Physics-constrained DeepONet for Surrogate CFD models: a curved backward-facing step case

TL;DR

This work addresses the challenge of efficiently surrogate CFD for parameterized curved backward-facing steps by introducing Physics-Constrained DeepONet (PC-DeepONet), which enforces incompressibility via the continuity equation $ \nabla \cdot \mathbf{u} = 0$ through a skew-symmetric operator $A$ with $ \mathbf{v} = \nabla \cdot A$ and $A = J_{\mathbf{b}} - J_{\mathbf{b}}^\top$. The method maps geometry parameters (via a cubic NURBS parameterization) to flow fields ($U$, $V$, $P$) using a DeepONet augmented to respect physics, ensuring divergence-free velocity even with sparse training data. Empirical results show PC-DeepONet outperforms a data-driven baseline, achieving a relative $L_2$ error around $4.45\times 10^{-3}$ with rapid convergence (50 iterations) on unseen geometries, while highlighting boundary-layer sampling as a source of error. The study demonstrates effective, data-efficient operator learning for CFD surrogates and points to future work on boundary conditions and turbulence at higher Reynolds numbers.

Abstract

The Physics-Constrained DeepONet (PC-DeepONet), an architecture that incorporates fundamental physics knowledge into the data-driven DeepONet model, is presented in this study. This methodology is exemplified through surrogate modeling of fluid dynamics over a curved backward-facing step, a benchmark problem in computational fluid dynamics. The model was trained on computational fluid dynamics data generated for a range of parameterized geometries. The PC-DeepONet was able to learn the mapping from the parameters describing the geometry to the velocity and pressure fields. While the DeepONet is solely data-driven, the PC-DeepONet imposes the divergence constraint from the continuity equation onto the network. The PC-DeepONet demonstrates higher accuracy than the data-driven baseline, especially when trained on sparse data. Both models attain convergence with a small dataset of 50 samples and require only 50 iterations for convergence, highlighting the efficiency of neural operators in learning the dynamics governed by partial differential equations.

Figures (5)

• Figure 1: Control points for parameterisation of the BFS slope.
• Figure 2: Illustration of the curved BFS used for training the DeepONet.
• Figure 3: Architecture of the DeepONet used to learn the operator between slope shape and flow field (adapted from moya2022fed).
• Figure 4: Comparison between the output of the DeepONet on an unseen test case with the ground truth for U, V and P.
• Figure 5: Comparison between the output of the Physics-Constrained DeepONet with the ground truth for U, V and P.