# A Physics-Informed Neural Network Framework For Partial Differential Equations on 3D Surfaces: Time-Dependent Problems

arXiv: https://arxiv.org/abs/2103.13878 · HTML: https://mlsanity.com/paper/2103.13878/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 33 references that are in the map.
- First posted on arXiv 19 Mar 2021.

## Builds on (most cited first, 7 of 7)

- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks](https://mlsanity.com/paper/1603.05279/) (2016)
- [hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition](https://mlsanity.com/paper/2003.05385/) (2020)
- [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018)
- [Variational Physics-Informed Neural Networks For Solving Partial Differential Equations](https://mlsanity.com/paper/1912.00873/) (2019)
- [Adversarial Uncertainty Quantification in Physics-Informed Neural Networks](https://mlsanity.com/paper/1811.04026/) (2018)

## Built on by (most cited first, 5 of 5)

- [A Machine Learning Approach to the Nirenberg Problem](https://mlsanity.com/paper/2602.12368/) (2026)
- [Deep learning for solution and inversion of structural mechanics and vibrations](https://mlsanity.com/paper/2105.09477/) (2021)
- [Physics-informed neural networks for transformed geometries and manifolds](https://mlsanity.com/paper/2311.15940/) (2023)
- [Minimal surfaces, Knots, and Neural Networks](https://mlsanity.com/paper/2605.26234/) (2026)
- [A shallow physics-informed neural network for solving partial differential equations on surfaces](https://mlsanity.com/paper/2203.01581/) (2022)

## Abstract (arXiv)

In this paper, we show a physics-informed neural network solver for the time-dependent surface PDEs. Unlike the traditional numerical solver, no extension of PDE and mesh on the surface is needed. We show a simplified prior estimate of the surface differential operators so that PINN's loss value will be an indicator of the residue of the surface PDEs. Numerical experiments verify efficacy of our algorithm.