Explain inverse-problem superiority of LM-PINNs and OVP-PINNs via global reduced-loss structure

Determine how the global structure of the reduced loss with respect to physical parameters governs accurate parameter recovery in inverse problems and thereby explains the superior performance of Lagrangian-Mechanics-informed PINNs (LM-PINNs) and Onsager-Variational-Principle-informed PINNs (OVP-PINNs), beyond what is captured by local flatness of the full loss landscape.

Background

The paper visualizes and compares loss landscapes for different thermodynamic structure-informed PINNs and finds that flatter local landscapes correlate with better recovery of physical quantities for HM-PINNs and EIT-PINNs. However, despite these observations, LM-PINNs and OVP-PINNs exhibit superior robustness and performance on inverse problems, which is not explained by the local loss landscape analysis.

The authors hypothesize that inverse-problem performance depends on the global structure of the reduced loss with respect to physical parameters rather than local flatness around trained weights, and they explicitly defer a detailed analysis of this global reduced-loss structure to future work.