{"ID":2896646,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2507.06967","arxiv_id":"2507.06967","title":"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.","short_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 abo...","url_abs":"https://arxiv.org/abs/2507.06967","url_pdf":"https://arxiv.org/pdf/2507.06967v2","authors":"[\"Sebastien Andre-Sloan\",\"Anirbit Mukherjee\",\"Matthew Colbrook\"]","published":"2025-07-09T15:58:26Z","proceeding":"cs.LG","tasks":"[\"cs.LG\",\"cs.AI\"]","methods":"[\"Large Language Model\"]","has_code":false}