Activity: Talk or presentation › Oral presentation › Research
Description
Physics-Informed Neural Networks (PINNs) are increasingly used to solve various partial differential equations (PDEs), especially in high dimensions. In real-world applications, data samples are noisy, making it essential to understand the conditions under which a predictor can achieve a small empirical risk. In this work, we present a first-of-its-kind lower bound on the size of neural networks required for the supervised PINN empirical risk to fall below the variance of noisy supervision labels. Specifically, we show that to achieve low training error, the number of parameters must be lowerbounded by a little less than one trainable parameter per training sample. Consequently, using more noisy training data alone does not provide a “free lunch” in reducing empirical risk. We investigate PINNs applied to the Hamilton–Jacobi–Bellman (HJB) PDE as a case study. Our findings lay the groundwork for a program on rigorously quantifying parameter requirements for effective PINN training under noisy conditions.