Proceedings of the PowerUp Conference

The ability to assess and control approximation errors is key to the success of numerical methods. In contrast, learned approximations must ensure during training that error requirements are met, typically relying on statistical loss func- tions that do not account for the underlying physics. In this work, we argue that the assessment of learning errors in neural solvers should be aligned with numerical methods by evaluating errors in the residual norm of the governing equations. We show how this residual-based error measure can be integrated into the learning process through a physics-informed non-dimensionalisation of inputs and outputs, yielding a training objective that reflects physical sensitivities. The proposed construction reduces reliance on dataset-dependent statistics, improves robustness for limited training data, and provides an interpretable connection between training loss and engineering error measures. We demonstrate the approach on learning power flow solutions as an example of neural solvers for algebraic systems.