A Locally Conservative Enriched Linearized Neural Network Approximation to Elliptic PDEs

This paper presents a locally conservative Enriched Linearized Neural Network (ENN) method for the Darcy flow model. A linearized shallow ReLU${}^k$ network, whose hidden-layer parameters are fixed on a quasi-uniform set of the sphere, carries the approximation, and piecewise constant functions on an auxiliary subdivision of the domain are added to ensure local mass conservation. The subdivision may consist of general, possibly curved, elements. Since neural network functions do not satisfy the inverse inequality, the stability and error analysis of the standard discontinuous Galerkin interior penalty formulation do not apply. The missing inverse inequality is remedied by an edge identity combined with a Galerkin least-squares (GLS) term, together with a single neural approximant that is optimal in the $L^2$, $H^1$ and $H^2$ norms simultaneously. We use the nonsymmetric interior penalty (NIPG) form, which is coercive for every positive penalty parameter. With $n$, the width of the shallow neural network, used in ENN, we prove an error estimate in the EG norm that is of the optimal $H^1$ order, i.e., $O(n^{-(r-1)\mathrm{/}d})$ for solutions in $H^r$, $r\ge 2$, and $O(n^{-(s-1)\mathrm{/}d})$ for solutions in $H^s$, $\frac{3}{2}<s\le 2$, for any shape-regular subdivision, when the GLS parameter satisfies $\min \{h,n^{-1\mathrm{/}d}{\}}^2\lesssim \tau \lesssim n^{-2\mathrm{/}d}$ and the edge terms are scaled with $\min \{h_e,n^{-1\mathrm{/}d}\}$, i.e., by the resolution of the network rather than of the mesh. A numerical Darcy flux is constructed, which is locally conservative, regardless of the size or the number of the elements. Sample numerical results demonstrate the correctness of the theory. In particular, a fixed $4\times 4$ subdivision gives the same accuracy as a subdivision refined with the network, and the mass loss stays at round-off unlike a non-conservative neural network method.