PINNs error estimates for nonlinear equations in $\mathbb{R}$-smooth Banach spaces
Jiexing Gao, Yurii Zakharian
TL;DR
This work develops an operator-theoretic framework for PINN error estimation in $\mathbb{R}$-smooth Banach spaces, introducing a real $p$-form and classes of operators such as $(p,\psi)$-submonotone, $(p,\epsilon)$-powered, and coercive, to bound PINN residuals across parabolic, hyperbolic, and elliptic PDEs. A Bramble–Hilbert type lemma is extended to $L^p$ spaces via a projection operator $\mathcal{J}_p$, enabling $L^p$ residual bounds and polyhedral error control in non-Hilbert spaces. The paper derives exact a posteriori error bounds that relate total error to residuals, operator constants, and training errors for a broad set of PDE types, and validates the theory through numerical experiments on 1D heat, KdV, Maxwell, Good Boussinesq, Rayleigh, and Poisson with piecewise forcing. Overall, the results broaden theoretical guarantees for PINNs beyond Hilbert spaces, clarifying how residuals govern approximation accuracy in diverse PDE settings and guiding residual-based diagnostics and algorithm design.
Abstract
In the paper, we describe in operator form classes of PDEs that admit PINN's error estimation. Also, for $L^p$ spaces, we obtain a Bramble-Hilbert type lemma that is a tool for PINN's residuals bounding.