# Bayesian Reasoning for Physics Informed Neural Networks

arXiv: https://arxiv.org/abs/2308.13222 · HTML: https://mlsanity.com/paper/2308.13222/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 32 of its 78 references that are in the map.
- First posted on arXiv 25 Aug 2023.

## Builds on (most cited first, 30 of 32)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantification](https://mlsanity.com/paper/1801.06879/) (2018)
- [Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisons](https://mlsanity.com/paper/2201.07766/) (2022)
- [Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10566/) (2017)
- [Machine Learning of Linear Differential Equations using Gaussian Processes](https://mlsanity.com/paper/1701.02440/) (2017)
- [Modeling the Dynamics of PDE Systems with Physics-Constrained Deep Auto-Regressive Networks](https://mlsanity.com/paper/1906.05747/) (2019)
- [Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networks](https://mlsanity.com/paper/2007.04542/) (2020)
- [A high-bias, low-variance introduction to Machine Learning for physicists](https://mlsanity.com/paper/1803.08823/) (2018)
- [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)
- [Physics-Informed Machine Learning: A Survey on Problems, Methods and Applications](https://mlsanity.com/paper/2211.08064/) (2022)
- [Physics-based Deep Learning](https://mlsanity.com/paper/2109.05237/) (2021)
- [When Physics Meets Machine Learning: A Survey of Physics-Informed Machine Learning](https://mlsanity.com/paper/2203.16797/) (2022)
- [NVIDIA SimNet^{TM}: an AI-accelerated multi-physics simulation framework](https://mlsanity.com/paper/2012.07938/) (2020)
- [DPM: A Novel Training Method for Physics-Informed Neural Networks in Extrapolation](https://mlsanity.com/paper/2012.02681/) (2020)
- [Inferring solutions of differential equations using noisy multi-fidelity data](https://mlsanity.com/paper/1607.04805/) (2016)
- [Physics Informed Neural Networks for Simulating Radiative Transfer](https://mlsanity.com/paper/2009.13291/) (2020)
- [Adaptive Self-supervision Algorithms for Physics-informed Neural Networks](https://mlsanity.com/paper/2207.04084/) (2022)
- [Physics-Guided, Physics-Informed, and Physics-Encoded Neural Networks in Scientific Computing](https://mlsanity.com/paper/2211.07377/) (2022)
- [Bayesian Learning for Neural Networks: an algorithmic survey](https://mlsanity.com/paper/2211.11865/) (2022)
- [Constraining Gaussian Processes to Systems of Linear Ordinary Differential Equations](https://mlsanity.com/paper/2208.12515/) (2022)
- [Predicting Porosity, Permeability, and Tortuosity of Porous Media from Images by Deep Learning](https://mlsanity.com/paper/2007.02820/) (2020)
- [Bayesian Deep Learning for Partial Differential Equation Parameter Discovery with Sparse and Noisy Data](https://mlsanity.com/paper/2108.04085/) (2021)

## Abstract (arXiv)

We introduce an evidence-driven Bayesian formulation of physics-informed neural networks that enables automatic optimization of loss weights between PDE residuals, boundary conditions, and observational data. Unlike existing Bayesian PINN approaches based on sampling or variational inference, the proposed method uses a Laplace approximation to compute model evidence analytically, enabling efficient hyperparameter tuning and model comparison without posterior sampling. We demonstrate the method on the heat, wave, and Burgers' equations, obtaining solutions in agreement with exact or reference results. In the Burgers' equation example, we further show that the framework naturally integrates information from governing equations and noisy measurements, providing predictive uncertainties within a unified Bayesian setting.