A novel distributed-order Cattaneo–Christov model is proposed to effectively characterize non-classical heat conduction processes with memory effect and time–space relaxation behaviors originating from distributed-order fractional derivatives. A fractional physics-informed neural networks (fPINN) algorithm is employed to address both the forward and inverse problems of the distributed-order heat conduction model. For the forward problem, we propose an SfPINN algorithm that incorporates a squared loss term and employs an adaptive updating strategy for the loss-term weights. First, the boundary conditions are embedded into the network output such that they are automatically satisfied. In addition, we design a two-stage training strategy to enhance computational efficiency: in the first stage, the squared loss term associated with the initial condition is incorporated into the loss function; in the second stage, the squared residual term of the governing equation is introduced into the loss function. Numerical results show that the proposed algorithm outperforms the standard fPINN method in both solution accuracy and training iteration speed. For the inverse problem, the numerical results demonstrate that as the iteration number increases, the estimated parameter values progressively converge to their true values and finally stabilize.
Chen et al. (Mon,) studied this question.