Task-aware evolution in physics-informed neural networks: Application to Saint-Venant torsion problems

초록

The Saint-Venant torsion theory is a classical theory for analyzing the torsional behavior of structural components. Conventional numerical methods, including the finite element method (FEM), typically rely on mesh-based approaches, which often result in significant increases in computational cost. The objective of this study is to develop a series of novel numerical methods based on physics-informed neural networks (PINN) for solving the Saint-Venant torsion equations. Utilizing the automatic differentiation capability of neural networks, the PINN can provide partial differential equations (PDEs) solvers without the need for intricate computational techniques. We present an integrated framework that simultaneously addresses single-instance stiffness (via VS-PINN) and multi-query parametric efficiency (via Parametric PINN) for torsion problems, which has not been explored in prior work. First, a PINN solver was developed to compute the torsional constant for bars with arbitrary cross-sectional geometries. This was followed by the development of a solver capable of handling cases with sharp geometric transitions; variable-scaling PINN (VS-PINN). Finally, a parametric PINN was constructed to address the limitations of conventional single-instance PINN. The results from all three solvers showed good agreement with reference solutions, demonstrating their accuracy and robustness. Specifically, we report 0.1% error for circular and square sections, 3.0% error for triangular sections with PINN, a reduction from 0.97% to 0.11% with VS-PINN in the stiff 1D case, and 1.0% error for the Parametric PINN across varying torque parameters. Once training has been completed, the parametric PINN can predict solutions with remarkable efficiency for varying PDE data or problem settings. Thanks to the retraining-free nature of the parametric PINN, the model can achieve over 100x faster speed for a large number of instances, compared to FEM. Each solver can be selectively employed in a task-aware manner, ensuring that its utilization aligns with the specific objectives, such as geometry-specific solving, handling stiffness, or parametric generalization.

키워드