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…
Read the paper
Related studies
- Preliminary Toxicity Analysis of 3DCRT versus IMRT on the High Dose Arm of the RTOG 0126 Prostate Cancer Trial
- Adaptive Radiotherapy for Head and Neck Cancer: Initial Clinical Outcomes from a Prospective Trial
- American Association of Physicists in Medicine Task Group 263: Standardizing Nomenclatures in Radiation Oncology
- Motion Management Strategies and Technical Issues Associated with Stereotactic Body Radiotherapy of Thoracic and Upper Abdominal Tumors: A Review from NRG Oncology
- High-dose-rate interstitial brachytherapy as a monotherapy for localized prostate cancer: treatment description and preliminary results of a phase I/II clinical trial
- Highly efficient training, refinement, and validation of a knowledge-based plan quality control system for radiotherapy clinical trials
Go deeper in Evidence Hub
Evidence Hub adds a study data sheet (design, sample size, evidence level, key numbers), an AI summary and a chat grounded in the paper's text.