# VI-PINNs: Variance-involved Physics-informed Neural Networks for Fast and Accurate Prediction of Partial Differential Equations

arXiv: https://arxiv.org/abs/2211.16753 · HTML: https://mlsanity.com/paper/2211.16753/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 55 references that are in the map.
- First posted on arXiv 30 Nov 2022.

## Builds on (most cited first, 30 of 31)

- [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)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Reinforcement Learning with Unsupervised Auxiliary Tasks](https://mlsanity.com/paper/1611.05397/) (2016)
- [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)
- [A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks](https://mlsanity.com/paper/2207.10289/) (2022)
- [Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanism](https://mlsanity.com/paper/2009.04544/) (2020)
- [Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisons](https://mlsanity.com/paper/2201.07766/) (2022)
- [Respecting causality is all you need for training physics-informed neural networks](https://mlsanity.com/paper/2203.07404/) (2022)
- [Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems](https://mlsanity.com/paper/1809.08327/) (2018)
- [Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equations](https://mlsanity.com/paper/2107.07871/) (2021)
- [Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networks](https://mlsanity.com/paper/2007.04542/) (2020)
- [Adversarial Uncertainty Quantification in Physics-Informed Neural Networks](https://mlsanity.com/paper/1811.04026/) (2018)
- [Auxiliary Tasks in Multi-task Learning](https://mlsanity.com/paper/1805.06334/) (2018)
- [A Method for Representing Periodic Functions and Enforcing Exactly Periodic Boundary Conditions with Deep Neural Networks](https://mlsanity.com/paper/2007.07442/) (2020)
- [Multi-Objective Loss Balancing for Physics-Informed Deep Learning](https://mlsanity.com/paper/2110.09813/) (2021)
- [A fast and accurate physics-informed neural network reduced order model with shallow masked autoencoder](https://mlsanity.com/paper/2009.11990/) (2020)
- [On The Effect of Auxiliary Tasks on Representation Dynamics](https://mlsanity.com/paper/2102.13089/) (2021)
- [Auxiliary Tasks Speed Up Learning PointGoal Navigation](https://mlsanity.com/paper/2007.04561/) (2020)
- [SoftAdapt: Techniques for Adaptive Loss Weighting of Neural Networks with Multi-Part Loss Functions](https://mlsanity.com/paper/1912.12355/) (2019)
- [A novel meta-learning initialization method for physics-informed neural networks](https://mlsanity.com/paper/2107.10991/) (2021)
- [Active learning of deep surrogates for PDEs: Application to metasurface design](https://mlsanity.com/paper/2008.12649/) (2020)
- [Accelerating Physics-Informed Neural Network Training with Prior Dictionaries](https://mlsanity.com/paper/2004.08151/) (2020)
- [An extended physics informed neural network for preliminary analysis of parametric optimal control problems](https://mlsanity.com/paper/2110.13530/) (2021)
- [Multi-Output Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Uncertainties](https://mlsanity.com/paper/2202.01710/) (2022)

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

- [MAD-NG: Meta-Auto-Decoder Neural Galerkin Method for Solving Parametric Partial Differential Equations](https://mlsanity.com/paper/2512.21633/) (2025)

## Abstract (arXiv)

Although physics-informed neural networks(PINNs) have progressed a lot in many real applications recently, there remains problems to be further studied, such as achieving more accurate results, taking less training time, and quantifying the uncertainty of the predicted results. Recent advances in PINNs have indeed significantly improved the performance of PINNs in many aspects, but few have considered the effect of variance in the training process. In this work, we take into consideration the effect of variance and propose our VI-PINNs to give better predictions. We output two values in the final layer of the network to represent the predicted mean and variance respectively, and the latter is used to represent the uncertainty of the output. A modified negative log-likelihood loss and an auxiliary task are introduced for fast and accurate training. We perform several experiments on a wide range of different problems to highlight the advantages of our approach. The results convey that our method not only gives more accurate predictions but also converges faster.