Beyond stability constraints: A CNN surrogate solver for transient heat equations in heterogeneous and anisotropic domains

Transient heat-transfer simulations with varying material properties, source terms, domain sizes, and boundary conditions remain computationally demanding when solved repeatedly with conventional discretization methods. This study presents a CNN-based surrogate solver for one- and two-dimensional heat equations with variable initial conditions, Dirichlet and adiabatic boundaries, heterogeneous diffusivity fields, anisotropy, and time-dependent inputs. Training data are generated using an explicit finite difference scheme over broad ranges of conductivity, density, specific heat, heat generation, domain length, and time-step size, while boundary conditions are encoded directly in the input channels. The proposed architecture avoids downsampling to preserve local thermal information, lifts the physical inputs into a high-dimensional latent space, and processes them through feature-aggregated residual blocks with replicate padding, directional convolutions, and PReLU activations. An auto-regressive refinement strategy and a multi-step mean-absolute-error loss are employed to improve temporal consistency during recursive prediction. In one-dimensional benchmarks, the refined model achieves the best auto-regressive performance among the tested baselines, reaching a mean relative accuracy of 93.39% with a mean absolute error of 0.32 K, while maintaining competitive single-step accuracy. In two dimensions, the model attains 81.09% relative accuracy for next-step prediction and 80.88% for auto-regressive prediction, while successfully reproducing heterogeneous, homogeneous, and anisotropic temperature fields. The results further show that the proposed solver handles evolving boundary conditions and unseen time-step values with good robustness. Compared with finite difference simulations, the method offers more predictable runtime and pronounced speed advantages for fine-grid and high-dimensional thermal analyses. This makes it promising for real-time thermal monitoring and control.