Physics-informed neural network for the simulation of turbulent jet grout flow

Abstract This study presents a physics-informed neural network (PINN) framework to simulate axisymmetric submerged jet grout flow using high-fidelity data generated from ANSYS Fluent simulations. The CFD model solves the Reynolds-averaged Navier–Stokes (RANS) equations coupled with the $$k - \varepsilon$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>-</mml:mo> <mml:mi>ε</mml:mi> </mml:mrow> </mml:math> turbulence closure model, converted into axisymmetric form to reflect jet grouting. The PINN incorporates continuity, RANS equations, and turbulence closure equations into the loss function, enabling it to learn turbulent flow physics without requiring mesh generation. Separate neural networks were used for each flow variables—axial velocity ( $$U$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>U</mml:mi> </mml:math> ), radial velocity ( $$V$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>V</mml:mi> </mml:math> ), pressure ( $$P$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>P</mml:mi> </mml:math> ), turbulent kinetic energy ( $$k$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>k</mml:mi> </mml:math> ), and dissipation rate ( $$\varepsilon$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ε</mml:mi> </mml:math> ). The model was trained using a curriculum learning strategy to enhance stability and avoid gradient explosion and the predicted results closely matched the CFD outputs. This study demonstrates that PINNs can serve as efficient and accurate surrogate models for complex, high-gradient flow systems like jet grouting, offering a promising tool for understanding submerged jet flow behavior and reducing computational demands in geotechnical engineering.