# Randomized Physics-Informed Neural Networks for Bayesian Data Assimilation

arXiv: https://arxiv.org/abs/2407.04617 · HTML: https://mlsanity.com/paper/2407.04617/

## Key facts

- Cited by 7 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 5 Jul 2024.

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

- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Loss Surfaces, Mode Connectivity, and Fast Ensembling of DNNs](https://mlsanity.com/paper/1802.10026/) (2018)
- [Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithm](https://mlsanity.com/paper/1608.04471/) (2016)
- [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 Conceptual Introduction to Hamiltonian Monte Carlo](https://mlsanity.com/paper/1701.02434/) (2017)
- [Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisons](https://mlsanity.com/paper/2201.07766/) (2022)
- [Integrating Scientific Knowledge with Machine Learning for Engineering and Environmental Systems](https://mlsanity.com/paper/2003.04919/) (2020)
- [Physics-Informed Neural Networks for Multiphysics Data Assimilation with Application to Subsurface Transport](https://mlsanity.com/paper/1912.02968/) (2019)
- [Quality of Uncertainty Quantification for Bayesian Neural Network Inference](https://mlsanity.com/paper/1906.09686/) (2019)
- [Bayesian Physics-Informed Neural Networks for real-world nonlinear dynamical systems](https://mlsanity.com/paper/2205.08304/) (2022)
- [Physics-Informed Neural Network Method for Parabolic Differential Equations with Sharply Perturbed Initial Conditions](https://mlsanity.com/paper/2208.08635/) (2022)
- [Randomized Physics-Informed Machine Learning for Uncertainty Quantification in High-Dimensional Inverse Problems](https://mlsanity.com/paper/2312.06177/) (2023)

## Built on by (most cited first, 7 of 7)

- [A Conformal Prediction Framework for Uncertainty Quantification in Physics-Informed Neural Networks](https://mlsanity.com/paper/2509.13717/) (2025)
- [Enhanced BPINN Training Convergence in Solving General and Multi-scale Elliptic PDEs with Noise](https://mlsanity.com/paper/2408.09340/) (2024)
- [VAE-DNN: Energy-Efficient Trainable-by-Parts Surrogate Model For Parametric Partial Differential Equations](https://mlsanity.com/paper/2508.03839/) (2025)
- [Statistical Learning Analysis of Physics-Informed Neural Networks](https://mlsanity.com/paper/2602.11097/) (2026)
- [Generative AI-enhanced Probabilistic Multi-Fidelity Surrogate Modeling Via Transfer Learning](https://mlsanity.com/paper/2602.00072/) (2026)
- [Uncertainty-aware Multi-fidelity Closure via Conditional Normalizing Flows](https://mlsanity.com/paper/2606.09857/) (2026)
- [Efficient Message Passing for Partial Differential Equation Priors](https://mlsanity.com/paper/2609.32956/) (2026)

## Abstract (arXiv)

We propose a randomized physics-informed neural network (PINN) or rPINN method for uncertainty quantification in inverse partial differential equation (PDE) problems with noisy data. This method is used to quantify uncertainty in the inverse PDE PINN solutions. Recently, the Bayesian PINN (BPINN) method was proposed, where the posterior distribution of the PINN parameters was formulated using the Bayes' theorem and sampled using approximate inference methods such as the Hamiltonian Monte Carlo (HMC) and variational inference (VI) methods. In this work, we demonstrate that HMC fails to converge for non-linear inverse PDE problems. As an alternative to HMC, we sample the distribution by solving the stochastic optimization problem obtained by randomizing the PINN loss function. The effectiveness of the rPINN method is tested for linear and non-linear Poisson equations, and the diffusion equation with a high-dimensional space-dependent diffusion coefficient. The rPINN method provides informative distributions for all considered problems. For the linear Poisson equation, HMC and rPINN produce similar distributions, but rPINN is on average 27 times faster than HMC. For the non-linear Poison and diffusion equations, the HMC method fails to converge because a single HMC chain cannot sample multiple modes of the posterior distribution of the PINN parameters in a reasonable amount of time.