Thermodynamically consistent physics-informed neural network-finite element method for viscoelastic fluids using square-root conformation

Viscoelastic fluids are commonly found in nature and industry, yet their accurate numerical simulation remains challenging due to complex constitutive relations and numerical instabilities. Here, we propose a hybrid PINN-FEM framework that combines Physics-Informed Neural Networks (PINN) with the finite element method (FEM) for simulating viscoelastic flows. The proposed method is strictly guided by the general equation for non-equilibrium reversible–irreversible coupling (GENERIC) formalism to ensure thermodynamic consistency and employs the square-root conformation representation to expose the intrinsic low-dimensional structure of the constitutive equations. Specifically, polymer entropy can be expressed as a scalar function of the eigenvalues of the square-root conformation tensor. Leveraging this fact, we transform the constitutive learning task from high-dimensional tensor regression into a simple scalar function approximation, thereby significantly reducing the dimensionality of the learning problem and the required training data size. We validate the method on Poiseuille flow, four-roll mill flow, and the flow around a periodic array of cylinders across a range of Weissenberg numbers. The results demonstrate that as the Weissenberg number increases, the viscoelastic fluids exhibit enhanced elasticity, leading to a nonlinear expansion of the regions covered by the eigenvalues of the square-root conformation tensor. By combining the physical rigor of GENERIC with the data-driven flexibility of PINN and the computational power of FEM, this work provides an efficient, accurate, and thermodynamically consistent approach for developing constitutive models, with promising extensibility to more complex fluid systems.