Quantification of modeling uncertainty in physics-informed neural networks by interval arithmetic
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The seemingly unrestrained advancements in machine learning technologies drive engineers to increasingly often rely on data-driven approaches in the design of complex systems. This not only requires extensive data to capture the underlying behavior, but it effectively ascribes complete trustworthiness to data. However, real-world scenarios frequently provide only sparse, noisy, or otherwise corrupted measurements. Compounding this challenge is the need to balance resource utilization and performance. To alleviate some of the burden related to the models exclusively relying on data, the class of physics-informed machine learning emerged, which integrates knowledge about the governing physical laws, with data from the system. The majority of physics-based machine learning models, however, remain black boxes and the epistemic uncertainty stemming from the model itself remains unquantified or quantified in a biased way. This work explores the extension of physics-informed neural networks with an interval description of the epistemic uncertainty, integral to the model, thereby avoiding subjective assignments of prior beliefs to components of the model. In the proposed implementation, data processing is done through numerical weights and two separate fully connected layers are used to estimate the center and interval uncertainty of the prediction, to alleviate the problem of interval repeated variables. In addition, the architecture leverages a physics-informed loss function considering whether the output of the model satisfies the underlying partial differential equation and boundary conditions. We validate the performance of the architecture on simplified synthetic data, representing a structural dynamics problem, in particular a noisy dataset generated using an analytical solution to the Euler-Lagrange equation. The interval PINN, proposed in this paper, provides a way for evaluating a cantilever beam's free oscillations in a way that considers both the underlying structural mechanics knowledge and the available measurement data. The architecture is more explainable than purely data-based models, while avoiding applying prior bias on the quantification epistemic uncertainty.
