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

Source abstract

Memory-dependent evolution equations replace the time derivative of the state by a convolution with a memory kernel K(t). They are used in anomalous diffusion, viscoelasticity and tumour-growth modelling, usually with K fixed in advance, most often as the power-law kernel of a Caputo derivative. We study the joint recovery of K and the state u with a physics-informed neural network (PINN) trained by a block-coordinate Variable Projection (VarPro) scheme, for four kernel parametrizations: a non-negative Prony series, a fractional (Caputo) power law, a stretched exponential and a shape-constrained spline. In the best-performing runs on a synthetic benchmark, the state is recovered with relative errors below 0.05% for all four families, while kernel recovery depends strongly on the family. The fractional kernel is recovered almost exactly, the stretched exponential to 5.7%, and the Prony series with a kernel-function reconstruction error of 2–4% in its best runs. A local sensitivity analysis at the exact solution, which requires no training, shows that the individual Prony parameters are severely ill conditioned: the column-normalized condition number of the sensitivity matrix is more than two orders of magnitude larger when the true decay rates are a factor of four apart than when they are well separated. Fitted against the exact state, without a network, the kernel and the three active Prony modes are recovered to a rounding error, while the redundant zero-weight modes of the over-parametrized representation remain non-identifiable. A perturbation of only 0.1% raises the active-mode parameter error to tens of percent while the kernel error stays near 0.1%, and this error tracks the condition number. A Gauss–Newton/Laplace approximation and an eight-replicate training ensemble agree that the rates are far less well determined than K(t). A spline that assumes only positivity, monotonicity and convexity reaches a kernel error of about 6%. The results show that an accurate state does not guarantee accurate recovery of the kernel or of its parameters, and that state accuracy, kernel-function accuracy and parameter identifiability should be assessed separately.