DMFS: Differentiable Modeling for Frozen Soil Thermodynamic Characteristics

Thermodynamic characteristics and constitutive relationships are used to predict frozen soil’s behaviors in cold regions. Physical models typically require robust thermodynamic constitutive relationships; however, estimating these relationships of frozen soils is challenging and often requires time-consuming and expensive field and laboratory testing. This study introduces a novel differentiable modeling for frozen soil to obtain its thermodynamic characteristics, termed as the DMFS model. DMFS infers key constitutive relationships that govern heat transfer in frozen soils using observed soil temperature data. In this model, three key thermodynamic constitutive relationships, i.e., the soil freezing characteristic curve, thermal conductivity curve, and heat capacity curve, are represented by physically constrained neural networks (PCNNs). These PCNN-represented constitutive relationships are embedded in heat transfer equations for forward modeling and yield temperature predictions, which are compared against observations to construct a loss function. The forward modeling was implemented via a differentiable numerical solver, which enables backpropagation of loss gradient to PCNNs. By minimizing the loss function with gradient backpropagation, PCNNs are optimized to accurately capture the unknown constitutive relationships. Results from both numerical experiments and in-situ borehole experiments demonstrated that the DMFS model could effectively infer constitutive relationships, even with sparse and noisy data. These findings indicate that DMFS holds a significant promise for characterizing frozen soil’s thermodynamic properties in cold regions.