General proof of local observability

Prove local observability for general coupled physical systems and sensor configurations, beyond the specific configurations for which the physics-constrained attentive neural operator framework provides numerical verification.

Background

The paper derives conditional stability and parameter-recovery estimates under local state-stability and sensor-observability assumptions. For the numerical benchmarks, local parameter-to-observation sensitivity is checked computationally, including a finite-difference diagnostic for the two-dimensional lid-driven cavity problem at Reynolds number 5000.

The authors do not establish observability theoretically for arbitrary coupled systems, sensor layouts, or parameter regimes. Consequently, extending the numerical evidence to a general proof of local observability remains unresolved and is important for determining when sparse single-field observations uniquely constrain unobserved fields and unknown physical parameters.