  All discretized numerical models contain modelling errors - this reality is
amplified when reduced-order models are used. The ability to accurately
approximate modelling errors informs statistics on model confidence and
improves quantitative results from frameworks using numerical models in
prediction, tomography, and signal processing. Further to this, the
compensation of highly nonlinear and non-Gaussian modelling errors, arising in
many ill-conditioned systems aiming to capture complex physics, is a
historically difficult task. In this work, we address this challenge by
proposing a neural network approach capable of accurately approximating and
compensating for such modelling errors in augmented direct and inverse
problems. The viability of the approach is demonstrated using simulated and
experimental data arising from differing physical direct and inverse problems.
