Learning Memory Operators with Physics-Informed Neural Networks: State Recovery, Kernel Reconstruction, and Parameter Identifiability

Abstract

Memory-dependent evolution equations replace the time derivative of the state with a convolution against a memory kernel \( K(t) \), and appear throughout anomalous diffusion, viscoelasticity, and tumour-growth modelling.\( K \) is usually fixed a priori, most often to the power-law kernel of a Caputo fractional derivative. We investigate the joint recovery of the memory kernel K and the state u using a physics-informed neural network (PINN) trained by a block-coordinate Variable Projection (VarPro) scheme% , and we do so across four structurally different parametrizations: a non-negative Prony series, a fractional (Caputo) power law, a stretched exponential, and a shape-constrained free spline. On a synthetic benchmark the state is recovered with relative\( L^2 \) errors below \( 0.05\% \) across all four families, while kernel recovery varies sharply with the family: \( 0.03\% \) for the single-parameter fractional kernel, a \( 2--4\% \) structural gap for the Prony series that a systematic ablation (mode count, parameter bounds, data volume, optimizer iterations) fails to close, and an intermediate \( 5.7\% \) for the stretched exponential. A local sensitivity (Fisher-information-type) analysis at the exact solution, requiring no training, shows that the persistent instability of the individual Prony parameters is primarily associated with poor parameter identifiability rather than solely with optimization failure: its column-normalized condition number is more than two orders of magnitude worse when the true decay rates sit a factor of \( 4 \)apart than when they are well separated. Remarkably, this parameter-level non-identifiability does not prevent accurate recovery of the aggregated kernel function \( K(t) \) itself. For the stretched-exponential kernel, the initial instability was traced to a regularization term inherited from the Prony formulation. Removing this term reduced the kernel error from 13% to 5.7%. We further test and reject the classical hypothesis that a wider observation window would close the remaining gap. A shape-constrained spline that assumes only positivity, monotonicity, and convexity, with no exponential or power-law structure at all, recovers the benchmark kernel's shape to within \( 6\% \), matching the correctly specified parametric families and showing that the difficulty above sits in each family's own parametrization rather than in the learning problem itself. Finally, a Gauss--Newton/Laplace uncertainty analysis and an \( 8 \)-replicate nonlinear training ensemble converge on the same conclusion: even in the badly conditioned Prony case, individual rates carry uncertainty of hundreds to millions of percent, while the aggregated function \( K(t) \) stays determined to within a few percent, a quantitative instance of the classical inverse-problem distinction between parameter- and function-level identifiability. Overall, accurate state recovery is shown, both empirically and analytically, not to guarantee a uniquely recoverable memory operator.