Neural network approximation in discrete dual norms with adaptive test spaces

In robust variational physics-informed neural networks (RVPINNs), the loss function is formulated in terms of the Riesz representative of the variational residual within a discrete test space. This approach guarantees that the loss function is robust with respect to the true error in the energy norm up to a remainder term that depends on both the neural network approximation and the discrete space configuration. However, in problems with localized singularities, steep gradients, or interface layers, a fixed coarse test space may fail to resolve the continuous Riesz representative of the residual during training. Although this can be avoided by using a sufficiently fine test space from the start, doing so may be computationally inefficient. We therefore propose an adaptive algorithm that enriches the test space only where the error between the discrete and continuous Riesz representatives is pronounced. We establish theoretical adaptive strategies within the RVPINN framework and derive their error bounds. Furthermore, we propose a computable refinement indicator and prove that, under the saturation assumption, it serves as a reliable and efficient error estimator for the non-computable discrepancy between the discrete and continuous Riesz representatives. Finally, we propose a practical adaptive algorithm and demonstrate its effectiveness through numerical experiments on elliptic Dirichlet problems.