# An unsupervised learning approach to solving heat equations on chip based on Auto Encoder and Image Gradient

arXiv: https://arxiv.org/abs/2007.09684 · HTML: https://mlsanity.com/paper/2007.09684/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Jul 2020.

## Built on by (most cited first, 13 of 13)

- [Conditional physics informed neural networks](https://mlsanity.com/paper/2104.02741/) (2021)
- [DeepOHeat: Operator Learning-based Ultra-fast Thermal Simulation in 3D-IC Design](https://mlsanity.com/paper/2302.12949/) (2023)
- [A composable machine-learning approach for steady-state simulations on high-resolution grids](https://mlsanity.com/paper/2210.05837/) (2022)
- [A Thermal Machine Learning Solver For Chip Simulation](https://mlsanity.com/paper/2209.04741/) (2022)
- [Deep learning approaches to surrogates for solving the diffusion equation for mechanistic real-world simulations](https://mlsanity.com/paper/2102.05527/) (2021)
- [A Latent space solver for PDE generalization](https://mlsanity.com/paper/2104.02452/) (2021)
- [Analyzing the Performance of Deep Encoder-Decoder Networks as Surrogates for a Diffusion Equation](https://mlsanity.com/paper/2302.03786/) (2023)
- [Feasibility Study on Active Learning of Smart Surrogates for Scientific Simulations](https://mlsanity.com/paper/2407.07674/) (2024)
- [A composable autoencoder-based iterative algorithm for accelerating numerical simulations](https://mlsanity.com/paper/2110.03780/) (2021)
- [Geometry encoding for numerical simulations](https://mlsanity.com/paper/2104.07792/) (2021)
- [DeepOHeat-v1: Efficient Operator Learning for Fast and Trustworthy Thermal Simulation and Optimization in 3D-IC Design](https://mlsanity.com/paper/2504.03955/) (2025)
- [Faster Optimization on Sparse Graphs via Neural Reparametrization](https://mlsanity.com/paper/2205.13624/) (2022)
- [A conservative hybrid physics-informed neural network method for Maxwell-Amp`ere-Nernst-Planck equations](https://mlsanity.com/paper/2312.05891/) (2023)

## Abstract (arXiv)

Solving heat transfer equations on chip becomes very critical in the upcoming 5G and AI chip-package-systems. However, batches of simulations have to be performed for data driven supervised machine learning models. Data driven methods are data hungry, to address this, Physics Informed Neural Networks (PINN) have been proposed. However, vanilla PINN models solve one fixed heat equation at a time, so the models have to be retrained for heat equations with different source terms. Additionally, issues related to multi-objective optimization have to be resolved while using PINN to minimize the PDE residual, satisfy boundary conditions and fit the observed data etc. Therefore, this paper investigates an unsupervised learning approach for solving heat transfer equations on chip without using solution data and generalizing the trained network for predicting solutions for heat equations with unseen source terms. Specifically, a hybrid framework of Auto Encoder (AE) and Image Gradient (IG) based network is designed. The AE is used to encode different source terms of the heat equations. The IG based network implements a second order central difference algorithm for structured grids and minimizes the PDE residual. The effectiveness of the designed network is evaluated by solving heat equations for various use cases. It is proved that with limited number of source terms to train the AE network, the framework can not only solve the given heat transfer problems with a single training process, but also make reasonable predictions for unseen cases (heat equations with new source terms) without retraining.