Li–Yau inequalities for general non-local diffusion equations via reduction to the heat kernel

We establish a reduction principle to derive Li–Yau inequalities for non-local diffusion problems in a very general framework, which covers both the discrete and continuous setting. Our approach is not based on curvature-dimension inequalities but on heat kernel representations of the solutions and consists in reducing the problem to the heat kernel. As an important application we solve a long-standing open problem by obtaining a Li–Yau inequality for positive solutions u to the fractional (in space) heat equation of the form $$(-\Delta )^{\beta /2}(\log u)\le C/t$$ ( - Δ ) β / 2 ( log u ) ≤ C / t , where $$\beta \in (0,2)$$ β ∈ ( 0 , 2 ) . We also show that this Li–Yau inequality allows to derive a Harnack inequality. We further illustrate our general result with an example in the discrete setting by proving a sharp Li–Yau inequality for diffusion on a complete graph.

Neural Network Based Deep Learning Method for Multi-Dimensional Neutron Diffusion Problems with Novel Treatment to Boundary

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.