Interpretable Symbolic Regression for PM1 Estimation from Routine Air Quality Measurements: Implications for Sustainable Air Quality Management

Fine particulate matter (PM1) is regarded as one of the most hazardous atmospheric aerosol fractions because of its ability to penetrate deeply into the respiratory system and its high contribution of secondary inorganic and organic components. Despite its environmental and public health significance, PM1 is not routinely monitored in most national air quality monitoring networks, creating a need for methods capable of estimating its concentrations from commonly available observations. This study developed interpretable symbolic regression models for estimating daily PM1 concentrations using routinely monitored particulate matter fractions and meteorological variables. Daily observations collected at the Warsaw University of Life Sciences (SGGW) monitoring station between 2023 and 2025 were analysed. Predictor variables included PM2.5, PM10, total suspended particles (TSP) and routinely measured meteorological parameters. Symbolic regression implemented using the PySR library (version 1.5.10) in Python 3.12 was employed to identify explicit mathematical equations without assuming a predefined mathematical structure. Because PM1 is physically nested within PM2.5, the study additionally examined the PM1/PM2.5 mass concentration ratio to distinguish the expected structural relationship between the two fractions from additional variability associated with atmospheric conditions. The PM1/PM2.5 ratio showed substantial variability, with an overall mean of 0.849 and daily values ranging from 0.520 to 0.962, while annual mean values varied between 0.831 and 0.863. Baseline models based on a fixed PM1/PM2.5 ratio and conventional linear regression were also evaluated. Among the symbolic regression models, the equation using PM2.5 and relative humidity showed the highest predictive performance, with R2 = 0.9946, MAE = 0.3658 µg·m−3 and MSE = 0.1984 (µg·m−3)2 for the holdout test dataset. The model based solely on PM2.5 also showed high predictive performance (R2 = 0.9917), confirming the strong structural relationship between PM1 and PM2.5. The fixed PM1/PM2.5 ratio provided lower predictive accuracy than the conventional linear regression models, indicating that the relationship between the two particulate matter fractions cannot be adequately described by a single constant ratio. On the same holdout testing subset, conventional linear regression using PM2.5 achieved R2 = 0.9932 and MAE = 0.4066 µg·m−3, while the addition of relative humidity increased R2 to 0.9956 and reduced MAE to 0.3126 µg·m−3. The results indicate that the high predictability of PM1 from PM2.5 is primarily associated with the intrinsic relationship between the two nested particulate matter fractions, whereas meteorological variables provide complementary information, particularly when broader particulate matter fractions such as PM10 and TSP are used. The main contribution of symbolic regression is therefore the data-driven discovery of explicit mathematical relationships and parsimonious predictor combinations rather than superior predictive accuracy compared with conventional regression. The developed equations should be regarded as empirical relationships requiring independent temporal and spatial validation before application beyond the investigated dataset.