# Posterior Concentration of Bayesian Physics-Informed Neural Networks for Elliptic PDEs

arXiv: https://arxiv.org/abs/2605.08672 · HTML: https://mlsanity.com/paper/2605.08672/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 26 of its 53 references that are in the map.
- First posted on arXiv 9 May 2026.

## Builds on (most cited first, 26 of 26)

- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Bayesian Deep Learning and a Probabilistic Perspective of Generalization](https://mlsanity.com/paper/2002.08791/) (2020)
- [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)
- [What Are Bayesian Neural Network Posteriors Really Like?](https://mlsanity.com/paper/2104.14421/) (2021)
- [A vector-contraction inequality for Rademacher complexities](https://mlsanity.com/paper/1605.00251/) (2016)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEs](https://mlsanity.com/paper/2006.16144/) (2020)
- [Error analysis for physics informed neural networks (PINNs) approximating Kolmogorov PDEs](https://mlsanity.com/paper/2106.14473/) (2021)
- [Bayesian Physics-Informed Neural Networks for real-world nonlinear dynamical systems](https://mlsanity.com/paper/2205.08304/) (2022)
- [Posterior Concentration for Sparse Deep Learning](https://mlsanity.com/paper/1803.09138/) (2018)
- [Machine Learning For Elliptic PDEs: Fast Rate Generalization Bound, Neural Scaling Law and Minimax Optimality](https://mlsanity.com/paper/2110.06897/) (2021)
- [Efficient Variational Inference for Sparse Deep Learning with Theoretical Guarantee](https://mlsanity.com/paper/2011.07439/) (2020)
- [Convergence Rates of Variational Inference in Sparse Deep Learning](https://mlsanity.com/paper/1908.04847/) (2019)
- [On the Representation of Solutions to Elliptic PDEs in Barron Spaces](https://mlsanity.com/paper/2106.07539/) (2021)
- [A Priori Generalization Error Analysis of Two-Layer Neural Networks for Solving High Dimensional Schr"odinger Eigenvalue Problems](https://mlsanity.com/paper/2105.01228/) (2021)
- [Adaptive variational Bayes: Optimality, computation and applications](https://mlsanity.com/paper/2109.03204/) (2021)
- [Some observations on high-dimensional partial differential equations with Barron data](https://mlsanity.com/paper/2012.01484/) (2020)
- [Posterior and variational inference for deep neural networks with heavy-tailed weights](https://mlsanity.com/paper/2406.03369/) (2024)
- [Masked Bayesian Neural Networks : Theoretical Guarantee and its Posterior Inference](https://mlsanity.com/paper/2305.14765/) (2023)
- [Asymptotic Properties for Bayesian Neural Network in Besov Space](https://mlsanity.com/paper/2206.00241/) (2022)
- [Posterior concentrations of fully-connected Bayesian neural networks with general priors on the weights](https://mlsanity.com/paper/2403.14225/) (2024)
- [Uncertainty Quantification in PINNs for Turbulent Flows: Bayesian Inference and Repulsive Ensembles](https://mlsanity.com/paper/2604.17156/) (2026)
- [On the estimation rate of Bayesian PINN for inverse problems](https://mlsanity.com/paper/2406.14808/) (2024)
- [Posterior Contraction for Sparse Neural Networks in Besov Spaces with Intrinsic Dimensionality](https://mlsanity.com/paper/2506.19144/) (2025)

## Abstract (arXiv)

We study the posterior contraction rate of Bayesian Physics-Informed Neural Networks (PINNs) for solving a general class of elliptic partial differential equations (PDEs). We focus on learning of the elliptic equation with a non-homogeneous Dirichlet boundary condition from independent and noisy measurements collected both inside the domain and on the boundary. Assuming that the PDE admits a strong solution in a H"older space and using with a suitably constructed prior on the neural network weights, we prove that the posterior distribution concentrates around the exact solution at a near-minimax rate. Furthermore, the chosen prior is rate-adaptive: the posterior contracts at an (almost) optimal rate without prior knowledge of the smoothness level of the exact solution. Our results provide statistical guarantees for uncertainty quantification of PDEs via Bayesian PINNs.