# Solutions to Elliptic and Parabolic Problems via Finite Difference Based Unsupervised Small Linear Convolutional Neural Networks

arXiv: https://arxiv.org/abs/2311.00259 · HTML: https://mlsanity.com/paper/2311.00259/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 47 references that are in the map.
- First posted on arXiv 1 Nov 2023.

## Builds on (most cited first, 24 of 24)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Densely Connected Convolutional Networks](https://mlsanity.com/paper/1608.06993/) (2016)
- [Identity Mappings in Deep Residual Networks](https://mlsanity.com/paper/1603.05027/) (2016)
- [3D U-Net: Learning Dense Volumetric Segmentation from Sparse Annotation](https://mlsanity.com/paper/1606.06650/) (2016)
- [Deep High-Resolution Representation Learning for Human Pose Estimation](https://mlsanity.com/paper/1902.09212/) (2019)
- [Visualizing the Loss Landscape of Neural Nets](https://mlsanity.com/paper/1712.09913/) (2017)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Deeply-Supervised Nets](https://mlsanity.com/paper/1409.5185/) (2014)
- [CAN-PINN: A Fast Physics-Informed Neural Network Based on Coupled-Automatic-Numerical Differentiation Method](https://mlsanity.com/paper/2110.15832/) (2021)
- [DAS-PINNs: A deep adaptive sampling method for solving high-dimensional partial differential equations](https://mlsanity.com/paper/2112.14038/) (2021)
- [Optimal control of PDEs using physics-informed neural networks](https://mlsanity.com/paper/2111.09880/) (2021)
- [MgNet: A Unified Framework of Multigrid and Convolutional Neural Network](https://mlsanity.com/paper/1901.10415/) (2019)
- [Physics-informed Convolutional Neural Networks for Temperature Field Prediction of Heat Source Layout without Labeled Data](https://mlsanity.com/paper/2109.12482/) (2021)
- [Error Estimates for the Deep Ritz Method with Boundary Penalty](https://mlsanity.com/paper/2103.01007/) (2021)
- [A Comprehensive Review on Deep Supervision: Theories and Applications](https://mlsanity.com/paper/2207.02376/) (2022)
- [On the Generalization of PINNs outside the training domain and the Hyperparameters influencing it](https://mlsanity.com/paper/2302.07557/) (2023)
- [PocketNet: A Smaller Neural Network for Medical Image Analysis](https://mlsanity.com/paper/2104.10745/) (2021)
- [Inversion of Time-Lapse Surface Gravity Data for Detection of 3D CO2 Plumes via Deep Learning](https://mlsanity.com/paper/2209.02850/) (2022)
- [FMG-Net and W-Net: Multigrid Inspired Deep Learning Architectures For Medical Imaging Segmentation](https://mlsanity.com/paper/2304.02725/) (2023)
- [Joint inversion of Time-Lapse Surface Gravity and Seismic Data for Monitoring of 3D CO2 Plumes via Deep Learning](https://mlsanity.com/paper/2310.04430/) (2023)

## Built on by (most cited first, 2 of 2)

- [Learning Discontinuous Galerkin Solutions to Elliptic Problems via Small Linear Convolutional Neural Networks](https://mlsanity.com/paper/2502.08783/) (2025)
- [Hybrid coupling with numerics-informed neural networks and the overlapping Schwarz alternating method](https://mlsanity.com/paper/2609.17841/) (2026)

## Abstract (arXiv)

In recent years, there has been a growing interest in leveraging deep learning and neural networks to address scientific problems, particularly in solving partial differential equations (PDEs). However, many neural network-based methods like PINNs rely on auto differentiation and sampling collocation points, leading to a lack of interpretability and lower accuracy than traditional numerical methods. As a result, we propose a fully unsupervised approach, requiring no training data, to estimate finite difference solutions for PDEs directly via small linear convolutional neural networks. Our proposed approach uses substantially fewer parameters than similar finite difference-based approaches while also demonstrating comparable accuracy to the true solution for several selected elliptic and parabolic problems compared to the finite difference method.