Source-linked AI summary
Physics-informed Convolutional Neural Networks for Temperature Field Prediction of Heat Source Layout without Labeled Data
Xiaoyu Zhao, Zhiqiang Gong, Yunyang Zhang, Wen Yao, Xiaoqian Chen
TL;DR
Repeated thermal PDE simulations make heat-source layout analysis costly, while labeled simulation data are expensive to prepare. The paper develops a physics-informed CNN trained with finite-difference physics and boundary constraints, and reports predictions comparable to numerical and data-driven methods.
Problem
Repeated PDE solves and costly simulation-data preparation limit deep surrogate modeling for thermal analysis and heat-source layout optimization.
Method
A CNN learns the mapping from heat-source layout to steady temperature field using a heat-conduction finite-difference loss, hard Dirichlet and Neumann boundary constraints, and task-specific architecture and hard-example mining.
Results
The experiments demonstrate predictions comparable to numerical methods and data-driven deep learning models.
Takeaways & Limitations
Physics-informed CNNs can serve as thermal simulation surrogates trained without labeled simulation data for this HSL-TFP setting.
Takeaways & Limitations
The paper notes that data-driven methods can be more effective in some applications and suggests combining physics knowledge with labeled data for verification and learning.
Abstract
from arXiv · showhide
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.
1 Introduction
The paper targets costly repeated PDE solves in thermal layout optimization with a physics-informed CNN surrogate trained without labeled simulation data. It combines physics-based loss, boundary constraints, CNN design, and pixel-level hard-example mining, achieving predictions comparable to numerical and data-driven methods.
- Motivation: Repeated PDE solves make thermal layout evaluation expensive and time-consuming within optimization loops.The motivation is thermal management of electronic systems, where heat-source layout affects temperature fields.
- Approach: The proposed surrogate learns the mapping from parametric heat-source layouts to PDE solution fields, rather than solving only one fixed PDE instance.CNNs are selected for parameter sharing and local connections in large-scale, high-dimensional problems.
- Approach: A heat-conduction-equation loss discretized by finite differences enables training without labeled simulation data.The loss incorporates the governing physics directly into network optimization.
- Approach: Hard constraints are imposed on Dirichlet and Neumann boundaries to support stable and fast network convergence.Boundary treatment is presented as a central component because the solution depends sensitively on boundary conditions.
- Contributions: A UNet-like architecture and pixel-level online hard-example mining address task-specific network design and uneven optimization difficulty across the domain.The mining method adaptively emphasizes positions that are harder to optimize.
- Results: Experiments report predictions comparable to both numerical methods and data-driven methods.The paper also studies the performance impact of different network components and training methods.
2 Temperature Field Prediction of Heat Source Layout (HSL-TFP)
The HSL-TFP problem predicts the steady temperature field in a rectangular conducting plate from the layout and intensity of heat sources. The field satisfies Poisson’s equation with an isothermal Dirichlet boundary and adiabatic Neumann boundaries, motivating a neural surrogate for repeated solver calls.
- Problem setup: HSL-TFP predicts the steady temperature distribution in a rectangular conducting plate containing several electronic-component heat sources.Heat-source layout affects the temperature field throughout the domain.
- Governing conditions: The steady temperature field satisfies Poisson’s equation with an isothermal bottom boundary and adiabatic remaining boundaries.The bottom boundary has temperature T0, while the other boundaries exchange no heat.
- Problem setup: The heat-source layout is represented by an intensity distribution function that specifies each source’s intensity and distribution area.The function φ(x, y) correlates the source representation with the resulting temperature field.
- Motivation: Traditional FDM, FEM, and FVM solvers can solve the problem but may be inefficient when repeatedly called for temperature predictions.The paper therefore investigates replacing repeated numerical solves with a neural network surrogate.
3 Physics-informed Convolutional Neural Networks for HSL-TFP
The method uses a CNN to learn heat-source-layout-to-temperature operators from discretized PDEs, training without labels through finite-difference physics constraints and hard boundary enforcement.
- 3.1 Physics-informed loss function: The physics-informed loss uses finite differences to enforce the heat-conduction equation, with a convolution kernel computing the required second differences.Equal mesh steps allow the finite-difference equation to be implemented efficiently through convolution operations.
- 3.1 Physics-informed loss function: The loss formulation improves training efficiency by introducing an intermediate T′ that avoids back-propagating its computation graph into neighboring temperature points.This formulation is derived from the finite-difference equation and is consistent with Jacobi iteration, guiding predictions toward successive iterative solutions.
- 3.2 Hard-contrainted boundary condition: Hard boundary constraints directly set Dirichlet boundary values and pad Neumann boundaries with corresponding inner-domain nodes before loss computation.The procedure applies boundary padding, computes T′, and excludes Dirichlet-boundary values from the loss.
- 3.3 CNN architecture and framework for HSL-TFP: A modified UNet learns the mapping from discretized heat-source intensity φ(x, y) to temperature field T as an image-to-image regression problem.The discretized input and solution are treated as images, enabling CNN-based operator learning for an entire family of heat-conduction equations.
- 3.3 CNN architecture and framework for HSL-TFP: The CNN uses gridded inputs and outputs because flattening high-dimensional fields would create scalability problems for fully connected networks.The architecture is based on UNet, whose encoder-decoder structure is suited to image-style data.
4 Experiment
PI-UNet predicts steady-state temperature fields without labeled data and is evaluated against finite-difference computations across two heat-source-layout cases. Ablations examine supervised-data requirements, loss design, padding, batch size, and network components.
- 4.2 Prediction Performance: MAE against FDM is below 0.03K in both cases, while Max-AE remains below 0.1K, indicating accurate and uniform predictions.The reported train–test metric gap is small, and prediction examples are close to FDM computations.
- 4.3 Comparisons with supervised learning: Physics-informed learning achieves precision close to supervised learning with 4000 labeled samples, with only a slight overall gap.Supervised accuracy improves with more than 4000 labels, but the improvement is not significant; physics-informed learning is substantially weaker on Max-AE.
- 4.4 Ablation Studies: P-OHEM outperforms MSE, whereas L1 cannot optimize properly under the physics-informed loss.P-OHEM assigns larger weights to positions with larger errors, especially near the isothermal hole where temperature changes sharply.
- 4.4 Ablation Studies: Reflect padding performs better than zero padding because boundary errors from finite differences diffuse into the whole domain.Zero padding produces relatively larger boundary loss values and reduces prediction precision across the domain.
- 4.4 Ablation Studies: GN, bilinear upsampling, and GELU form the most effective component combination, while smaller batch sizes improve convergence under fixed epochs.Bilinear upsampling outperforms transpose convolution, and GN is slightly better than IN.
5 Conclusions
The proposed physics-informed CNN predicts heat-source-layout temperature fields without simulation data and can serve rapid, high-precision layout optimization. The authors identify broader physics problems and irregular domains or large meshes as future directions, while noting that data-driven methods may be more effective in some applications.
- The proposed network produces high-precision HSL-TFP predictions comparable to numerical and data-driven models without solving the PDE at inference time.It uses a physics-informed loss and can directly output the temperature-field solution.
- The trained model can act as a surrogate for rapid temperature prediction within heat-source-layout optimization.This supports layout optimization that requires repeated temperature-field evaluations.
- The approach may extend to parameterized PDEs in elasticity, fluid mechanics, and electromagnetism.
- Irregular computation domains and large-scale meshes remain proposed directions for further investigation.
- The authors note that data-driven methods can be more effective in some applications and suggest combining physics knowledge with labeled data for verification and learning.