD modes in the Newtonian and viscoelastic jets. The yellow-translucent surfaces mark the average location of the jet edges.
+
+
+
+Since the DMD modes are space-dependent, their interpretation can be improved by applying HODMD in one of the spatial directions. By applying the decomposition in the spanwise-homogeneous direction, we now describe the dynamics as a superposition of steady and traveling waves. In doing this, we find that a potential mechanism sustaining the turbulent regime in the viscoelastic jet is given by the interaction between short streaks and wave packets located in both edges in the near-field of the jet; we did not find an equivalent mechanism in the Newtonian jet, where similar streaks are found after the potential core, and transition occurs because of the modal growth of the instability.
+
+
+ 
+
+
+
+
+ Flow structures in the near-field. The streaks modify the bulk flow, where perturbations grow along the jet edges because of the interaction of the streaks with wavepacket structures at the same location.
+
+
+
+In the viscoelastic jet, both near-field streaks and instability are purely elastic, since the Reynolds number is too small and turbulence is rather sustained by polymer elasticity. This finding supposes, to the authors' knowledge, the first description of a transition mechanism to elastic turbulence in viscoelastic jets.
+
+---
+
+## Reduced-order modeling
+
+We have demonstrated that coherent structures offer a compact and interpretable representation of the complex dynamics in the viscoelastic turbulent planar jet; we now show that they can also be used as a good basis for building reduced-order models in the same case.
+
+We combine proper orthogonal decomposition (POD) with a deep neural network for forecasting future states from historic data. This is done in an autoregressive way, meaning that the model uses its own prediction to condition the next step. The model is optimized during training for finding the best solution by means of minimizing the mean-squared error between the true and predicted fields in the subspace of POD modes. In doing so, we reduce the size of the model, since POD is a non-parametric method, and the amount of training data. The optimal subspace from POD is physically-interpretable, and using it during training constitutes a method for weakly enforcing the physics underlying in the system into the machine learning model.
+
+
+ 
+
+
+
+
+ POD decomposition of the viscoelastic jet. The reconstruction with 25 POD modes yields $\sim$ 50$\%$ of energy, and $\sim$ 85$\%$ with 125 POD modes.
+
+
+
+We adopt the POD with deep learning, or [POD-DL](https://modelflows.github.io/modelflowsapp/software/notebooks/2026-deeplearning/), for our predictor. POD-DL first reduces the dimensionality of the data by means of POD, and it implements a deep neural network for modeling the temporal evolution of the POD coefficients. The neural network couples a long short-term memory (LSTM) network that learns the long-range temporal structure of the input, with a non-linear feed-forward network (FFN) that focuses on non-linear relations between the input and output sequences. In this study, we also explore the effect of layer depth in the prediction accuracy. To do so, we introduce the POD-DL with residual, or POD-rDL, which implements skip connections and stronger regularization for stabilizing the training of deeper neural networks.
+
+
+ 
+
+
+
+
+ Sketch of the prediction models. The symbols N and H denote the dimensionality and T the length of the output for each layer.
+
+
+
+We find that depth indeed improves the accuracy of the model, with skip connections the most efficient strategy for building deep and generalizable neural networks. In addition, the POD-rDL outperforms the POD-DL predicting more complex input spaces, which are characterized by higher-order POD modes. As layer depth increases, deep learning models are able to learn more complex temporal abstractions; this allows models to learn better the multi-scale behavior of turbulent flows. However, architectural choices matter the most if, for instance, models are trained for multi-step prediction. In this case, the choice of output layer is crucial, where the POD-DL, specifically the feed-forward network that transforms non-linearly each candidate prediction step mapped from the output from the LSTM network, shows a performance similar to the POD-rDL.
+
+
+ 
+
+
+
+
+ Prediction error over the temporal horizon, measured as the cumulative average of the L2 error norm to the temporal horizon of the velocity field reconstructed from POD.
+
+
+
+
+ 
+
+
+
+
+ Prediction from POD-DL and POD-rDL for multi-step and higher-dimensional input space predictions, respectively.
+
+
+
+---
+
+## Summary
+
+This post shows using modal decompositions for modeling viscoelastic turbulent planar jets. First, we showed HODMD for identifying the dominant coherent structures, revealing that viscoelastic dynamics are governed by fewer modes than Newtonian turbulent planar jets, and uncovering a purely elastic near-field transition mechanism driven by the interaction of streamwise streaks and wave packets at the jet edges — to the authors' knowledge. Second, a hybrid POD and deep learning architecture (POD-DL and POD-DL with residual or POD-rDL) is introduced to forecast the temporal evolution of POD mode coefficients in an autoregressive manner, with the POD subspace serving as a physically interpretable, low-dimensional basis that implicitly encodes flow physics into the learning objective. Results show that residual connections and increased network depth improve generalization, particularly for higher-order mode dynamics, establishing this hybrid framework as an efficient and interpretable surrogate for viscoelastic turbulent planar jets, that is extensible to other non-Newtonian flows.
+
+---
+
+## Reference
+
+- Amor, C., Corrochano, A., Soligo, G., Le Clainche, S., & Rosti, M. E. **Coherent structures in Newtonian and viscoelastic turbulent planar jets**. *Journal of Fluid Mechanics* 1036, A7 (2026). [Link](https://doi.org/10.1017/jfm.2026.11610).
+
+- Amor, C., Corrochano, A., Rosti, M. E., & Le Clainche, S. **Reduced-order modeling of a viscoelastic turbulent jet with hybrid machine learning models**. *Journal of Physics: Conference Series* 3230, 012001 (2026). [Link](https://doi.org/10.1088/1742-6596/3230/1/012001)
+
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