Adaptively balanced Poisson-constrained physics-informed neural networks for robust displacement integration in background-oriented Schlieren

Displacement integration in background-oriented Schlieren (BOS) is a critical step in the reconstruction of physical fields. This process typically employs high-order fitting integration or discrete Poisson solvers. This paper examines the strengths and limitations of these conventional approaches and proposes a novel physics-informed neural network (PINN) framework constrained by the Poisson equation, termed the adaptively balanced Poisson-constrained PINN (AB-PoissonPINN). The proposed method incorporates relative loss balancing with random backtracking (ReLoBRaLo) to dynamically balance the contributions of different loss components. The integration performance of AB-PoissonPINN is evaluated through both simulated and experimental numerical integration and is benchmarked against established techniques, including weighted cubic spline least squares integration (WCSLI), discrete Poisson solvers, and standard PINN. Experimental results demonstrate that AB-PoissonPINN consistently achieves higher accuracy than WCSLI, discrete Poisson solvers, and standard PINN under both noise-free conditions and various noise levels.