# Noisy PDE Training Requires Bigger PINNs

arXiv: https://arxiv.org/abs/2507.06967 · HTML: https://mlsanity.com/paper/2507.06967/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 43 references that are in the map.
- First posted on arXiv 9 Jul 2025.

## Builds on (most cited first, 30 of 32)

- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Towards Physics-informed Deep Learning for Turbulent Flow Prediction](https://mlsanity.com/paper/1911.08655/) (2019)
- [A Universal Law of Robustness via Isoperimetry](https://mlsanity.com/paper/2105.12806/) (2021)
- [Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems](https://mlsanity.com/paper/1809.08327/) (2018)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEs](https://mlsanity.com/paper/2006.16144/) (2020)
- [Error bounds for approximations with deep ReLU neural networks in Ws,p norms](https://mlsanity.com/paper/1902.07896/) (2019)
- [Algorithms for Solving High Dimensional PDEs: From Nonlinear Monte Carlo to Machine Learning](https://mlsanity.com/paper/2008.13333/) (2020)
- [A Mathematical Guide to Operator Learning](https://mlsanity.com/paper/2312.14688/) (2023)
- [A Theoretical Analysis of Deep Neural Networks and Parametric PDEs](https://mlsanity.com/paper/1904.00377/) (2019)
- [Physics-informed neural networks for solving Reynolds-averaged Navierunicodex2013Stokes equations](https://mlsanity.com/paper/2107.10711/) (2021)
- [Physics-informed Autoencoders for Lyapunov-stable Fluid Flow Prediction](https://mlsanity.com/paper/1905.10866/) (2019)
- [Generic bounds on the approximation error for physics-informed (and) operator learning](https://mlsanity.com/paper/2205.11393/) (2022)
- [Data-driven discovery of Koopman eigenfunctions for control](https://mlsanity.com/paper/1707.01146/) (2017)
- [When Do Extended Physics-Informed Neural Networks (XPINNs) Improve Generalization?](https://mlsanity.com/paper/2109.09444/) (2021)
- [Neural Stochastic PDEs: Resolution-Invariant Learning of Continuous Spatiotemporal Dynamics](https://mlsanity.com/paper/2110.10249/) (2021)
- [Solving the Kolmogorov PDE by means of deep learning](https://mlsanity.com/paper/1806.00421/) (2018)
- [Numerical analysis of physics-informed neural networks and related models in physics-informed machine learning](https://mlsanity.com/paper/2402.10926/) (2024)
- [Uncertainty quantification for noisy inputs-outputs in physics-informed neural networks and neural operators](https://mlsanity.com/paper/2311.11262/) (2023)
- [Numerical Solution of the Parametric Diffusion Equation by Deep Neural Networks](https://mlsanity.com/paper/2004.12131/) (2020)
- [Teaching the Incompressible Navier-Stokes Equations to Fast Neural Surrogate Models in 3D](https://mlsanity.com/paper/2012.11893/) (2020)
- [Elliptic PDE learning is provably data-efficient](https://mlsanity.com/paper/2302.12888/) (2023)
- [Active Training of Physics-Informed Neural Networks to Aggregate and Interpolate Parametric Solutions to the Navier-Stokes Equations](https://mlsanity.com/paper/2005.05092/) (2020)
- [Learning Physics-Informed Neural Networks without Stacked Back-propagation](https://mlsanity.com/paper/2202.09340/) (2022)
- [Is L2 Physics-Informed Loss Always Suitable for Training Physics-Informed Neural Network?](https://mlsanity.com/paper/2206.02016/) (2022)
- [Hutchinson Trace Estimation for High-Dimensional and High-Order Physics-Informed Neural Networks](https://mlsanity.com/paper/2312.14499/) (2023)
- [Numerically Solving Parametric Families of High-Dimensional Kolmogorov Partial Differential Equations via Deep Learning](https://mlsanity.com/paper/2011.04602/) (2020)
- [Bias-Variance Trade-off in Physics-Informed Neural Networks with Randomized Smoothing for High-Dimensional PDEs](https://mlsanity.com/paper/2311.15283/) (2023)

## Built on by (most cited first, 2 of 2)

- [Generalization Bounds for Physics-Informed Neural Networks for the Incompressible Navier-Stokes Equations](https://mlsanity.com/paper/2603.23072/) (2026)
- [Overcoming the Limits of Finite Difference Method; Physics-Informed Neural Network for Noisy High-Dimensional Heat Diffusion](https://mlsanity.com/paper/2606.07982/) (2026)

## Abstract (arXiv)

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 σ2 (the variance of the supervision data), then necessarily dNlog dNgtrsim Ns η2, where Ns is the number of samples and dN 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 σ2. These results provide a quantitative foundation for understanding parameter requirements when training PINNs in the presence of noisy data.