Machine learning with differentiable modeling in geotechnical engineering: a case study of bearing capacity of shallow foundation on cohesionless soil

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Machine learning models can capture empirical relationships but often generalize poorly under data scarcity and distribution shift, whereas classical bearing-capacity equations are interpretable but do not adapt to data. We develop a differentiable modeling neural network (DMNN) for ultimate bearing capacity of shallow foundations on cohesionless soils. The general bearing-capacity equation serves as a differentiable structural prior, with analytical factors remaining explicit and the neural network restricted to learning positive, dimensionless shape and depth modifiers. The DMNN is compared with a fully connected neural network (FCNN) using synthetic datasets generated by the Vesic, Meyerhof, and Hansen formulations, noise robustness tests, and 97 laboratory and field observations. Leave-one-range-out tests based on friction angle ϕ and footing width B assess transfer to unseen input domains under matched and mismatched label-generating formulations. On the Vesic dataset, the DMNN achieved mean R 2 values of 0.83 and 1.00 for ϕ - and B -range transfer, versus -0.23 and -0.08 for the FCNN, and retained its advantage under formulation mismatch and controlled label noise. When evaluated on the database of 97 laboratory and field observations, the DMNN generally outperformed the FCNN across training sizes of 10–70 records, approached parity with Meyerhof at approximately 30 records, and surpassed all four classical formulas from 40 records onward. In laboratory-to-field transfer, the DMNN achieved R 2 = 0.50 versus -0.57 for the FCNN; reverse transfer remained difficult for both. SHAP analysis showed a more physically consistent feature-response trend for soil unit weight. Across these tests, embedding explicit mechanics improved sample efficiency, robustness, transferability, and interpretability while retaining data-driven adaptability.