TY - JOUR

T1 - Fast parametric three-dimensional physics-informed neural network approach for fluid flows in complex geometries

AU - Tsgoev, Chermen

AU - Sakharov, Danil

AU - Bratenkov, Miron

AU - Mullyadzhanov, Rustam

N1 - Chermen Tsgoev, Danil Sakharov, Miron Bratenkov, Rustam Mullyadzhanov, Fast parametric three-dimensional physics-informed neural network approach for fluid flows in complex geometries, Results in Engineering, Volume 30, 2026, 110582, ISSN 2590-1230, https://doi.org/10.1016/j.rineng.2026.110582. This work was supported by a grant for research centers, provided by the Ministry of Economic Development of the Russian Federation in accordance with the subsidy agreement with the Novosibirsk State University dated April 17, 2025 No. 139-15-2025-006: IGK 000000C313925P3S0002.

PY - 2026/6

Y1 - 2026/6

N2 - 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. These results are obtained within the MultiPINN framework available via https://github.com/labadt/multipinn.

AB - 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. These results are obtained within the MultiPINN framework available via https://github.com/labadt/multipinn.

KW - Geometry-informed input features

KW - Navier–Stokes equations

KW - Parameterized PINN

KW - Physics-informed neural networks

KW - Нейронные сети, учитывающие физические закономерности

KW - Уравнения Навье — Стокса

KW - Входные признаки, учитывающие геометрию

KW - Параметризованные нейронные сети, учитывающие физические закономерности (PINN)

UR - https://www.mendeley.com/catalogue/6b10864d-6b93-3408-82cb-8d2c2fa18edf/

UR - https://www.scopus.com/pages/publications/105038049500

U2 - 10.1016/j.rineng.2026.110582

DO - 10.1016/j.rineng.2026.110582

M3 - Article

VL - 30

JO - Results in Engineering

JF - Results in Engineering

SN - 2590-1230

M1 - 110583

ER -