Comparison of Trefftz-Based PINNs and Standard PINNs Focusing on Structure Preservation

TL;DR

This paper compares Trefftz-based PINNs with standard PINNs, demonstrating that Trefftz-PINNs better preserve global physical structures like magnetic field lines and fluid streamlines, even at low error levels.

Contribution

The study introduces a Trefftz-based PINN framework that enhances structure preservation in physics-informed neural networks compared to standard approaches.

Findings

Trefftz-PINNs maintain magnetic field topology better than standard PINNs.

Trefftz-PINNs outperform standard PINNs in CFD streamline preservation.

Structure preservation is improved by constraining the solution space prior to learning.

Abstract

In this study, we investigate the capability of physics-informed neural networks (PINNs) to preserve global physical structures by comparing standard PINNs with a Trefftz-based PINN (Trefftz-PINN). The target problem is the reproduction of mag-netic field-line structures in a helical fusion reactor configuration. Using identical training data sampled from exact solutions, we perform comparisons under matched mean squared error (MSE) levels. Visualization of magnetic field lines reveals that standard PINNs may exhibit structural collapse across magnetic surfaces even when the MSE is sufficiently small, whereas Trefftz-PINNs successfully preserve the global topology of magnetic field lines. Furthermore, the proposed framework is extended to computational fluid dynamics (CFD) problems, where streamline structures of veloc-ity fields are analyzed. Similar tendencies are observed, demonstrating that Trefftz-PINNs provide superior structure preservation compared to standard PINNs. These results indicate that minimizing numerical error alone does not guarantee physical consistency, and that constraining the solution space prior to learning is an effective strategy for physics-consistent surrogate modeling.