# A Dimension-Augmented Physics-Informed Neural Network (DaPINN) with High Level Accuracy and Efficiency

arXiv: https://arxiv.org/abs/2210.13212 · HTML: https://mlsanity.com/paper/2210.13212/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 41 references that are in the map.
- First posted on arXiv 19 Oct 2022.

## Builds on (most cited first, 21 of 21)

- [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)
- [Simple Copy-Paste is a Strong Data Augmentation Method for Instance Segmentation](https://mlsanity.com/paper/2012.07177/) (2020)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [On the Expressive Power of Deep Neural Networks](https://mlsanity.com/paper/1606.05336/) (2016)
- [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021)
- [hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition](https://mlsanity.com/paper/2003.05385/) (2020)
- [Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanism](https://mlsanity.com/paper/2009.04544/) (2020)
- [Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems](https://mlsanity.com/paper/2111.02801/) (2021)
- [Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems](https://mlsanity.com/paper/1809.08327/) (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)
- [On the number of response regions of deep feed forward networks with piece-wise linear activations](https://mlsanity.com/paper/1312.6098/) (2013)
- [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)
- [Multi-scale Deep Neural Network (MscaleDNN) for Solving Poisson-Boltzmann Equation in Complex Domains](https://mlsanity.com/paper/2007.11207/) (2020)
- [Physics-informed neural networks for inverse problems in supersonic flows](https://mlsanity.com/paper/2202.11821/) (2022)
- [Physics-informed neural network for ultrasound nondestructive quantification of surface breaking cracks](https://mlsanity.com/paper/2005.03596/) (2020)
- [Physics-Informed Neural Networks for Multiphysics Data Assimilation with Application to Subsurface Transport](https://mlsanity.com/paper/1912.02968/) (2019)
- [A Phase Shift Deep Neural Network for High Frequency Approximation and Wave Problems](https://mlsanity.com/paper/1909.11759/) (2019)
- [Deep Neural Network Approach to Forward-Inverse Problems](https://mlsanity.com/paper/1907.12925/) (2019)
- [Physics-informed Neural Network for Nonlinear Dynamics in Fiber Optics](https://mlsanity.com/paper/2109.00526/) (2021)

## Built on by (most cited first, 7 of 7)

- [From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learning](https://mlsanity.com/paper/2410.13228/) (2024)
- [Residual-based attention in physics-informed neural networks](https://mlsanity.com/paper/2307.00379/) (2023)
- [Learning in PINNs: Phase transition, diffusion equilibrium, and generalization](https://mlsanity.com/paper/2403.18494/) (2024)
- [A comprehensive analysis of PINNs: Variants, Applications, and Challenges](https://mlsanity.com/paper/2505.22761/) (2025)
- [Two-scale Neural Networks for Partial Differential Equations with Small Parameters](https://mlsanity.com/paper/2402.17232/) (2024)
- [Perception-Informed Neural Networks: Beyond Physics-Informed Neural Networks](https://mlsanity.com/paper/2505.03806/) (2025)
- [Multi-Scale Separable Fourier Neural Networks for Solving High-Frequency PDEs](https://mlsanity.com/paper/2605.31027/) (2026)

## Abstract (arXiv)

Physics-informed neural networks (PINNs) have been widely applied in different fields due to their effectiveness in solving partial differential equations (PDEs). However, the accuracy and efficiency of PINNs need to be considerably improved for scientific and commercial use. To address this issue, we systematically propose a novel dimension-augmented physics-informed neural network (DaPINN), which simultaneously and significantly improves the accuracy and efficiency of the PINN. In the DaPINN model, we introduce inductive bias in the neural network to enhance network generalizability by adding a special regularization term to the loss function. Furthermore, we manipulate the network input dimension by inserting additional sample features and incorporating the expanded dimensionality in the loss function. Moreover, we verify the effectiveness of power series augmentation, Fourier series augmentation and replica augmentation, in both forward and backward problems. In most experiments, the error of DaPINN is 1~2 orders of magnitude lower than that of PINN. The results show that the DaPINN outperforms the original PINN in terms of both accuracy and efficiency with a reduced dependence on the number of sample points. We also discuss the complexity of the DaPINN and its compatibility with other methods.