PINNs error estimates for nonlinear equations in $\mathbb{R}$-smooth Banach spaces
arXiv:2305.11915v32 citations
Originality Synthesis-oriented
AI Analysis
This work addresses the challenge of theoretical error analysis for PINNs in PDE solving, but it appears incremental as it extends existing frameworks to more general spaces.
The authors tackled the problem of providing error estimates for Physics-Informed Neural Networks (PINNs) applied to nonlinear PDEs in smooth Banach spaces, resulting in a Bramble-Hilbert type lemma for bounding PINN residuals in L^p spaces.
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.