aiqbench analysis
This paper introduces a certification-and-selection framework for physics-informed inverse learning that accepts model outputs only when their uncertainty bounds do not exceed a safe baseline threshold. By combining data, physics, and optimization residuals into a deterministic error bound, the method effectively filters out hallucinated or ill-posed reconstructions without altering the underlying architecture. For practitioners working on PDE-governed problems, this approach offers a rigorous mechanism to ensure reliability and prevent degradation in strongly ill-posed regimes.
No-Harm Physics-Informed Inverse Learning with Residual-Calibrated Uncertainty
arXiv:2606.07153v1 Announce Type: cross 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 $$R_{\mathrm{learn}}\le R_{\mathrm{base}}+\varepsilon_{\mathrm{safe}};$$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.
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 $$R_{\mathrm{learn}}\le R_{\mathrm{base}}+\varepsilon_{\mathrm{safe}};$$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.
Numerical Analysis (math.NA); Machine Learning (cs.LG); Optimization and Control (math.OC)
arXiv:2606.07153 [math.NA]
(or arXiv:2606.07153v1 [math.NA] for this version)
https://doi.org/10.48550/arXiv.2606.07153 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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