No-Harm Physics-Informed Inverse Learning with Residual-Calibrated Uncertainty
Abstract
Physics-informed learning is increasingly used for partial differential equation (PDE)-governed inverse problems, but its reliability remains difficult to certify. This paper develops a no-harm certification-and-selection framework for physics-informed inverse learning. A learned reconstruction is accepted only when its residual-calibrated radius is no worse than the baseline radius, namely when Rlearn≤ Rbase+εsafe;otherwise, the method returns the baseline. The certificate combines data, physics, boundary or initial-condition, and optimization residuals. Under a conditional stability estimate, these residuals yield an a posteriori reconstruction-error bound and a deterministic uncertainty radius. A high-probability certificate is also derived for physics residuals estimated from independent random collocation points. Numerical tests on Poisson source recovery, inverse heat reconstruction, limited-angle tomography, elliptic coefficient identification, and stochastic residual validation show that the selector accepts certified improvements, rejects shifted, hallucinated, or unfinished candidates, and becomes conservative in strongly ill-posed regimes. The framework is therefore a certification-and-selection layer, not another reconstruction architecture.
Related
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BibTeX
@misc{katende2026noharm,
title = {{No-Harm Physics-Informed Inverse Learning with Residual-Calibrated Uncertainty}},
author = {Ronald Katende},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.07153},
eprint = {2606.07153},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.07153},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - No-Harm Physics-Informed Inverse Learning with Residual-Calibrated Uncertainty
AU - Ronald Katende
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2606.07153
UR - https://arxiv.org/abs/2606.07153
AB - Physics-informed learning is increasingly used for partial differential equation (PDE)-governed inverse problems, but its reliability remains difficult to certify. This paper develops a no-harm certification-and-selection framework for physics-informed inverse learning. A learned reconstruction is accepted only when its residual-calibrated radius is no worse than the baseline radius, namely when R_learn≤ R_base+ε_safe;otherwise, the method returns the baseline. The certificate combines data, physics, boundary or initial-condition, and optimization residuals. Under a conditional stability estimate, these residuals yield an a posteriori reconstruction-error bound and a deterministic uncertainty radius. A high-probability certificate is also derived for physics residuals estimated from independent random collocation points. Numerical tests on Poisson source recovery, inverse heat reconstruction, limited-angle tomography, elliptic coefficient identification, and stochastic residual validation show that the selector accepts certified improvements, rejects shifted, hallucinated, or unfinished candidates, and becomes conservative in strongly ill-posed regimes. The framework is therefore a certification-and-selection layer, not another reconstruction architecture.
ER -