How to Avoid Trivial Solutions in Physics-Informed Neural Networks
This addresses a core problem in PINNs for simulation science, particularly in data-scarce settings, though it appears incremental as it builds on existing PINN frameworks.
The paper tackles the problem of physics-informed neural networks (PINNs) learning trivial solutions that satisfy physics constraints but are incorrect, by developing an alternative sampling approach and a new penalty term. This method reduces the number of collocation points needed by up to 80% for benchmark problems while achieving competitive results.
The advent of scientific machine learning (SciML) has opened up a new field with many promises and challenges in the field of simulation science by developing approaches at the interface of physics- and data-based modelling. To this end, physics-informed neural networks (PINNs) have been introduced in recent years, which cope for the scarcity in training data by incorporating physics knowledge of the problem at so-called collocation points. In this work, we investigate the prediction performance of PINNs with respect to the number of collocation points used to enforce the physics-based penalty terms. We show that PINNs can fail, learning a trivial solution that fulfills the physics-derived penalty term by definition. We have developed an alternative sampling approach and a new penalty term enabling us to remedy this core problem of PINNs in data-scarce settings with competitive results while reducing the amount of collocation points needed by up to 80 \% for benchmark problems.