Noisy PDE Training Requires Bigger PINNs

Abstract

Physics-Informed Neural Networks (PINNs) are increasingly used to approximate solutions of partial differential equations (PDEs), particularly in high dimensions. In real-world settings, data are often noisy, making it crucial to understand when a predictor can still achieve low empirical risk. Yet, little is known about the conditions under which a PINN can do so effectively. We analyse PINNs applied to the Hamilton--Jacobi--Bellman (HJB) PDE and establish a lower bound on the network size required for the supervised PINN empirical risk to fall below the variance of noisy supervision labels. Specifically, if a predictor achieves empirical risk $O(η)$ below $σ^2$ (the variance of the supervision data), then necessarily $d_N\log d_N\gtrsim N_s η^2$, where $N_s$ is the number of samples and $d_N$ the number of trainable parameters. A similar constraint holds in the fully unsupervised PINN setting when boundary labels are noisy. Thus, simply increasing the number of noisy supervision labels does not offer a ``free lunch'' in reducing empirical risk. We also give empirical studies on the HJB PDE, the Poisson PDE and the the Navier-Stokes PDE set to produce the Taylor-Green solutions. In these experiments we demonstrate that PINNs indeed need to be beyond a threshold model size for them to train to errors below $σ^2$. These results provide a quantitative foundation for understanding parameter requirements when training PINNs in the presence of noisy data.