Stable heat jet approach for temperature control of Fermi-Pasta-Ulam beta chain

In this paper,we propose a stable heat jet approach for accurate temperature control of the nonlinear Fermi-Pasta-Ulam beta(FPU-β)chain.First,we design a stable nonlinear boundary condition,with co-efficients determined by a machine learning technique.Its stability can be proved rigorously.Based on this stable boundary condition,we derive a two-way boundary condition complying with phonon heat source,and construct stable heat jet approach.Numerical tests illustrate the stability of the boundary condition and the effectiveness in eliminating boundary reflections.Furthermore,we extend the bound-ary condition formulation with more atoms,and train the coefficients to eliminate extreme short waves by machine learning technique.Under this extended boundary condition,the heat jet approach is effec-tive for high temperature,and may be adopted for multiscale computation of atomic motion at finite temperature.