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Description
We introduce two different neural-network-based strategies for the construction of conforming approximation spaces on general polygonal meshes for the numerical solution of partial differential equations. The proposed methodologies build upon the Virtual Element Method (VEM) paradigm, but replace the implicit definition of local basis functions with explicit neural representations.
In the original Neural Approximated Virtual Element Method (NAVEM), neural networks are trained to approximate the VEM basis functions on each element through a linear combination of harmonic functions. These learned representations are then employed within a standard finite element assembly. This strategy avoids the computation of problem-dependent VEM projection and stabilization operators, while preserving the geometric flexibility of polygonal discretizations. However, this original formulation generally produces basis functions that are exactly harmonic, as in the VEM framework, but discontinuous across element interfaces.
To overcome this issue, we develop two conforming neural variants, termed B-NAVEM and P-NAVEM, designed to enforce exact continuity of basis functions. Both approaches rely on fully connected feed-forward architectures but differ in their training principles.
The B-NAVEM formulation is based on a Physics-Informed Neural Network (PINN) with exact imposition of Dirichlet boundary conditions. This approach is employed to solve the local Laplace problems that define the virtual basis functions. The resulting functions are polynomial on element boundaries and approximately harmonic in the interior, thereby mimicking the structure of the VEM space while restoring conformity.
Conversely, rather than targeting the VEM space itself, P-NAVEM directly constructs continuous basis functions by training neural networks to enforce polynomial reproducibility, i.e., the key property that ensures optimal convergence rates.
Numerical experiments on linear and nonlinear model problems demonstrate that the proposed neural formulations achieve optimal convergence rates (up to neural-network accuracy) and exhibit competitive performance with respect to the standard VEM, while preserving its geometric flexibility.