Certified machine learning: A posteriori error estimation for physics-informed neural networks
This work provides a certification method for PINNs, addressing the need for error estimation in physics-informed machine learning, though it is incremental as it builds on existing PINN frameworks.
The paper tackles the problem of assessing the reliability of Physics-Informed Neural Networks (PINNs) by deriving a rigorous a posteriori error bound for PINN predictions, applicable even to unseen data without prior knowledge of the true solution, and demonstrates it on two academic toy problems, including one in model-predictive control.
Physics-informed neural networks (PINNs) are one popular approach to incorporate a priori knowledge about physical systems into the learning framework. PINNs are known to be robust for smaller training sets, derive better generalization problems, and are faster to train. In this paper, we show that using PINNs in comparison with purely data-driven neural networks is not only favorable for training performance but allows us to extract significant information on the quality of the approximated solution. Assuming that the underlying differential equation for the PINN training is an ordinary differential equation, we derive a rigorous upper limit on the PINN prediction error. This bound is applicable even for input data not included in the training phase and without any prior knowledge about the true solution. Therefore, our a posteriori error estimation is an essential step to certify the PINN. We apply our error estimator exemplarily to two academic toy problems, whereof one falls in the category of model-predictive control and thereby shows the practical use of the derived results.