# Physics-informed ConvNet: Learning Physical Field from a Shallow Neural Network

arXiv: https://arxiv.org/abs/2201.10967 · HTML: https://mlsanity.com/paper/2201.10967/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 83 references that are in the map.
- First posted on arXiv 26 Jan 2022.

## Builds on (most cited first, 27 of 27)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Image Super-Resolution Using Deep Convolutional Networks](https://mlsanity.com/paper/1501.00092/) (2014)
- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Generative Adversarial Networks: An Overview](https://mlsanity.com/paper/1710.07035/) (2017)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [A Survey on Deep Transfer Learning](https://mlsanity.com/paper/1808.01974/) (2018)
- [A Survey of Convolutional Neural Networks: Analysis, Applications, and Prospects](https://mlsanity.com/paper/2004.02806/) (2020)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networks](https://mlsanity.com/paper/1901.06523/) (2019)
- [On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks](https://mlsanity.com/paper/2012.10047/) (2020)
- [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018)
- [PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domain](https://mlsanity.com/paper/2004.13145/) (2020)
- [Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems](https://mlsanity.com/paper/1809.08327/) (2018)
- [Modeling the Dynamics of PDE Systems with Physics-Constrained Deep Auto-Regressive Networks](https://mlsanity.com/paper/1906.05747/) (2019)
- [Parallel Physics-Informed Neural Networks via Domain Decomposition](https://mlsanity.com/paper/2104.10013/) (2021)
- [The Limits and Potentials of Deep Learning for Robotics](https://mlsanity.com/paper/1804.06557/) (2018)
- [PhyCRNet: Physics-informed Convolutional-Recurrent Network for Solving Spatiotemporal PDEs](https://mlsanity.com/paper/2106.14103/) (2021)
- [Lift & Learn: Physics-informed machine learning for large-scale nonlinear dynamical systems](https://mlsanity.com/paper/1912.08177/) (2019)
- [Shallow and Deep Convolutional Networks for Saliency Prediction](https://mlsanity.com/paper/1603.00845/) (2016)
- [Physics-Guided Deep Neural Networks for Power Flow Analysis](https://mlsanity.com/paper/2002.00097/) (2020)
- [Physics-informed Convolutional Neural Networks for Temperature Field Prediction of Heat Source Layout without Labeled Data](https://mlsanity.com/paper/2109.12482/) (2021)
- [Conditional physics informed neural networks](https://mlsanity.com/paper/2104.02741/) (2021)

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

- [KANtrol: A Physics-Informed Kolmogorov-Arnold Network Framework for Solving Multi-Dimensional and Fractional Optimal Control Problems](https://mlsanity.com/paper/2409.06649/) (2024)
- [Do physics-informed neural networks (PINNs) need to be deep? Shallow PINNs using the Levenberg-Marquardt algorithm](https://mlsanity.com/paper/2602.08515/) (2026)
- [MLPs and KANs for data-driven learning in physical problems: A performance comparison](https://mlsanity.com/paper/2504.11397/) (2025)
- [FlashPDE: A Drop-In Fused Triton Operator Library for Neural PDE Solvers](https://mlsanity.com/paper/2607.18020/) (2026)
- [Reconstruction of three-dimensional shapes of normal and disease-related erythrocytes from partial observations using multi-fidelity neural networks](https://mlsanity.com/paper/2511.14962/) (2025)

## Abstract (arXiv)

Big-data-based artificial intelligence (AI) supports profound evolution in almost all of science and technology. However, modeling and forecasting multi-physical systems remain a challenge due to unavoidable data scarcity and noise. Improving the generalization ability of neural networks by "teaching" domain knowledge and developing a new generation of models combined with the physical laws have become promising areas of machine learning research. Different from "deep" fully-connected neural networks embedded with physical information (PINN), a novel shallow framework named physics-informed convolutional network (PICN) is recommended from a CNN perspective, in which the physical field is generated by a deconvolution layer and a single convolution layer. The difference fields forming the physical operator are constructed using the pre-trained shallow convolution layer. An efficient linear interpolation network calculates the loss function involving boundary conditions and the physical constraints in irregular geometry domains. The effectiveness of the current development is illustrated through some numerical cases involving the solving (and estimation) of nonlinear physical operator equations and recovering physical information from noisy observations. Its potential advantage in approximating physical fields with multi-frequency components indicates that PICN may become an alternative neural network solver in physics-informed machine learning.