A novel adaptive weighting scheme for physics-informed neural networks with uncertainty prediction in presence of noisy data and outliers

Physics-Informed Neural Networks (PINNs) have recently emerged as a powerful approach, integrating data-driven deep learning with physical knowledge, and are increasingly utilised in fields such as physics, engineering, chemistry and mathematics. While PINNs offer advantages like resilience to noisy data, high-resolution predictions, and mesh-free simulation capabilities, they face challenges in balancing data-driven and physics-based losses. This work introduces a novel adaptive weighting scheme to enhance PINN convergence when dealing with noisy measurements combined with a PINN architecture capable of predicting uncertainty variables, using a model designed to estimate parameters (e.g., mean and standard deviation) for assumed distributions of predicted quantities. The methodology, based on Negative Log Likelihood (NLL) expectations, allows for different interesting features. At first, it weights the various diagnostics accordingly with their uncertainties, giving more importance to more accurate and reliable measurements. Second, the loss to physics is increased once a target NLL on the boundary is achieved, allowing for very accurate reconstructions. Last, the adaptive weighting scheme requires the selection of a new hyperparameter that it is easy to estimate. Moreover, if the hyperparameter is wrongly estimated, it is very easy to detect such an error and by a simple and general procedure it is possible to reach the best convergence in few iterations Additionally, the proposed methodology effectively detects problematic boundaries and diagnostics, offering robust performance across diverse applications. In this work applications of the presented method to synthetic cases simulating real experimental and engineering problems are proposedincluding heat transfer, fluid dynamics, and magnetohydrodynamics scenarios. This adaptive weighting framework demonstrates significant promise for inverse problem-solving, diagnostics assessment, and applications where data and physical models must be harmonised.