# Multi-Output Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Uncertainties

arXiv: https://arxiv.org/abs/2202.01710 · HTML: https://mlsanity.com/paper/2202.01710/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Feb 2022.

## Built on by (most cited first, 14 of 14)

- [From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learning](https://mlsanity.com/paper/2410.13228/) (2024)
- [L-HYDRA: Multi-Head Physics-Informed Neural Networks](https://mlsanity.com/paper/2301.02152/) (2023)
- [Uncertainty quantification for noisy inputs-outputs in physics-informed neural networks and neural operators](https://mlsanity.com/paper/2311.11262/) (2023)
- [PhyMPGN: Physics-encoded Message Passing Graph Network for spatiotemporal PDE systems](https://mlsanity.com/paper/2410.01337/) (2024)
- [Enhanced BPINN Training Convergence in Solving General and Multi-scale Elliptic PDEs with Noise](https://mlsanity.com/paper/2408.09340/) (2024)
- [MRF-PINN: A Multi-Receptive-Field convolutional physics-informed neural network for solving partial differential equations](https://mlsanity.com/paper/2209.03151/) (2022)
- [Physics-informed neural networks for operator equations with stochastic data](https://mlsanity.com/paper/2211.10344/) (2022)
- [VI-PINNs: Variance-involved Physics-informed Neural Networks for Fast and Accurate Prediction of Partial Differential Equations](https://mlsanity.com/paper/2211.16753/) (2022)
- [Solving PDEs with Unmeasurable Source Terms Using Coupled Physics-Informed Neural Network with Recurrent Prediction for Soft Sensors](https://mlsanity.com/paper/2301.08618/) (2023)
- [Physics informed cell representations for variational formulation of multiscale problems](https://mlsanity.com/paper/2405.16770/) (2024)
- [Repulsive Ensembles for Bayesian Inference in Physics-informed Neural Networks](https://mlsanity.com/paper/2505.17308/) (2025)
- [BridgeNet: A Hybrid, Physics-Informed Machine Learning Framework for Solving High-Dimensional Fokker-Planck Equations](https://mlsanity.com/paper/2506.04354/) (2025)
- [Investigating Nonlinear Quenching Effects on Polar Field Buildup in the Sun Using Physics-Informed Neural Networks](https://mlsanity.com/paper/2602.16656/) (2026)
- [Physics-Informed Neural Networks for Complex Eigenfrequency Identification and Mode Structure Reconstruction of the Ground-State ITG Branch](https://mlsanity.com/paper/2608.01850/) (2026)

## Abstract (arXiv)

Physics-informed neural networks (PINNs) have recently been used to solve various computational problems which are governed by partial differential equations (PDEs). In this paper, we propose a multi-output physics-informed neural network (MO-PINN) which can provide solutions with uncertainty distributions for both forward and inverse PDE problems with noisy data. In this framework, the uncertainty arising from the noisy data is first translated into multiple measurements regarding the prior noise distribution using the bootstrap method, and then the outputs of neural networks are designed to satisfy the measurements as well as the underlying physical laws.The posterior estimation of target parameters can be obtained at the end of training, which can be further used for uncertainty quantification and decision making. In this paper, MO-PINNs are demonstrated with a series of numerical experiments including both linear and nonlinear, forward and inverse problems. The results show that MO-PINN is able to provide accurate predictions with noisy data.In addition, we also demonstrate that the prediction and posterior distributions from MO-PINNs are consistent with the solutions from traditional a finite element method (FEM) solver and Monte Carlo methods given the same data and prior knowledge. Finally, we show that additional statistical knowledge can be incorporated into the training to improve the prediction if available.