Visualizing the loss landscapes of physics-informed neural networks

Training a neural network requires navigating a high-dimensional, non-convex loss surface to find parameters that minimize this loss. In many ways, it is surprising that optimizers such as stochastic gradient descent and ADAM can reliably locate minima which perform well on both the training and test data. To understand the success of training, a "loss landscape" community has emerged to study the geometry of the loss function and the dynamics of optimization, often using visualization techniques. However, these loss landscape studies have mostly been limited to machine learning for image classification. In the newer field of physics-informed machine learning, little work has been conducted to visualize the landscapes of losses defined not by regression to large data sets, but by differential operators acting on state fields discretized by neural networks. In this work, we provide a comprehensive review of the loss landscape literature, as well as a discussion of the few existing physics-informed works which investigate the loss landscape. We then use a number of the techniques we survey to empirically investigate the landscapes defined by the Deep Ritz and squared residual forms of the physics loss function. We find that the loss landscapes of physics-informed neural networks have many of the same properties as the data-driven classification problems studied in the literature. Unexpectedly, we find that the two formulations of the physics loss often give rise to similar landscapes, which appear smooth, well-conditioned, and convex in the vicinity of the solution. The purpose of this work is to introduce the loss landscape perspective to the scientific machine learning community, compare the Deep Ritz and the strong form losses, and to challenge prevailing intuitions about the complexity of the loss landscapes of physics-informed networks.