Reliable and efficient inverse analysis using physics-informed neural networks with normalized distance functions and adaptive weight tuning

TL;DR

The paper tackles the persistent challenge of enforcing boundary conditions in physics-informed neural networks (PINNs) for both forward and inverse PDE problems. It introduces a framework that combines R-function–based normalized distance fields for hard boundary imposition with bias-corrected adaptive weight tuning to balance competing losses, enabling robust inverse analysis even on non-convex geometries. Numerical results on Poisson forward problems and incompressible-flow inversions show that the proposed method outperforms penalty-based approaches in accuracy and convergence speed, and remains effective without curriculum training. The work provides a practical, reliable approach for inverse PINN analysis applicable to complex engineering problems, with potential extensions to time-dependent domains and advanced network architectures.

Abstract

Physics-informed neural networks have attracted significant attention in scientific machine learning for their capability to solve forward and inverse problems governed by partial differential equations. However, the accuracy of PINN solutions is often limited by the treatment of boundary conditions. Conventional penalty-based methods, which incorporate boundary conditions as penalty terms in the loss function, cannot guarantee exact satisfaction of the given boundary conditions and are highly sensitive to the choice of penalty parameters. This paper demonstrates that distance functions, specifically R-functions, can be leveraged to enforce boundary conditions, overcoming these limitations. R-functions provide normalized distance fields, enabling flexible representation of boundary geometries, including non-convex domains, and facilitating various types of boundary conditions. Nevertheless, distance functions alone are insufficient for accurate inverse analysis in PINNs. To address this, we propose an integrated framework that combines the normalized distance field with bias-corrected adaptive weight tuning to improve both accuracy and efficiency. Numerical results show that the proposed method provides more accurate and efficient solutions to various inverse problems than penalty-based approaches, even in the presence of non-convex geometries with complex boundary conditions. This approach offers a reliable and efficient framework for inverse analysis using PINNs, with potential applications across a wide range of engineering problems.

Figures (55)

• Figure 1: SDF, $s_{PQ}$
• Figure 2: TF, $t_{PQ}$
• Figure 3: ADF, $\phi_{PQ}$
• Figure 4: Gradient, $\nabla \phi_{PQ}$
• Figure 6: ADF, $\phi^{(1)}$