Machine Learning-Driven Scattering Efficiency Prediction in Passive Daytime Radiative Cooling

Abstract

Passive daytime radiative cooling (PDRC) has emerged as a promising, electricity-free cooling approach that reflects sunlight while radiating heat through the atmospheric transparent window. However, the design and optimization of PDRC materials remain challenging, requiring significant time and resources for experimental and numerical modeling efforts. In this work, we developed a machine learning (ML)-driven approach to predict scattering efficiency in the wavelength of 0.3–2.5 μm, with the aim of eventually optimizing the microstructural design of PDRC materials. By employing ML models such as linear regression, neural networks, and random forests, we aimed to predict and optimize the scattering efficiency across different pore sizes and mixed-pore-size configurations. As a result, the random forest model demonstrated superior prediction performance with minimal error, effectively capturing complex, non-linear interactions between material features. We also leveraged data transformation techniques such as one-hot encoding for generative predictions in mixed-pore-size configurations. The presented ML-driven platform serves as a valuable open resource for PDRC researchers, facilitating the rapid and cost-effective optimization of PDRC materials and accelerating the development of sustainable cooling technologies.

1. Introduction

The demand for sustainable cooling solutions has surged, driven by the urgent need to combat climate change and reduce greenhouse gas emissions from conventional cooling technologies, such as air conditioning [1,2,3,4,5]. Air conditioning alone accounts for approximately 12% of global electricity consumption, and as urban populations grow and temperatures rise, this energy demand is expected to increase substantially [6,7]. For example, buildings consume around 40% of the world’s total energy, and approximately 36% of global CO₂ emissions come from buildings [8]. Equally concerning, air conditioning cooling systems are energy-intensive and environmentally unsustainable, relying on refrigerants that significantly contribute to global warming. Consequently, there is a pressing need for alternative cooling methods that are both efficient and sustainable.

Passive daytime radiative cooling (PDRC) has emerged as a promising approach to address these challenges [9,10,11,12]. It offers an electricity-free cooling solution by reflecting solar radiation (in the wavelength range of 0.3–2.5 μm) and emitting thermal radiation through the atmospheric transparent window (ATW, in the wavelength range of 8–13 μm) into the cold outer space [13,14,15,16,17]. This approach enables spontaneous cooling to temperatures below the ambient environment [18,19,20,21,22,23]. Recent studies suggest that effective PDRC could lower ambient temperatures by more than 5 °C during the day, contributing to substantial energy savings and carbon emission reductions [24,25,26].

To achieve scalable and cost-effective PDRC solutions, researchers have explored the development of polymer-based PDRC coatings that incorporate (hierarchical) micro- and nanostructures to enhance solar reflection and infrared emission [27,28,29,30,31]. Experimentally, methods such as crystallinity alternation [1], phase inversion [32,33,34], and electrospinning [35,36,37] have been employed to fabricate these structures with high radiative cooling efficiency. Numerical models have also been constructed to show the relationship between microstructure and optical properties, providing insights into optimizing cooling performance [38]. However, both experimental optimization and numerical modeling are time-consuming and may take years of effort. Experimental design requires substantial iterative adjustments to microstructural parameters, while numerical modeling faces complex structural and material property challenges that can limit rapid advancement.

To overcome the challenges above, in this work, we developed novel, data-driven approaches using machine learning (ML) to create predictive models for the efficient design and optimization of PDRC materials. Numerical modeling studies have shown that while structural design substantially impacts solar reflection, it has minimal influence on thermal radiation emission in ATW [38]. As noted in the literature [1,27], there is a strong correlation between scattering efficiency and reflection in the solar reflection range (wavelength: 0.3–2.5 μm). Leveraging these insights, our ML models focus on predicting scattering efficiency within such wavelength range, with the goal of ultimately optimizing the microstructural design of PDRC materials. At a broad level, our ML models serve as a valuable open resource for researchers, enabling them to simulate and refine PDRC microstructures with minimal experimental input. Eventually, we aim to significantly expedite the development of the full range of high-performance PDRC materials.

2. Results and Discussion

To enable practical-level prediction and guidance, we employed the same pore geometry published in past studies [1,27]. Specifically, a core–shell structure model was used when calculating scattering efficiency from the Mie theory and Mie calculator [39]. Within the core–shell structure, the same pore geometry factors were employed (i.e., the radius of the core and the shell is 500 nm and 150 nm, respectively; the refractive index of the core and the shell is 1.39 and 1, respectively). To achieve our data-driven prediction, we selected three ML models (i.e., linear regression, neural network, and random forest) that are among the most popular ML models for scientific application. The detailed logic and mathematical explanations are included in the Method section.

(i). Linear Regression Model (Supplementary Code 1): This is one of the simplest and most interpretable ML models, primarily used for the predictive modeling of continuous outcomes. It establishes a relationship between the input features (e.g., pore sizes and wavelengths) and the output (e.g., scattering efficiency) by fitting a straight line through the data points. The model assumes a linear correlation between the variables, making it easy to understand and implement. Linear regression is particularly useful when the relationship between the variables is straightforward and does not exhibit complex interactions. However, when dealing with more intricate patterns in the data, the linear model may underperform compared to more sophisticated algorithms.

(ii). Neural Networks (Supplementary Code 2): These are highly flexible machine learning models inspired by the structure of the human brain. They consist of multiple layers of interconnected nodes (neurons) that process input data through a series of transformations to capture complex patterns. In predicting scattering efficiency, neural networks can learn intricate relationships between pore sizes, wavelengths, and scattering behavior, making them particularly suitable for capturing non-linearities and subtle interactions in the data. The model’s strength lies in its ability to generalize well to large datasets and capture multi-dimensional dependencies, but it requires careful tuning of hyperparameters and sufficient data to prevent overfitting.

(iii). Random Forest Model (Supplementary Code 3): This is a robust ensemble learning method that combines the predictions of multiple decision trees to improve accuracy and control overfitting. Each decision tree is trained on a random subset of the data, and the final output is derived from averaging the predictions of all trees (for regression tasks). In the context of predicting scattering efficiency, random forest is effective at capturing non-linear relationships and interactions between variables, such as pore size and wavelength, which may influence scattering in complex ways. The model is also highly flexible, performing well even when the relationship between variables is not easily captured by simpler models like linear regression.

To achieve our data-driven scattering efficiency prediction, we constructed the following logical steps (Figure 1): (i). We first split the dataset with a split ratio of 4:1 into a training set and a test set. The training set was used to train models, which were later evaluated on the test set. The data splitting process was performed using the scikit-learn package’s “train_test_split” method with a fixed random seed of 42 to ensure consistent splitting for each model. (ii). After data splitting, we one-hot encoded the column pore size. Using the scikit-learn package’s “OneHotEncoder” class, we transformed each unique pore size value into an individual feature. (iii). Next, we trained the three models on the same training set and evaluated them on the test set. The best-performing model was the random forest regressor, which is discussed later. (iv). Finally, we used the best model (i.e., random forest model) to perform all subsequent prediction. The model is capable of predicting the scattering efficiency of single pore sizes under different wavelengths, as well as that of mixed pore sizes.

The training data and testing data were collected by recording the scattering efficiency of single-pore-size samples under wavelengths spanning from 0.4 $\mathsf{\mu }$m to 2.5 $\mathsf{\mu }$m. Consequently, models trained on these data excel at predicting the scattering efficiency of untested single-pore-size samples. The comparison between the original results and predicted results using different ML models is shown in Figure 2. The linear regression (Figure 2a) and neural network (Figure 2b) models showed significant discrepancies, with a mean squared error (MSE) of 0.2722 and 0.01892, respectively. In contrast, the random forest model (Figure 2c) provided exceptionally accurate predictions, with a mean squared error of 0.002948.

The poor fitting observed in the linear regression and neural network models can be attributed to the following reasons. After one-hot encoding 1182 unique pore sizes, the data dimensionality increased drastically from 2 to 1183. However, the training data includes only single-pore-size data points, leading to a sparsely populated high-dimensional feature space, where all data points are located at the boundaries. This situation introduces the “curse of dimensionality” [40], where data points appear sparse and dissimilar, making it difficult for models like linear regression and neural networks to capture meaningful relationships, especially when the number of data points is limited relative to the number of features.

In high-dimensional spaces, data points become increasingly sparse. If we have features and N data points, as the number of dimensions increases, the volume of the feature space grows exponentially. Consequently, the density of data points decreases, as shown in the following Equation (1).

$$\rho ={\displaystyle \frac{N}{V_d}}$$

(1)

where $V_d$ is the volume of the $d$-dimensional feature space. Since $V_d\propto C^d$, where $C$ is a constant, as $d$ increases, $\rho$ decreases exponentially, making the data sparse.

Additionally, the linear regression does not handle feature interactions well, limiting its ability to capture complex relationships in the data. While neural networks have a higher capacity for modeling complex, non-linear relationships, they may fail to learn meaningful patterns without sufficient data to populate the feature space. Although regularization techniques, such as weight decay, can mitigate overfitting to some extent, neural networks still face challenges posed by the high dimensionality and sparsity of the data. In the future, as we collect more data to fill the feature space, we expect the neural network performance to improve, potentially surpassing that of random forest models.

In contrast, random forests perform well in high-dimensional spaces due to their ensemble structure. Each tree in a random forest is trained on a random subset of features, reducing the risk of overfitting and allowing the model to capture diverse relationships within the data. By combining predictions from multiple trees, random forests average out the errors of individual trees, producing a more stable and robust model. Additionally, random forests handle irrelevant or noisy features effectively, as each split considers only a subset of features, reducing the model’s sensitivity to the total number of features. From the perspective of ensemble learning [41], random forests mitigate the sparsity challenge by averaging the predictions of multiple decision trees. Assuming the trees are uncorrelated (which is approximated by random feature selection and bootstrapping), the variance of the ensemble decreases as the total number of trees increases.

Although the random forest model is the most accurate model for predicting scattering efficiency in our study, we have shared the code for all three models to make our platform open access. Moving forward, we will primarily focus on the predicted results from the random forest model. Based on the excellent prediction in scattered data points (Figure 2), we also predicted the scattering efficiency distribution across the entire wavelength range (0.4 $\mathsf{\mu }$m to 2.5 $\mathsf{\mu }$m) for different single pore sizes. The results are shown in Figure 3 and the Supplementary Video. These predictions indicate that the majority of the scattering efficiency was contributed by pore sizes between 1 $\mathsf{\mu }$m and 5 $\mathsf{\mu }$m.

Based on the above scattering efficiency predictions for a single pore size at different wavelengths, we performed a generative-ML-based scattering efficiency prediction for mixed pore sizes at different wavelengths (Figure 4). This was conducted using the random forest model with simple data transformation. The transformation involved one-hot encoding the pore sizes, treating each unique pore size as a distinct feature. For example, if a data point previously had a pore size of 0.1 $\mathsf{\mu }$m, it now has a value of 1 for the feature of “pore size = 0.1 $\mathsf{\mu }$m”. During model inference, a data point that had values of 0.5 for both “pore size = 1.0 μm” and “pore size = 5.0 μm” would represent a composite material containing equal parts (50% each) of two distinct pore sizes. Note that these predicted data are produced by this generative-ML-based model; future numerical modeling is necessary to cross-check that model prediction accuracy.

To further accelerate the experimental design of PDRC materials, we have shared the generative random forest model and other models as an open access resource. By making this tool available, we aim to empower researchers to streamline the design process, potentially reducing the need for extensive trial-and-error experiments and efficiently exploring the design space of microstructured materials for enhanced PDRC performance. Although we recommend using numerical simulations to validate the accuracy of our generative ML-based predictions for scattering efficiency, our generative random forest model serves as an excellent reference for PDRC researchers. This model can guide the exploration of various material synthesis processes to tune microstructures, potentially achieving enhanced scattering efficiency in designed materials. Thus, this approach would accelerate the development of high-performance PDRC materials.

3. Future Work

We acknowledge that our current implementation is limited to spherical scatterers embedded in an infinite matrix, which does not fully represent real-world PDRC materials with hierarchical or anisotropic structures. However, there are no current data available for training ML models to predict diverse geometries. We have made our coding an open access resource. In this case, when diverse geometry data become available, our ML is a very good resource to accelerate advancement in the PDRC field. Therefore, we provided two possible future focuses aligned with this work for reference: (i). When more training datasets (i.e., datasets with diverse pore geometries), either from experiments, numerical simulation, or both, become available, expanding the training dataset to include more diverse geometries and refractive index combinations is necessary. (ii). ML predictions should be compared with validated numerical simulations for polydisperse and non-spherical systems.

4. Conclusions

In this work, the major contribution to the PDRC field is that we addressed the challenges associated with designing efficient PDRC materials by introducing a novel data-driven platform utilizing ML. By focusing on the relationship between material microstructures and scattering efficiency, we effectively demonstrated the utility of our approach in predicting the scattering efficiency of PDRC materials. Among the tested models, the random forest model proved to be the most effective for predicting scattering efficiency in single-pore-size configurations. Furthermore, by leveraging the prediction success of the random forest model in a single-pore-size configuration, we employed a data transformation strategy, using one-hot encoding, to generatively model the scattering efficiency in mixed-pore-size configurations.

By making our ML models and code openly accessible, we aim to support the broader research community in exploring innovative PDRC material designs while reducing the experimental burden. Ultimately, our approach has the potential to contribute significantly towards the development of scalable, high-performance PDRC solutions that reduce global energy consumption and mitigate the environmental impacts associated with conventional cooling methods.