Convergence criteria for iterative formats in high-dimensional optimisation problems discretised from mathematical partial differential equation

High-dimensional optimisation of partial differential equations often involves large-scale variables and complex constraints, leading to convergence issues during iterative solution processes.To address this, this paper first discretises the original equation using the finite element method to construct an approximate optimisation model, then proposes an improved alternating direction method of multipliers for efficient solution.To further enhance accuracy, second-order schemes and compact difference schemes are employed for temporal and spatial discretisation, respectively, with rigorous convergence proofs established for the discretisation formats.Addressing diffusion-wave phenomena in the problem, a fractional-order physical information neural network is applied to investigate its direct and inverse problems, thereby providing high-quality predicted solutions for the high-dimensional optimisation.Experimental results indicate that the suggested approach converges in just 5.37 seconds, significantly outperforming comparison methods and validating its superior convergence performance for high-dimensional optimisation problems.