Explainable AI yields a five-parameter formula for nonlinear concrete creep

Concrete that carries a heavy load for years does not simply hold still. It keeps deforming, slowly and persistently, in a time-dependent process known as creep. At stress levels below roughly 0.4 of the material’s compressive strength, this deformation grows roughly in proportion to the applied stress, a regime engineers call linear creep. Push the stress ratio above 0.4, however, and the relationship breaks down: creep deformation becomes nonlinear, the creep rate climbs sharply, and microcracks begin to propagate in ways that can end in material failure. Large-span arch bridges, super-high-rise buildings, and high arch dams all endure sustained high stresses over long service periods, which makes accurate prediction of this nonlinear regime a genuine safety concern rather than an academic curiosity.

A research team now reports a computational framework that tackles this problem by fusing three tools: a hyperparameter-optimized gradient-boosting model, a game-theoretic explanation technique, and an automated equation-discovery algorithm. The pipeline is summarized as prediction, explainability analysis, and data fitting. First, a LightGBM model whose settings were tuned by the Dung Beetle Optimizer maps eleven input parameters onto concrete creep compliance. Then SHAP analysis identifies which parameters actually drive the predictions. Finally, the PySR symbolic regression tool distills the learned relationship into a compact mathematical formula that engineers can evaluate by hand, with no black-box model in sight.

The data foundation comes from the Northwest University database, a collection of 29,177 measured observations of concrete creep flexibility spanning different compressive strengths and loading conditions. Because the study targets the nonlinear regime, the researchers filtered the database to retain only creep tests with stress levels exceeding 0.4 and complete records for all input parameters. That screening left 1,169 valid samples, which were randomly divided into 80 percent for training and 20 percent for testing. The eleven input variables cover material composition, mechanical properties, specimen geometry, environmental conditions, and loading history, ranging from the aggregate-cement ratio and cement content to the 28-day strength, elastic modulus, volume-surface ratio, loading age, temperature, relative humidity, applied stress, stress level, and loading duration.

LightGBM, a gradient-boosted decision tree framework, was chosen for its fast training speed and low memory consumption, and for its ability to capture nonlinear responses and interactions among variables without a complex neural network architecture. Its performance, however, depends heavily on hyperparameters such as the learning rate, tree depth, and number of estimators. The Dung Beetle Optimizer addresses this by mimicking four behaviors observed in dung beetles, including ball rolling, reproduction, foraging, and stealing, to search the hyperparameter space. Using the coefficient of determination on the training set as the fitness function, the team ran the optimization ten times independently. The resulting model achieved a mean test R² of 0.96 with an RMSE of 10.05 and an MAE of 6.74, and the small standard deviations across runs indicated a stable optimization process.

The benchmark comparison was decisive. Against standard LGBM, XGBoost, a back-propagation neural network, a general artificial neural network, and linear regression, the optimized model reduced the mean absolute error by 42.6, 51.7, 56.6, 55.7, and 66.3 percent, respectively. Linear regression fared worst, a result the authors attribute to its assumption of a linear input-output relationship, which cannot represent creep behavior shaped by multiple interacting factors. To probe generalizability more rigorously, the team also performed leave-one-study-out validation, holding out every specimen from one literature source at a time so that no data from the same study appeared in both training and test sets. Even under this stricter regime, the model maintained an overall R² of 0.93, with an RMSE of 13.81 and an MAE of 9.69.

Accuracy alone, though, was not the goal. Machine learning models in civil engineering often operate as black boxes, and this lack of interpretability constrains their credibility and adoption. The SHAP method, which approximates a complex model with stacked linear models and quantifies each feature’s contribution, was applied to all six machine learning models. Across every architecture, the stress level and the applied stress ranked as the two most important features, confirming that the stress state of concrete is the decisive factor in nonlinear creep. The optimized model stood out for placing loading duration third, a ranking consistent with the physical essence of creep, which develops over time. The neural network models, by contrast, elevated the volume-surface ratio and downweighted loading duration, while linear regression produced the most divergent ranking of all.

The SHAP analysis identified seven features with the greatest influence: stress level, applied stress, loading duration, relative humidity, 28-day compressive strength, loading age, and elastic modulus. The first three correlate positively with nonlinear creep, the remaining four negatively. Local, single-sample explanations added nuance that global rankings alone cannot capture. In one pair of scenarios identical except for loading duration, extending the load from seven days to 296 days pushed the predicted creep compliance above the baseline, illustrating creep’s cumulative nature. In another pair, raising the temperature from 20 degrees Celsius to 93 degrees Celsius shifted the prediction from 105.7 to 150.8, turning temperature from a negative factor into the most critical positive one, an effect the authors link to altered water molecule mobility and reduced resistance to stress relaxation within the concrete matrix.

The final step converted these insights into an explicit equation. Because symbolic regression is highly sensitive to input dimensionality, with redundant features causing an explosion in the search space, the team first used the SHAP rankings to construct candidate feature subsets: a tree-based key set, a neural-network-based key set, a linear-model-based key set, and the complete feature set. PySR then evolved mathematical expressions using genetic algorithms with mutation, crossover, and simulated annealing, balancing prediction error against formula complexity. The winning formula, drawn from the tree-based key set, uses only five parameters: loading duration, loading age, relative humidity, stress level, and 28-day strength. It achieved a mean ratio of 0.99, an RMSE of 0.21, an MAE of 0.17, and an R² of 0.81, outperforming the B4 model by 14.1 percent in R² and the classical MC 2010, ACI 209, and AASHTO models by 22.7, 35.0, and 39.7 percent respectively.

A one-at-a-time sensitivity analysis lent the formula physical plausibility. Creep compliance rose steeply during the initial loading phase before the growth rate gradually declined, matching the typical time-logarithmic character of concrete creep. Loading age showed a rapid initial decrease in creep that stabilized around 28 days, consistent with the mid-to-late stages of cement hydration producing a more stable microstructure. Higher relative humidity reduced predicted creep, higher stress levels increased it, and higher 28-day strength suppressed it monotonically, with the trend flattening beyond 60 megapascals as denser high-strength microstructures yield diminishing returns. The authors caution that this analysis reflects local response trends around dataset mean values and does not capture interactions among input features.

The framework’s scope has clear boundaries. The proposed formula applies to predicting nonlinear creep of concrete within the parameter ranges of the filtered dataset, at stress levels greater than 0.4, under sustained uniaxial compressive loading. It does not explicitly characterize intrinsic mechanisms such as temperature-humidity coupling or the evolution of micro-aggregate interfaces, and its generalizability across different material systems and environmental conditions requires further validation. The authors suggest that integrating mechanism-based models with data-driven approaches, and incorporating cross-domain transfer learning to exploit shared features among concrete systems, could enhance both explainability and extrapolation in future work. Even so, the study offers a template for how black-box prediction, rigorous explanation, and symbolic equation discovery can be chained together to produce models that are simultaneously accurate, transparent, and practical enough for engineering design.

Subject of Research: An explainable machine learning framework for explicit modeling of nonlinear creep behavior in concrete

Article Title: A novel explainable data-driven method for explicit modeling of nonlinear creep behavior in concrete

Article References: Chen, M., Tang, Q., Chen, F., Jiang, Y., Zhou, J., & Xin, J. (2026). A novel explainable data-driven method for explicit modeling of nonlinear creep behavior in concrete. Case Studies in Construction Materials, 25, Article e06638. https://doi.org/10.1016/j.cscm.2026.e06638

Image Credits: AI Generated

DOI: 10.1016/j.cscm.2026.e06638

Keywords: concrete creep, nonlinear creep, machine learning, LightGBM, dung beetle optimizer, SHAP, symbolic regression, creep compliance, structural engineering, explainable AI, NU database, high stress concrete