Recipes for when Physics Fails: Recovering Robust Learning of Physics Informed Neural Networks
This addresses robustness issues in PINNs for solving PDEs, which is important for researchers and practitioners in scientific computing and machine learning, though it is incremental as it builds on existing PINN methods.
The paper tackles the problem of Physics-Informed Neural Networks (PINNs) being sensitive to errors in training data and overfitting, which can propagate errors and cause convergence to physics-obeying local minima. It introduces Gaussian Process-based smoothing and sparse inducing points to recover robust PINN performance, demonstrating results on time-dependent Schrödinger and Burgers' equations with comparisons to benchmark models.
Physics-informed Neural Networks (PINNs) have been shown to be effective in solving partial differential equations by capturing the physics induced constraints as a part of the training loss function. This paper shows that a PINN can be sensitive to errors in training data and overfit itself in dynamically propagating these errors over the domain of the solution of the PDE. It also shows how physical regularizations based on continuity criteria and conservation laws fail to address this issue and rather introduce problems of their own causing the deep network to converge to a physics-obeying local minimum instead of the global minimum. We introduce Gaussian Process (GP) based smoothing that recovers the performance of a PINN and promises a robust architecture against noise/errors in measurements. Additionally, we illustrate an inexpensive method of quantifying the evolution of uncertainty based on the variance estimation of GPs on boundary data. Robust PINN performance is also shown to be achievable by choice of sparse sets of inducing points based on sparsely induced GPs. We demonstrate the performance of our proposed methods and compare the results from existing benchmark models in literature for time-dependent Schrödinger and Burgers' equations.