# Physics-informed Information Field Theory for Modeling Physical Systems with Uncertainty Quantification

arXiv: https://arxiv.org/abs/2301.07609 · HTML: https://mlsanity.com/paper/2301.07609/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 75 references that are in the map.
- First posted on arXiv 18 Jan 2023.

## Builds on (most cited first, 25 of 25)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Stochastic Gradient Hamiltonian Monte Carlo](https://mlsanity.com/paper/1402.4102/) (2014)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [A Conceptual Introduction to Hamiltonian Monte Carlo](https://mlsanity.com/paper/1701.02434/) (2017)
- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [Composable Effects for Flexible and Accelerated Probabilistic Programming in NumPyro](https://mlsanity.com/paper/1912.11554/) (2019)
- [Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems](https://mlsanity.com/paper/1809.08327/) (2018)
- [Physics-Informed Generative Adversarial Networks for Stochastic Differential Equations](https://mlsanity.com/paper/1811.02033/) (2018)
- [Solving and Learning Nonlinear PDEs with Gaussian Processes](https://mlsanity.com/paper/2103.12959/) (2021)
- [Discovering physical concepts with neural networks](https://mlsanity.com/paper/1807.10300/) (2018)
- [Machine learning in cardiovascular flows modeling: Predicting arterial blood pressure from non-invasive 4D flow MRI data using physics-informed neural networks](https://mlsanity.com/paper/1905.04817/) (2019)
- [Physics-Informed Neural Networks for Nonhomogeneous Material Identification in Elasticity Imaging](https://mlsanity.com/paper/2009.04525/) (2020)
- [Learning Functional Priors and Posteriors from Data and Physics](https://mlsanity.com/paper/2106.05863/) (2021)
- [Metric Gaussian Variational Inference](https://mlsanity.com/paper/1901.11033/) (2019)
- [Physics-informed neural networks for solving parametric magnetostatic problems](https://mlsanity.com/paper/2202.04041/) (2022)
- [Hamiltonian Monte Carlo with Energy Conserving Subsampling](https://mlsanity.com/paper/1708.00955/) (2017)
- [Embedded Model Error Representation for Bayesian Model Calibration](https://mlsanity.com/paper/1801.06768/) (2018)
- [Learning without Data: Physics-Informed Neural Networks for Fast Time-Domain Simulation](https://mlsanity.com/paper/2106.15987/) (2021)
- [Information field theory](https://mlsanity.com/paper/1301.2556/) (2013)
- [Probabilistic simulation of partial differential equations](https://mlsanity.com/paper/2010.06583/) (2020)
- [Information Field Theory and Artificial Intelligence](https://mlsanity.com/paper/2112.10133/) (2021)

## Built on by (most cited first, 9 of 9)

- [A Variational Framework for Residual-Based Adaptivity in Neural PDE Solvers and Operator Learning](https://mlsanity.com/paper/2509.14198/) (2025)
- [DGenNO: A Novel Physics-aware Neural Operator for Solving Forward and Inverse PDE Problems based on Deep, Generative Probabilistic Modeling](https://mlsanity.com/paper/2502.06250/) (2025)
- [Conservation-informed Graph Learning for Spatiotemporal Dynamics Prediction](https://mlsanity.com/paper/2412.20962/) (2024)
- [An information field theory approach to Bayesian state and parameter estimation in dynamical systems](https://mlsanity.com/paper/2306.02150/) (2023)
- [Auto-weighted Bayesian Physics-Informed Neural Networks and robust estimations for multitask inverse problems in pore-scale imaging of dissolution](https://mlsanity.com/paper/2308.12864/) (2023)
- [Bayesian neural networks with interpretable priors from Mercer kernels](https://mlsanity.com/paper/2510.23745/) (2025)
- [Coding-agents can replicate scientific machine learning papers](https://mlsanity.com/paper/2607.02134/) (2026)
- [Learning Latent Hardening (LLH): Enhancing Deep Learning with Domain Knowledge for Material Inverse Problems](https://mlsanity.com/paper/2501.10481/) (2025)
- [An interpretation of the Brownian bridge as a physics-informed prior for the Poisson equation](https://mlsanity.com/paper/2503.00213/) (2025)

## Abstract (arXiv)

Data-driven approaches coupled with physical knowledge are powerful techniques to model systems. The goal of such models is to efficiently solve for the underlying field by combining measurements with known physical laws. As many systems contain unknown elements, such as missing parameters, noisy data, or incomplete physical laws, this is widely approached as an uncertainty quantification problem. The common techniques to handle all the variables typically depend on the numerical scheme used to approximate the posterior, and it is desirable to have a method which is independent of any such discretization. Information field theory (IFT) provides the tools necessary to perform statistics over fields that are not necessarily Gaussian. We extend IFT to physics-informed IFT (PIFT) by encoding the functional priors with information about the physical laws which describe the field. The posteriors derived from this PIFT remain independent of any numerical scheme and can capture multiple modes, allowing for the solution of problems which are ill-posed. We demonstrate our approach through an analytical example involving the Klein-Gordon equation. We then develop a variant of stochastic gradient Langevin dynamics to draw samples from the joint posterior over the field and model parameters. We apply our method to numerical examples with various degrees of model-form error and to inverse problems involving nonlinear differential equations. As an addendum, the method is equipped with a metric which allows the posterior to automatically quantify model-form uncertainty. Because of this, our numerical experiments show that the method remains robust to even an incorrect representation of the physics given sufficient data. We numerically demonstrate that the method correctly identifies when the physics cannot be trusted, in which case it automatically treats learning the field as a regression problem.