🤖 AI Summary
This study addresses the lack of systematic understanding of the loss landscape in physics-informed neural networks (PINNs), particularly under differential operator–based physics losses. For the first time in scientific machine learning, the authors introduce loss landscape analysis by integrating multiple visualization and geometric techniques to systematically compare the optimization landscapes defined by the Deep Ritz method and strong-form residual losses. Their findings reveal that both types of physics-informed losses yield locally smooth, well-conditioned, and nearly convex loss landscapes whose geometric properties closely resemble those of conventional data-driven problems. Notably, these landscapes exhibit favorable structural characteristics near the true solution, challenging the prevailing view that PINN loss landscapes are inherently highly complex. This work thus provides a theoretical foundation for the observed stability and effectiveness of PINN optimization.
📝 Abstract
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.