# On the estimation rate of Bayesian PINN for inverse problems

arXiv: https://arxiv.org/abs/2406.14808 · HTML: https://mlsanity.com/paper/2406.14808/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 40 references that are in the map.
- First posted on arXiv 21 Jun 2024.

## Builds on (most cited first, 16 of 16)

- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Nonparametric regression using deep neural networks with ReLU activation function](https://mlsanity.com/paper/1708.06633/) (2017)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Physics-informed learning of governing equations from scarce data](https://mlsanity.com/paper/2005.03448/) (2020)
- [Deep Network Approximation for Smooth Functions](https://mlsanity.com/paper/2001.03040/) (2020)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEs](https://mlsanity.com/paper/2006.16144/) (2020)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEs](https://mlsanity.com/paper/2007.01138/) (2020)
- [The phase diagram of approximation rates for deep neural networks](https://mlsanity.com/paper/1906.09477/) (2019)
- [DeepMoD: Deep learning for Model Discovery in noisy data](https://mlsanity.com/paper/1904.09406/) (2019)
- [Convergence Rates for Learning Linear Operators from Noisy Data](https://mlsanity.com/paper/2108.12515/) (2021)
- [Machine Learning For Elliptic PDEs: Fast Rate Generalization Bound, Neural Scaling Law and Minimax Optimality](https://mlsanity.com/paper/2110.06897/) (2021)
- [Simultaneous approximation of a smooth function and its derivatives by deep neural networks with piecewise-polynomial activations](https://mlsanity.com/paper/2206.09527/) (2022)
- [Statistical Guarantees for Regularized Neural Networks](https://mlsanity.com/paper/2006.00294/) (2020)

## Built on by (most cited first, 3 of 3)

- [Damper-B-PINN: Damper Characteristics-Based Bayesian Physics-Informed Neural Network for Vehicle State Estimation](https://mlsanity.com/paper/2502.20772/) (2025)
- [DBPnet: Damper Characteristics-Based Bayesian Physics-Informed Neural Network for Wheel Load Estimation](https://mlsanity.com/paper/2605.24860/) (2026)
- [Posterior Concentration of Bayesian Physics-Informed Neural Networks for Elliptic PDEs](https://mlsanity.com/paper/2605.08672/) (2026)

## Abstract (arXiv)

Solving partial differential equations (PDEs) and their inverse problems using Physics-informed neural networks (PINNs) is a rapidly growing approach in the physics and machine learning community. Although several architectures exist for PINNs that work remarkably in practice, our theoretical understanding of their performances is somewhat limited. In this work, we study the behavior of a Bayesian PINN estimator of the solution of a PDE from n independent noisy measurement of the solution. We focus on a class of equations that are linear in their parameters (with unknown coefficients θ_star). We show that when the partial differential equation admits a classical solution (say u_star), differentiable to order β, the mean square error of the Bayesian posterior mean is at least of order n-2β/(2β + d). Furthermore, we establish a convergence rate of the linear coefficients of θ_star depending on the order of the underlying differential operator. Last but not least, our theoretical results are validated through extensive simulations.