New Instrument Detects Model Misspecification in Hybrid PDE-Parameter Learning

In the field of hybrid PDE-parameter learning, where models combine partial differential equations (PDEs) with learned parameters, it's crucial to know if the underlying mathematical operator assumed by the model is incorrect, or if the parameters are simply hard to identify. Traditional accuracy checks, like RMSE, often fail to reveal these fundamental issues, as a misspecified model can still appear accurate within its training domain while yielding significantly wrong coefficients. This paper presents a novel, reference-free instrument designed to diagnose these problems from a single model fit, without needing an external oracle. The instrument uses an information-matrix statistic to detect operator misspecification and a rank statistic to identify unidentifiable parameters. This dual approach allows for a clear separation of these two distinct types of model failure. The instrument was tested on a self-adjoint parabolic inverse problem, where it successfully distinguished between correct specification, two types of misspecification, and non-identifiability. For instance, under correct specification, the statistic remained low, but it dramatically increased under misspecification, firing in every replicate. Conversely, it stayed mute for non-identifiable designs. This ability to separate failures is a significant contribution, as it provides a robust diagnostic tool for scientific machine learning models, even when standard accuracy metrics are misleading.