Virtual domain extension for imposing boundary conditions in flow simulation using pre-trained local neural operator

TL;DR

The paper tackles the challenge of reusing a pre-trained local neural operator (LNO) for flow predictions across varying boundary conditions by introducing Virtual Domain Extension (VDE). VDE treats BC imposition as determining values on an extended region beyond the physical domain, leveraging the LNO’s inherent domain corrosion to encode boundary information. It proposes three BC-imposition strategies for non-extendable BCs (direct imposition, pressure symmetry, and backpropagation-based optimization) and shows that the synchronous optimization approach delivers the most accurate predictions across test cases, while extendable BCs are handled via padding. Numerical results for Poiseuille flow, flow around a cylinder, and flow around a vehicle demonstrate substantial improvements in boundary accuracy and overall flow fields when using optimization-based BC imposition, validating VDE as a practical pathway to reuse pre-trained LNOs in diverse CFD applications. The work thus provides a general framework to reliably impose BCs in LNO-based flow predictions and points to future enhancements in LNO accuracy and BC generalization, with code available for reproducibility.

Abstract

This paper builds up a virtual domain extension (VDE) framework for imposing boundary conditions (BCs) in flow simulation using pre-trained local neural operator (LNO). It creates extended virtual domains to the input function to compensate for the corrosion nature of computational domains during LNO inference, thus turns the implementation of BC into the determination of field values on the extended domain. Several strategies to calculate the field values are proposed and validated in solving numerical examples, including padding operation, direct imposition, pressure symmetry, and optimization by backpropagation, and compared with boundary imposition in traditional solvers. It is found that the large time interval of LNO induces a relatively wide near-boundary domain to be processed, thus imposing BC on only a few nodes near the boundary following the immersed boundary conception in traditional solvers can hardly achieve high accuracy. With appropriate values assigned on the extended virtual domains, VDE can accurately impose BCs and lead to reasonable flow field predictions. This work provides a guidance for imposing BCs reliably in LNO prediction, which could facilitate the reuse of pre-trained LNO in more applications.

Figures (24)

• Figure 1: Comparison between the boundary treatment in FDM and LNO. (a) Ghost point in a 5-point central finite difference scheme. (b) Extended domain to predict the output on the desired computational domain in LNO.
• Figure 2: The time-marching local operator $\mathcal{G}_\text{L}$. $\boldsymbol{u}_t,\boldsymbol{u}_{t+\Delta t}$ are the physical fields at time level $t$ and $t+\Delta t$, respectively. $\Delta t$ is the time interval. $\Omega_\text{out}/\Omega_\text{in}$ is the output/input computational domain. $D_\text{out}/D_\text{in}$ is the unit output/input domain. $\boldsymbol{X}$ is an arbitrary shifting vector.
• Figure 3: The training process of LNO. $C$ is the extra extended size to ensure the input is exactly covered by an integer number of unit input domain.
• Figure 4: Time-marching prediction process of pre-trained LNO with boundary treatment
• Figure 5: Schematic of the input computational domain before and after padding operation. Each colored block represents one node in the space. Here the extension width is set as $2$ just for demonstration purposes.