• Modified physics informed neural networks solve 3D flows in complex pipes without data • Geometry based input features improve accuracy in curved three dimensional domains • Single neural network predicts steady 3D flows over a Reynolds number interval • Training uses varying Reynolds numbers without adding a separate parameter dimension • Finite difference derivatives reduce memory use and training time in 3D problems. We introduce a set of modifications to Physics-informed neural networks (PINNs) for three-dimensional Navier–Stokes equations in complex geometries, demonstrated by viscous incompressible flow through a curved pipe with a rectangular cross-section. To enhance training stability and reduce computational cost, numerical derivatives based on finite differences are employed instead of automatic differentiation, along with gradient accumulation during the optimization process. Geometry-informed input features are introduced and incorporated via a dedicated preprocessing layer. Numerical experiments, benchmarked against finite volume simulations, show that these modifications substantially decrease relative approximation errors and accelerate convergence, despite the geometric complexity of the domain. Furthermore, by training a single PINN model parametrically over Reynolds numbers 1 ≤ Re ≤ 200, we achieve accurate solutions across multiple flow regimes. Finally, we benchmark our geometry-informed embeddings against Fourier feature mappings to assess their relative effectiveness in both fixed- Re and parametric PINN settings.
Tsgoev et al. (Wed,) studied this question.