analiza aiqbench
This paper gives theoretical guarantees for Physics-Informed Neural Networks, which train on PDE residuals at sampled points. It proves convergence and derives sample-complexity and stability bounds using operator coercivity and McDiarmid's inequality, and it reports that the bounds track empirical behavior on elliptic, parabolic, and nonlinear PDEs. For ML readers, the useful part is the identified conditions (coercivity, activation smoothness, sampling admissibility) that plausibly govern when PINN training generalizes, though the results are only as strong as their stated regularity assumptions.
Non-Asymptotic Stability and Consistency Guarantees for Physics-Informed Neural Networks via Coercive Operator Analysis
arXiv:2506.13554v3 Announce Type: replace Abstract: We present a unified theoretical framework for analyzing the stability and consistency of Physics-Informed Neural Networks (PINNs), grounded in operator coercivity, variational formulations, and non-asymptotic perturbation theory. PINNs approximate solutions to partial differential equations (PDEs) by minimizing residual losses over sampled collocation and boundary points. We formalize both operator-level and variational notions of consistency, proving that residual minimization in Sobolev norms leads to convergence in energy and uniform norms under mild regularity. Deterministic stability bounds quantify how bounded perturbations to the network outputs propagate through the full composite loss, while probabilistic concentration results via McDiarmid's inequality yield sample complexity guarantees for residual-based generalization. A unified generalization bound links residual consistency, projection error, and perturbation sensitivity. Empirical results on elliptic, parabolic, and nonlinear PDEs confirm the predictive accuracy of our theoretical bounds across regimes. The framework identifies key structural principles, such as operator coercivity, activation smoothness, and sampling admissibility, that underlie robust and generalizable PINN training, offering principled guidance for the design and analysis of PDE-informed learning systems.
Abstract:We present a unified theoretical framework for analyzing the stability and consistency of Physics-Informed Neural Networks (PINNs), grounded in operator coercivity, variational formulations, and non-asymptotic perturbation theory. PINNs approximate solutions to partial differential equations (PDEs) by minimizing residual losses over sampled collocation and boundary points. We formalize both operator-level and variational notions of consistency, proving that residual minimization in Sobolev norms leads to convergence in energy and uniform norms under mild regularity. Deterministic stability bounds quantify how bounded perturbations to the network outputs propagate through the full composite loss, while probabilistic concentration results via McDiarmid's inequality yield sample complexity guarantees for residual-based generalization. A unified generalization bound links residual consistency, projection error, and perturbation sensitivity. Empirical results on elliptic, parabolic, and nonlinear PDEs confirm the predictive accuracy of our theoretical bounds across regimes. The framework identifies key structural principles, such as operator coercivity, activation smoothness, and sampling admissibility, that underlie robust and generalizable PINN training, offering principled guidance for the design and analysis of PDE-informed learning systems.
Machine Learning (cs.LG); Functional Analysis (math.FA); Numerical Analysis (math.NA)
arXiv:2506.13554 [cs.LG]
(or arXiv:2506.13554v3 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2506.13554 Focus to learn more arXiv-issued DOI via DataCite
https://doi.org/10.1016/j.cnsns.2026.109909 Focus to learn more DOI(s) linking to related resources
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