# Physics-informed Convolutional Neural Networks for Temperature Field Prediction of Heat Source Layout without Labeled Data

arXiv: https://arxiv.org/abs/2109.12482 · HTML: https://mlsanity.com/paper/2109.12482/

## Key facts

- Cited by 31 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 40 references that are in the map.
- First posted on arXiv 26 Sep 2021.

## Builds on (most cited first, 19 of 19)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Group Normalization](https://mlsanity.com/paper/1803.08494/) (2018)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (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)
- [Deep Fluids: A Generative Network for Parameterized Fluid Simulations](https://mlsanity.com/paper/1806.02071/) (2018)
- [Physics-Informed Neural Networks for Power Systems](https://mlsanity.com/paper/1911.03737/) (2019)
- [A physics-informed operator regression framework for extracting data-driven continuum models](https://mlsanity.com/paper/2009.11992/) (2020)
- [A Physics-Informed Machine Learning Approach for Solving Heat Transfer Equation in Advanced Manufacturing and Engineering Applications](https://mlsanity.com/paper/2010.02011/) (2020)
- [Deep Learning of Subsurface Flow via Theory-guided Neural Network](https://mlsanity.com/paper/1911.00103/) (2019)
- [A Deep Neural Network Surrogate Modeling Benchmark for Temperature Field Prediction of Heat Source Layout](https://mlsanity.com/paper/2103.11177/) (2021)
- [FEA-Net: A Physics-guided Data-driven Model for Efficient Mechanical Response Prediction](https://mlsanity.com/paper/2002.01893/) (2020)

## Built on by (most cited first, 30 of 31)

- [Physics-Informed Computer Vision: A Review and Perspectives](https://mlsanity.com/paper/2305.18035/) (2023)
- [MAgNET: A Graph U-Net Architecture for Mesh-Based Simulations](https://mlsanity.com/paper/2211.00713/) (2022)
- [A finite element-based physics-informed operator learning framework for spatiotemporal partial differential equations on arbitrary domains](https://mlsanity.com/paper/2405.12465/) (2024)
- [A Unified Hard-Constraint Framework for Solving Geometrically Complex PDEs](https://mlsanity.com/paper/2210.03526/) (2022)
- [Physics-informed MeshGraphNets (PI-MGNs): Neural finite element solvers for non-stationary and nonlinear simulations on arbitrary meshes](https://mlsanity.com/paper/2402.10681/) (2024)
- [A finite operator learning technique for mapping the elastic properties of microstructures to their mechanical deformations](https://mlsanity.com/paper/2404.00074/) (2024)
- [Multi-fidelity surrogate modeling for temperature field prediction using deep convolution neural network](https://mlsanity.com/paper/2301.06674/) (2023)
- [Physics Informed Neural Network for Dynamic Stress Prediction](https://mlsanity.com/paper/2211.16190/) (2022)
- [Physics-informed ConvNet: Learning Physical Field from a Shallow Neural Network](https://mlsanity.com/paper/2201.10967/) (2022)
- [Physics-informed Convolutional Recurrent Surrogate Model for Reservoir Simulation with Well Controls](https://mlsanity.com/paper/2305.09056/) (2023)
- [Integration of physics-informed operator learning and finite element method for parametric learning of partial differential equations](https://mlsanity.com/paper/2401.02363/) (2024)
- [Physics-informed MTA-UNet: Prediction of Thermal Stress and Thermal Deformation of Satellites](https://mlsanity.com/paper/2209.01009/) (2022)
- [Solving Maxwell's Equation in 2D with Neural Networks with Local Converging Inputs](https://mlsanity.com/paper/2302.02860/) (2023)
- [Physics-informed UNets for Discovering Hidden Elasticity in Heterogeneous Materials](https://mlsanity.com/paper/2306.01204/) (2023)
- [Smooth and Sparse Latent Dynamics in Operator Learning with Jerk Regularization](https://mlsanity.com/paper/2402.15636/) (2024)
- [A physics and data co-driven surrogate modeling approach for temperature field prediction on irregular geometric domain](https://mlsanity.com/paper/2203.08150/) (2022)
- [Uncertainty Guided Ensemble Self-Training for Semi-Supervised Global Field Reconstruction](https://mlsanity.com/paper/2302.11940/) (2023)
- [Solutions to Elliptic and Parabolic Problems via Finite Difference Based Unsupervised Small Linear Convolutional Neural Networks](https://mlsanity.com/paper/2311.00259/) (2023)
- [Position Paper on Materials Design -- A Modern Approach](https://mlsanity.com/paper/2312.10996/) (2023)
- [Physics-informed Machine Learning for Battery Pack Thermal Management](https://mlsanity.com/paper/2411.09915/) (2024)
- [A Reward-Directed Diffusion Framework for Generative Design Optimization](https://mlsanity.com/paper/2508.01509/) (2025)
- [Physics-Informed Convolutional Neural Networks for Corruption Removal on Dynamical Systems](https://mlsanity.com/paper/2210.16215/) (2022)
- [Adapting U-Net for linear elastic stress estimation in polycrystal Zr microstructures](https://mlsanity.com/paper/2303.08541/) (2023)
- [Physics-informed Neural Network Combined with Characteristic-Based Split for Solving Navier-Stokes Equations](https://mlsanity.com/paper/2304.10717/) (2023)
- [Learning Discontinuous Galerkin Solutions to Elliptic Problems via Small Linear Convolutional Neural Networks](https://mlsanity.com/paper/2502.08783/) (2025)
- [Deep Learning Meets Process-Based Models: A Hybrid Approach to Agricultural Challenges](https://mlsanity.com/paper/2504.16141/) (2025)
- [Physics-Informed Modeling for Wood Thermal Analysis and Prediction](https://mlsanity.com/paper/2606.23402/) (2026)
- [Learning to Reconstruct Temperature Field from Sparse Observations with Implicit Physics Priors](https://mlsanity.com/paper/2512.01196/) (2025)
- [Physics-informed convolutional neural networks for fluid flow through porous media](https://mlsanity.com/paper/2605.20250/) (2026)
- [MTL-FNO: A Lightweight Multi-Task Fourier Neural Operator for Sparse Field Reconstruction](https://mlsanity.com/paper/2605.26718/) (2026)

## Abstract (arXiv)

Recently, surrogate models based on deep learning have attracted much attention for engineering analysis and optimization. As the construction of data pairs in most engineering problems is time-consuming, data acquisition is becoming the predictive capability bottleneck of most deep surrogate models, which also exists in surrogate for thermal analysis and design. To address this issue, this paper develops a physics-informed convolutional neural network (CNN) for the thermal simulation surrogate. The network can learn a mapping from heat source layout to the steady-state temperature field without labeled data, which equals solving an entire family of partial difference equations (PDEs). To realize the physics-guided training without labeled data, we employ the heat conduction equation and finite difference method to construct the loss function. Since the solution is sensitive to boundary conditions, we properly impose hard constraints by padding in the Dirichlet and Neumann boundary conditions. In addition, the neural network architecture is well-designed to improve the prediction precision of the problem at hand, and pixel-level online hard example mining is introduced to overcome the imbalance of optimization difficulty in the computation domain. The experiments demonstrate that the proposed method can provide comparable predictions with numerical method and data-driven deep learning models. We also conduct various ablation studies to investigate the effectiveness of the network component and training methods proposed in this paper.