In this paper, the artificial neural networks (ANN) based deep learning (DL) techniques were developed to solve the neutron diffusion problems for the continuous neutron flux distribution without domain discretization in advance. Due to its mesh-free property, the DL solution can easily be extended to complicated geometries. Two specific realizations of DL methods with different boundary treatments are developed and compared for accuracy and efficiency, including the boundary independent method (BIM) and boundary dependent method (BDM). The performance comparison on analytic benchmark indicates BDM being the preferred DL method. Novel constructions of trial function are proposed to generalize the application of BDM. For a more in-depth understanding of the BDM on diffusion problems, the influence of important hyper-parameters is further investigated. Numerical results indicate that the accuracy of BDM can reach hundreds of times higher than that of BIM on diffusion problems. This work can provide a new perspective for applying the DL method to nuclear reactor calculations.

The inverse heat conduction problem (IHCP), which recovers unknown heat-source parameters from sparsely placed temperature measurements, underpins process monitoring, welding, and thermographic non-destructive testing. Two difficulties stand in the way of its real-time solving: the problem is fundamentally ill-posed, so a single point estimate cannot be trusted, and every new observation traditionally triggers an expensive re-solution of the governing PDE.To address these difficulties, this work introduces a physics-informed conditional variational autoencoder (PI-CVAE) that embeds a POD-Galerkin weak-form physics residual as a differentiable bottleneck in a reduced-order latent space, mapping sparse sensor observations directly to the full Bayesian posterior. To counter the ill-posedness, the model's probabilistic output provides, for each observation, not a single point estimate but the full posterior spread for uncertainty quantification; to remove the per-query PDE burden, the inversion is fully amortised across the training set, requiring no re-training at deployment and inverting any new observation in approximately 4 ms by a single forward pass---three to four orders of magnitude faster than MCMC or per-sample PINN optimisation while remaining close to MCMC accuracy. Validated across five progressively challenging benchmarks, the framework demonstrates that amortised inference approaches MCMC-level fidelity, transforming Bayesian IHCP from a computationally expensive per-query routine into a real-time, uncertainty-aware diagnostic tool.