Quantifying uncertainty in neural network predictions of forced vibrations

The prediction of forced vibrations in nonlinear systems is a common task in science and engineering, which can be tackled using various methodologies. A classical approach is based on solving differential (algebraic) equations derived from physical laws ('first principles'). Alternatively, Artificial Neural Networks (ANNs) may be applied, which rely on learning the dynamics of a system from given data. However, a fundamental limitation of ANNs is their lack of transparency, making it difficult to understand and trust the model's predictions. In this contribution, we follow a hybrid modelling approach combining a data‐based prediction using a stabilised Autoregressive Neural Network (s‐ARNN) with a priori knowledge from first principles. Moreover, aleatoric and epistemic uncertainty is quantified by a combination of mean‐variance estimation (MVE) and deep ensembles. Validating this approach for a classical duffing oscillator suggests that the MVE ensemble is the most accurate and reliable method for prediction accuracy and uncertainty quantification. These findings underscore the significance of understanding uncertainties in deep ANNs and the potential of our method in improving the reliability of predictive nonlinear system modelling. We also demonstrate that including partially known dynamics can further increase accuracy, highlighting the importance of combining ANNs and physical laws.