Physics-Driven Thermal Conduction Module

• PDTM is a computational framework that integrates heat conduction physics, using PDE-based methods and deep learning surrogates to model temperature fields.
• It enables rapid design optimization and high-throughput materials screening while ensuring physical consistency through embedded boundary and loss mechanisms.
• Validated against high-fidelity methods like FEM, PDTMs incorporate uncertainty quantification to achieve robust, scalable, and real-time thermal analyses.

A Physics-Driven Thermal Conduction Module (PDTM) is a computational framework for inferring, predicting, or simulating temperature fields and heat transfer in physical systems by embedding the governing physics—typically the heat equation or its variants—directly into the algorithmic structure. PDTMs span a continuum from traditional numerical PDE solvers to deep learning surrogates and hybrid physics–data-driven networks, unified by the explicit incorporation of physical laws into their formulation. Recent research has established PDTMs as powerful surrogates for design optimization, high-throughput materials screening, inverse problem learning, and physics-constrained computer vision.

1. Theoretical Foundations and Governing Equations

At the core of every PDTM lies a formulation of the heat conduction partial differential equation (PDE), which serves as both the target of inference and the constraint mechanism. For isotropic, steady-state conduction in a domain $\Omega\subset\mathbb R^d$, the canonical form is the Laplace or Poisson equation,

$\nabla\cdot k\nabla T = Q$

where $T$ is temperature, $k$ is (possibly spatially varying or anisotropic) thermal conductivity, and $Q$ is a volumetric source. For time-dependent (transient) settings, the equation generalizes to

$$\rho C \frac{\partial T}{\partial t} = \nabla\cdot(k\nabla T) + Q$$

with $\rho$ the density and $C$ the specific heat capacity.

Advanced modules may introduce

• Anisotropic conduction: $k$ is replaced by a tensor $\mathbf{K}$ with principal axes aligned to microstructure or magnetic field lines (Kannan et al., 2015, Pellissier et al., 2023).
• Phase transitions and moving boundary problems (Stefan-type): energy conservation across interfaces with latent heat and interface tracking (Hassanzadeh et al., 30 Nov 2025).
• Nonlocal or flux-limited models for kinetic regimes: flux-capping, mean free path-based suppression, and quasi-nonlocal closure (Chapman et al., 2021, Romano, 2021).
• Boundary conditions: Dirichlet, Neumann, Robin, or combinations; also, domain-specific treatments for interfaces or convective surfaces.

2. Algorithmic Architectures and Physics-Driven Learning

PDTMs are distinguished by architecture and the locus of physical law enforcement.

Classical Solvers

Traditional approaches discretize the governing PDE via finite difference, finite volume, or finite element schemes, optionally with acceleration from mathematical transforms (e.g., fast cosine or Fourier methods for homogeneous or separable domains) (Ye et al., 2024). For anisotropic or topology-varying conditions, geometric flexibility is achieved via (un)structured mesh frameworks.

Physics-Driven Neural Surrogates

Recent research has popularized the use of deep neural networks as PDE surrogates. Representative designs include:

• U-Net CNNs, where the input consists of geometry masks and prescribed boundary conditions encoded as image channels. The output is a field approximation $T(x,y)$, trained by minimizing a PDE residual loss alongside optional data-driven terms (Ma et al., 2022, Ma et al., 2020).
• Physics-Driven Deep Learning frameworks for moving boundary (Stefan) problems, employing multi-network structures (e.g., parallel DNNs for distinct regions/phases) with domain-specific constraints and interface losses (Hassanzadeh et al., 30 Nov 2025).
• Bayesian PINNs and uncertainty-quantified BNNs, which fulfill physics constraints (via residual loss terms) and model data noise/inverse inference natively (Jiang et al., 2021).

All architectures share the key principle of enforcing physics explicitly: the network loss function includes $\mathcal{L}_{\text{phys}}$, a term penalizing PDE violation, with gradients computed via automatic differentiation or convolutional stencils (Ma et al., 2022, Ma et al., 2020).

3. Physics-Driven Loss Mechanisms and Boundary Enforcement

The defining feature of a PDTM is the integration of the physics loss, typically via finite-difference or convolutional approximations of PDE operators:

$$\mathcal{L}_{\text{phys}} = \| \nabla^2 \hat{T} \|^2_2$$

for steady-state Laplace settings, and more generally, variational terms reflecting the full transient or nonlinear operator (Ma et al., 2022, Hassanzadeh et al., 30 Nov 2025).

Boundary conditions are encoded via input channels, direct clamping, or penalty terms $L_{\mathrm{BC}}$, ensuring satisfaction on domain boundaries and geometric interfaces.

Combined Data–Physics Loss: In hybrid modules, a data loss $\mathcal{L}_{\text{data}}$ quantifies the fit to reference solutions, while a physics loss $\mathcal{L}_{\mathrm{PDE}}$ enforces consistency with the underlying equations. A weighted sum (or phase-wise switching) enables rapid convergence and physical plausibility with reduced data (Ma et al., 2020).

Uncertainty Quantification: Bayesian variants propagate epistemic and aleatoric uncertainties, crucial for inverse problems or when data quality varies (Jiang et al., 2021).

4. Integration with Optimization, Inverse Design, and Applications

PDTMs serve as computational engines for high-throughput and inverse design by replacing expensive forward solvers in optimization loops:

• Layout and Topology Optimization: With U-Net-based PDTMs as fast surrogates, global solution spaces (e.g., hole arrangement in a conduction plate) are explored via metaheuristics (e.g., particle swarm optimization), enabling solution of design problems entirely without explicit FEM evaluations (Ma et al., 2022).
• Multiphysics Coupling: The core architecture is extensible to multi-region, multi-material, or coupled processes (e.g., solidification fronts with geometric features such as fins in TES) (Hassanzadeh et al., 30 Nov 2025).
• Nanostructured Materials and Multiscale Modeling: Modules such as the anisotropic MFP-BTE (aMFP-BTE) encode the Boltzmann Transport Equation for phononic transport, employing coarse graining in MFP space for tractable, mode-resolved conductivity predictions (Romano, 2021).
• Computer Vision and Imaging: In thermal image super-resolution, PDTM modules are integrated as inductive biases in neural architectures (e.g., PCNet) to enforce proper diffusion and prevent artefacts arising from naively transferring high-frequency priors from other modalities (Zhao et al., 7 Jan 2026, Chen et al., 2024).

5. Numerical Implementation and Computational Performance

Numerical realization varies by application and scale:

• Convolutional and U-Net Surrogates: Rapid inference (<<0.1 s per forward pass on GPU) enables real-time evaluation in optimization. Training converges on the order of $10^5$ iterations, with physics-only surrogates achieving mean squared errors $\sim10^{-3}$ relative to full FEM solvers (Ma et al., 2022, Ma et al., 2020).
• Classical Solvers with Transform Acceleration: Fast cosine transform (FCT) and batched tridiagonal solvers provide orders-of-magnitude speedup for high-resolution RVEs, exploiting GPU parallelism (Ye et al., 2024).
• Model Order Reduction: Proper orthogonal decomposition with Galerkin projection (PODTherm-GP) achieves 5–6 orders of magnitude DoF reduction and $10^3$–$10^4\times$ speedup over standard FEM, maintaining $\lesssim 2\%$ error in full-chip thermal simulations (Jiang et al., 2023).
• Astrophysical MHD and ICF Codes: Explicit and implicit FV modules for anisotropic conduction on moving, unstructured meshes (e.g., AREPO, RAMSES) scale efficiently to $10^{13}$ zones. Treatment of electron/ion coupling, saturation limiters, and flux limiting are critical for accurate reproduction of high-energy-density physics and galaxy cluster thermodynamics (Kannan et al., 2015, Pellissier et al., 2023, Chapman et al., 2021).

Module Type	Key Physics Mechanism	Computational Gains
Physics-driven U-Net CNN	Laplacian PDE loss	$10^2$–$10^3\times$ faster
FCT-accelerated FV solver	TPFA + DCT/FFT preconditioning	$5\times$ (GPU/CPU), scalable to $512^3$ DoF
aMFP-BTE deterministic BTE	Vectorial MFP interpolation	$50\times$ speedup, multiscale capability
POD–Galerkin ROM	Optimal modal projection	$10^3$–$10^4\times$ faster, $\sim$2% error
PINN/BNN with physics loss	PDE residuals + data fit	Reliable with low data, UQ

6. Validation, Limitations, and Best Practices

Validation Approaches:

• Direct comparison to high-fidelity FEM, finite-volume, or BTE solutions (Ma et al., 2022, Romano, 2021).
• Analytical or benchmark problems (1D/2D test cases, Stefan solutions, step/ring/Marshak problems) (Hassanzadeh et al., 30 Nov 2025, Kannan et al., 2015, Chapman et al., 2021).
• Integrated application metrics: design objective convergence, super-resolution performance (PSNR, SSIM), material property prediction, or scientific observables (e.g., ICM entropy, fusion yield).

Best Practices:

• Embed accurate stencils or PDE residual operators as non-trainable layers for stable learning.
• For data–physics hybrids, begin with coarse reference solutions and transfer to pure physics-driven loss as solutions converge (Ma et al., 2020).
• In multiphysics or high-gradient regimes, deploy physically motivated limiters, harmonic averaging for positive definiteness, and monitor for nonlocal/kinetic effects (Kannan et al., 2015, Chapman et al., 2021, Pellissier et al., 2023).

Limitations:

• Surrogate accuracy depends on network capacity and the expressiveness of input channels, as well as on the grid resolution and representativeness of the geometric encoding.
• For complex phase-change or coupled systems, multiple DNN submodules and carefully balanced loss weights are essential (Hassanzadeh et al., 30 Nov 2025).
• Material or geometry changes outside the training manifold require retraining for projection or reduced-order approaches (Jiang et al., 2023).
• The correct treatment of nonlocality and boundary fluxes is vital in ICF, nanostructured, or astrophysical applications (Romano, 2021, Chapman et al., 2021).

7. Impact and Application Domains

PDTMs are impactful in several emerging and mature disciplines:

• Thermal design and topological optimization: Enabling real-time inverse design in electronics cooling, energy systems, and structural engineering (Ma et al., 2022).
• Thermal management for electronic circuits: Rapid and accurate chip-level simulations for power-aware scheduling and DVFS systems (Jiang et al., 2023).
• Nanophononics and thermoelectrics: Rapid multiscale modeling for nanostructure screening and material discovery (Romano, 2021).
• Astrophysical and HEDP simulations: High-fidelity, robust modules for capturing anisotropic, flux-limited, and saturated conduction in complex, multi-material, moving domains (Kannan et al., 2015, Pellissier et al., 2023, Chapman et al., 2021).
• Thermal image processing and super-resolution: Physics-constrained neural modules that strictly enforce realistic temperature field behavior in vision, detection, and surveillance systems (Zhao et al., 7 Jan 2026, Chen et al., 2024).

In summary, the Physics-Driven Thermal Conduction Module constitutes a versatile family of algorithmic tools that unify physics, computational efficiency, and modern machine learning techniques for a broad spectrum of scientific and engineering thermal transport problems. The salient features are the direct embedding of the PDE into loss functions and algorithmic workflows, strict enforcement of boundary and interface conditions, and the capability to generalize beyond training or reference data, enabling accuracy and speed previously unattainable with purely numerical or data-driven approaches. (Ma et al., 2022, Ma et al., 2020, Hassanzadeh et al., 30 Nov 2025, Ye et al., 2024, Romano, 2021, Zhao et al., 7 Jan 2026, Kannan et al., 2015, Pellissier et al., 2023, Jiang et al., 2023, Chapman et al., 2021, Jiang et al., 2021)