Ivan Stebakov

Physics-informed neural networks (PINNs) have demonstrated great promise in solving partial differential equations without labeled data, yet their performance often deteriorates for highly nonlinear systems — particularly steady non-Newtonian flows governed by power-law or yield-stress rheologies. In this work, we present a systematic comparative study of PINN training strategies for Couette flow of pseudoplastic fluids between coaxial cylinders, governed by the Ostwald – de Waele and Herschel – Bulkley constitutive laws. We evaluate three established differential-form PINN variants — baseline (fixed loss weights), curriculum learning, and adaptive loss weighting — and introduce a novel variational PINN (vPINN) derived from the dissipation potential of Herschel – Bulkley fluids. Crucially, the proposed vPINN embeds the principle of minimum dissipation directly into the loss functional via the analytically integrated shear-dependent potential $\Phi(\dot{\gamma}) = q_0^{}\dot{\gamma}^2 + \frac{2q_1^{}\dot{\gamma}^{\,z+1}}{z+1}$, thereby enforcing physics through a variational principle rather than residual minimization. Using an exact analytical solution as ground truth, we benchmark all models on velocity and pressure reconstruction across varying gap geometries $\Bigl($dimensionless parameter $\gamma=\frac{R_1^{}}{R_2^{}-R_1^{}}\Bigr)$. While adaptive weighting improves pressure recovery by $6$–$8\,\%$ (MAE, $L_\infty^{}$, RSD), all differential PINNs exhibit nearly identical — and limited — accuracy for velocity prediction, with no benefit from curriculum scheduling. In contrast, the vPINN achieves a substantial and consistent gain in velocity accuracy: for $\gamma=4.0$, RSD drops from $1.09\,\%$ (all PINNs) to $0.85\,\%$ ($-22\,\%$); for the widest gap ($\gamma=0.67$), RSD falls from $6.0\,\%$ to $4.07\,\%$ ($-32\,\%$). MAE and $L_\infty^{}$ errors decrease by $27\,\%$ and $23\,\%$, respectively. These improvements arise because the variational formulation naturally mitigates spectral bias and avoids ill-conditioned gradients inherent to the highly nonlinear Navier – Stokes equation. The vPINN network also trains approximately twice as fast. Although vPINN currently predicts only velocity (pressure is recovered a posteriori), its consistent accuracy gains — especially where non-Newtonian effects dominate — establish variational PINNs as a compelling, physics-based alternative to residual-based approaches for complex rheological modeling.