Theory

The point of pinnrl is to estimate unknown physics from measurements and to say how far the estimate can be trusted. That claim rests on conditions, and the pages here state them. Each module of the inverse-modelling core cites the condition it relies on, so that a number the package reports comes with the assumption that makes it meaningful.

Page	Question it answers	Used by
Well-posedness and stability	When does a small change in the data give a small change in the estimate, and what does the physics do to an ill-posed problem?	the disturbance suite, the stability constants
Identifiability and conditioning	Can the sensors determine the parameters at all, and which combinations do they determine badly?	pinnrl.analysis, experiments F01–F05 and K01
The Bayesian view and uncertainty	Why is the PINN loss a negative log-posterior, what does that make the loss weights, and what counts as a calibrated error bar?	pinnrl.uq, K39, F06, K10
The numerical-analysis bridge	What do consistency, stability and convergence mean for a PINN, and how does its error relate to the training residual?	the error estimates, K06–K08
Equivalence regions and swarm sampling	Why report the set of parameters that fit the data rather than one estimate, and how to sample it when the problem is nonlinear?	pinnrl.uq, K10, K45–K48

The pages are short on purpose. They give the statement, the condition, the consequence for a PINN, and the reference where the proof lives. The notation is shared: \(u\) is the state, \(\theta\) the parameters to identify, \(\mathcal{N}_\theta[u] = 0\) the governing equation, \(y = \mathcal{H}u + \varepsilon\) the observations through a sensor operator \(\mathcal{H}\) with noise \(\varepsilon\), and \(u_w\) the network with weights \(w\).

Reference papers

The maintainer's reference papers on nonlinear uncertainty (the Oviedo particle-swarm work on sampling equivalence regions) are summarized in Equivalence regions and swarm sampling and cited by DOI. The PDFs stay out of the repository.