# Examining the robustness of Physics-Informed Neural Networks to noise for Inverse Problems

arXiv: https://arxiv.org/abs/2509.20191 · HTML: https://mlsanity.com/paper/2509.20191/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 39 references that are in the map.
- First posted on arXiv 24 Sep 2025.

## Builds on (most cited first, 13 of 13)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Respecting causality is all you need for training physics-informed neural networks](https://mlsanity.com/paper/2203.07404/) (2022)
- [Understanding and mitigating gradient pathologies in physics-informed neural networks](https://mlsanity.com/paper/2001.04536/) (2020)
- [Weak baselines and reporting biases lead to overoptimism in machine learning for fluid-related partial differential equations](https://mlsanity.com/paper/2407.07218/) (2024)
- [Tackling the Curse of Dimensionality with Physics-Informed Neural Networks](https://mlsanity.com/paper/2307.12306/) (2023)
- [Can Physics-Informed Neural Networks beat the Finite Element Method?](https://mlsanity.com/paper/2302.04107/) (2023)
- [CFDBench: A Large-Scale Benchmark for Machine Learning Methods in Fluid Dynamics](https://mlsanity.com/paper/2310.05963/) (2023)
- [Eighty Years of the Finite Element Method: Birth, Evolution, and Future](https://mlsanity.com/paper/2107.04960/) (2021)
- [Experience report of physics-informed neural networks in fluid simulations: pitfalls and frustration](https://mlsanity.com/paper/2205.14249/) (2022)
- [Optimizing the optimizer for data driven deep neural networks and physics informed neural networks](https://mlsanity.com/paper/2205.07430/) (2022)
- [A Two-Stage Imaging Framework Combining CNN and Physics-Informed Neural Networks for Full-Inverse Tomography: A Case Study in Electrical Impedance Tomography (EIT)](https://mlsanity.com/paper/2407.17721/) (2024)
- [Identification of Physical Properties in Acoustic Tubes Using Physics-Informed Neural Networks](https://mlsanity.com/paper/2406.11119/) (2024)

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

- [Automated Reproducibility Has a Problem Statement Problem](https://mlsanity.com/paper/2601.04226/) (2025)
- [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)

Approximating solutions to partial differential equations (PDEs) is fundamental for the modeling of dynamical systems in science and engineering. Physics-informed neural networks (PINNs) are a recent machine learning-based approach, for which many properties and limitations remain unknown. PINNs are widely accepted as less computationally efficient and accurate than traditional methods for solving PDEs, such as the finite element method. However, PINNs are commonly claimed to show promise in solving inverse problems and handling noisy or incomplete data. We compare the performance of PINNs in solving inverse problems with that of a traditional approach using the finite element method combined with a numerical optimizer. The models are tested on viscosity identification in 1D Burgers' equation and in 2D/3D Taylor-Green Vortex, in all cases with additive Gaussian noise applied to training and validation data. We find that while PINNs may require less human effort and specialized knowledge, they are outperformed by the traditional approach. For example, for 2D Taylor-Green Vortex with σ=1 noise, the baseline has a mean prediction RMSE of 0.0013 compared to 0.01 for the best PINN variation. However, PINNs scale better than the baseline with the computational complexity of the problem. We identify failures during training to be addressed if the PINN performance on noisy inverse problems is to become more competitive.