Linearized uniqueness of space dependent coefficients in a non-autonomous evolution equation from non-local observations

What the math gives to ML

The paper's transferable asset is a linearized observability principle: coefficient perturbations are identifiable when their induced state sensitivities cannot remain invisible under a family of time-trace or weighted spatial observations. This suggests a conditioning-aware training objective for neural inverse models, rather than merely fitting the final observation mismatch. A practical adaptation is to compute parameter-to-observation Jacobians through the differentiable simulator or latent dynamics and penalize poorly observed parameter directions using an observability Gramian. The same machinery can also select controls or measurement weights that make coefficient recovery better conditioned.