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

arXiv 2605.08672 ↗First posted 9 May 2026math.ST cs.NA math.NAcited by 0

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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.

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References · 26

  1. 2016Variational Inference: A Review for Statisticianscited by 1,466
  2. 2017DGM: A deep learning algorithm for solving partial differential equationscited by 674
  3. 2017The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problemscited by 581
  4. 2020Bayesian Deep Learning and a Probabilistic Perspective of Generalizationcited by 480
  5. 2017Nonparametric regression using deep neural networks with ReLU activation functioncited by 431
  6. 2020B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Datacited by 329
  7. 2021What Are Bayesian Neural Network Posteriors Really Like?cited by 307
  8. 2016A vector-contraction inequality for Rademacher complexitiescited by 141
  9. 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEscited by 115
  10. 2021Error analysis for physics informed neural networks (PINNs) approximating Kolmogorov PDEscited by 56
  11. 2022Bayesian Physics-Informed Neural Networks for real-world nonlinear dynamical systemscited by 54
  12. 2018Posterior Concentration for Sparse Deep Learningcited by 50
14 more
  1. 2021Machine Learning For Elliptic PDEs: Fast Rate Generalization Bound, Neural Scaling Law and Minimax Optimalitycited by 40
  2. 2020Efficient Variational Inference for Sparse Deep Learning with Theoretical Guaranteecited by 37
  3. 2019Convergence Rates of Variational Inference in Sparse Deep Learningcited by 27
  4. 2021On the Representation of Solutions to Elliptic PDEs in Barron Spacescited by 25
  5. 2021A Priori Generalization Error Analysis of Two-Layer Neural Networks for Solving High Dimensional Schr"odinger Eigenvalue Problemscited by 19
  6. 2021Adaptive variational Bayes: Optimality, computation and applicationscited by 17
  7. 2020Some observations on high-dimensional partial differential equations with Barron datacited by 16
  8. 2023Masked Bayesian Neural Networks : Theoretical Guarantee and its Posterior Inferencecited by 9
  9. 2024Posterior and variational inference for deep neural networks with heavy-tailed weightscited by 9
  10. 2022Asymptotic Properties for Bayesian Neural Network in Besov Spacecited by 7
  11. 2024Posterior concentrations of fully-connected Bayesian neural networks with general priors on the weightscited by 7
  12. 2024On the estimation rate of Bayesian PINN for inverse problemscited by 3
  13. 2026Uncertainty Quantification in PINNs for Turbulent Flows: Bayesian Inference and Repulsive Ensemblescited by 3
  14. 2025Posterior Contraction for Sparse Neural Networks in Besov Spaces with Intrinsic Dimensionalitycited by 2

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