Structural identifiability of constitutive relations from indirect field observations

Constitutive relations-functions linking a local state variable to a material or system property such as thermal conductivity, hydraulic permeability, or reaction rate constant-appear across the physical, engineering, and life sciences. They must often be inferred from sparse, indirect observations of the field they govern rather than from direct assays. Unlike the well-studied spatially varying parameter $\varphi \left(x\right)$ , the scalar state-dependent function $\varphi \left(u\right):R\to R^{+}$ of a single local state variable has received no systematic Bayesian treatment, and a basic question remains open: how much can sparse indirect observations actually inform $\varphi \left(u\right)$ ? We show the answer is governed by a two-factor attenuation: information must pass first through the inverse of the linearized partial differential equation (PDE) Jacobian and then through the sparse observation mask. The product of these factors lies far below unity for practical configurations, leaving the posterior structurally prior-dominated regardless of the number of indirect observations of the same type and configuration. The primary determinant is not the PDE class but the coupling mechanism-whether $\varphi \left(u\right)$ enters the residual through a spatial gradient (as for diffusion coefficients) or pointwise (as for source terms). We instantiate the analysis in a sparse variational Gaussian process with an explicit, checkable prior and a closed-form Laplace posterior, and validate it across four PDE classes, four constitutive shapes, and two prior conditions. A small number of direct constitutive measurements at well-placed state values, bypassing the PDE entirely, collapse the posterior where arbitrarily many indirect observations of the same type cannot.