Fine-tuning physics-informed neural networks for cavity flows using coordinate transformation

TL;DR

This work tackles the high training cost of physics-informed neural networks (PINNs) for incompressible Navier–Stokes cavity flows by introducing a coordinate transformation-based fine-tuning strategy. A pre-trained PINN, possibly trained on a different geometry or Reynolds number, is used to initialize a target network, while a deformation gradient $\mathbf{F}$ maps the reference configuration to the target geometry, embedding the transformation into the PDE loss. Numerical experiments across square, rectangular, shear-deformed, and nonlinear inflated cavities show that fine-tuning accelerates convergence and maintains accuracy, with the greatest gains when the pre-training geometry resembles the target. The approach holds potential for real-world, geometry-rich applications such as intra-aneurysmal blood flow modeling, where rapid, accurate estimates are essential. Extensions to 3D and automated pre-training selection are identified as future directions.

Abstract

Physics-informed neural networks (PINNs) have attracted attention as an alternative approach to solve partial differential equations using a deep neural network (DNN). Their simplicity and capability allow them to solve inverse problems for many applications. Despite the versatility of PINNs, it remains challenging to reduce their training cost. Using a DNN pre-trained with an arbitrary dataset with transfer learning or fine-tuning is a potential solution. However, a pre-trained model using a different geometry and flow condition than the target may not produce suitable results. This paper proposes a fine-tuning approach for PINNs with coordinate transformation, modelling lid-driven cavity flows with various shapes. We formulate the inverse problem, where the reference data inside the domain and wall boundary conditions are given. A pre-trained PINN model with an arbitrary Reynolds number and shape is used to initialize a target DNN. To reconcile the reference shape with different targets, governing equations as a loss of the PINNs are given with coordinate transformation using a deformation gradient tensor. Numerical examples for various cavity flows with square, rectangular, shear deformed and inflated geometries demonstrate that the proposed fine-tuning approach improves the training convergence compared with a randomly-initialized model. A pre-trained model with a similar geometry to the target further increases training efficiency. These findings are useful for real-world applications such as modelling intra-aneurysmal blood flows in clinical use.

Figures (23)

• Figure 1: A difficulty of a fine-tuning approach that the pre-trained DNN is applied to a new DNN for a cavity flow with a different aspect ratio. The black arrows denote flow vectors at the same coordinate point in the reference configuration (square domain) and target configuration (rectangular domain), whereas the blue arrow indicates the correct flow vector in the target configuration.
• Figure 2: Overview of the PINN in the current configuration for the stationary incompressible NS equations.
• Figure 3: A schematic of the coordinate transformation between the reference configuration ${\bf x}^R=x^R_i{\bf e}^R_i$ and target configuration ${\bf x}=x_i{\bf e}_i$ ($i\in[1,d]$).
• Figure 4: The proposed fine-tuning model for PINNs with coordinate transformation for cavity flows.
• Figure 5: Construction of reference data (and ground truth) obtained by the computational fluid dynamics for two-dimensional cavity flow problems.