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About this paper
Ismail Huseynov, Arzu Ahmadova, Agamirza Bashirov
This work develops a framework for two-sided error estimation for Physics-Informed Neural Networks (PINNs) solving ordinary differential equations, providing computable a posteriori lower and upper bounds. It introduces localized one-sided growth conditions for sharper certificates and addresses challenges in obtaining nontrivial lower bounds for hard-constrained initial conditions.
Key insights
- Using one-sided Lipschitz conditions for upper error bounds can yield sharper and less conservative certificates compared to traditional global Lipschitz constants.
- Scalar a posteriori lower bounds can be uninformative (trivial) for hard-constrained PINNs with zero initial error, requiring directional probing to provide meaningful lower error diagnostics.
- The framework enables the first rigorous two-sided error enclosures for PINNs applied to ODEs, significantly enhancing the reliability and interpretability of their predictions.
Summary by alphaXiv.
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