xLSTM for constitutive modelling of sand: From data-driven extrapolation to thermodynamic consistency

¹ Department of Civil and Environmental Engineering, Indian Institute of Technology, Delhi, New Delhi
² Department of Applied Mechanics, Indian Institute of Technology, Delhi, New Delhi

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Abstract

Constitutive modeling of granular soils is inherently a path-dependent problem in which the stress response at any loading step depends on the entire history of deformation and on the initial state of the specimen. Long Short-Term Memory (LSTM) networks have become the dominant recurrent architecture for data-driven constitutive models of soil, yet their scalar gating mechanism is architecturally mismatched to the covariance-rich loading histories observed in triaxial stress paths. This paper proposes the use of xLSTM [1] (Extended Long Short-Term Memory) for constitutive modeling of sand, and demonstrates its superiority over a standard LSTM baseline on a challenging extrapolation benchmark of drained triaxial compression tests on Karlsruhe sands. The mLSTM block at the core of xLSTM uses a matrix-valued memory cell updated by a covariance rule, enabling it to represent second-order interactions between loading-history features that scalar-gated LSTMs fundamentally cannot. Further a ThermoXLSTM, a thermodynamics-informed extension is introduced, in which effective stresses are derived from a learnable Helmholtz free-energy potential via automatic differentiation, enforcing the first and second laws of thermodynamics by construction. Model testing is done for the smallest and highest initial confining pressures (50 and 400 kPa) for different relative-density groups – conditions absent from training – creating a genuine extrapolation scenario. On this benchmark, xLSTM achieves R²>0.92 for mean effective stress p', deviatoric stress q, and volumetric strain ε_ν, improving NMSE by 34% and 18% over LSTM for p' and q respectively. ThermoXLSTM yields slightly lower accuracy metrics but guarantees thermodynamically admissible stress predictions regardless of the input conditions. Together, these models establish xLSTM as the preferred recurrent backbone for path-dependent geomechanical constitutive modeling and demonstrate that physics constraints can be embedded without sacrificing architectural advantage.