# Physics-informed neural networks for operator equations with stochastic data

arXiv: https://arxiv.org/abs/2211.10344 · HTML: https://mlsanity.com/paper/2211.10344/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 43 references that are in the map.
- First posted on arXiv 15 Nov 2022.

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

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems](https://mlsanity.com/paper/1809.08327/) (2018)
- [Adversarial Uncertainty Quantification in Physics-Informed Neural Networks](https://mlsanity.com/paper/1811.04026/) (2018)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEs](https://mlsanity.com/paper/2006.16144/) (2020)
- [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)
- [Deep UQ: Learning deep neural network surrogate models for high dimensional uncertainty quantification](https://mlsanity.com/paper/1802.00850/) (2018)
- [Learning in Modal Space: Solving Time-Dependent Stochastic PDEs Using Physics-Informed Neural Networks](https://mlsanity.com/paper/1905.01205/) (2019)
- [NeuralUQ: A comprehensive library for uncertainty quantification in neural differential equations and operators](https://mlsanity.com/paper/2208.11866/) (2022)
- [Hyper-parameter tuning of physics-informed neural networks: Application to Helmholtz problems](https://mlsanity.com/paper/2205.06704/) (2022)
- [Multi-Output Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Uncertainties](https://mlsanity.com/paper/2202.01710/) (2022)
- [Deep neural network surrogates for non-smooth quantities of interest in shape uncertainty quantification](https://mlsanity.com/paper/2101.07023/) (2021)
- [A matrix-free isogeometric Galerkin method for Karhunen-Lo`eve approximation of random fields using tensor product splines, tensor contraction and interpolation based quadrature](https://mlsanity.com/paper/2011.13861/) (2020)

## Built on by (most cited first, 1 of 1)

- [h-analysis and data-parallel physics-informed neural networks](https://mlsanity.com/paper/2302.08835/) (2023)

## Abstract (arXiv)

We consider the computation of statistical moments to operator equations with stochastic data. We remark that application of PINNs -- referred to as TPINNs -- allows to solve the induced tensor operator equations under minimal changes of existing PINNs code, and enabling handling of non-linear and time-dependent operators. We propose two types of architectures, referred to as vanilla and multi-output TPINNs, and investigate their benefits and limitations. Exhaustive numerical experiments are performed; demonstrating applicability and performance; raising a variety of new promising research avenues.