NRNN: A physics-informed neural framework for non-linear rock modelling and deep tunnel analysis

Modelling the highly non-linear deformation of rocks remains a formidable challenge for purely data-driven neural networks. To overcome these limitations, this study proposes the Non-linear Analytical Rock Neural Network (NRNN), a novel physics-informed deep learning framework. By explicitly embedding physical constraints into the forward network architecture, the NRNN incorporates mathematical constitutive laws as rigorous structural regularisers, thereby enhancing both stability and physical admissibility. When evaluated against comprehensive independent experimental datasets encompassing limestone, marble, sandstone, and shale, the NRNN effectively eliminates the physically inadmissible predictions observed in baseline Artificial Neural Networks (ANNs), reducing principal stress prediction errors (MSE) by up to 75%. Furthermore, when extended to boundary value problems, such as deeply buried tunnels, the framework employs an autoregressive deduction engine to simulate the mechanical response of deep circular tunnels. It successfully reconstructs complex full-field stress–strain distributions without the accumulation of spatial errors. The physics-informed architecture thus proves essential not only for eliminating physically inadmissible predictions, but also for achieving superior generalisation and physical consistency across diverse lithologies, offering a robust and computationally efficient tool for real-time back-analysis in deep underground engineering.