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We propose the differential equation based path integral (DEBPI) method tosimulate the real-time evolution of open quantum systems. In this method, asystem of partial differential equations is derived based on the continuationof a classical numerical method called iterative quasi-adiabatic propagatorpath integral (i-QuAPI). While the resulting system has infinite equations, weintroduce a reasonable closure to obtain a series of finite systems. Newnumerical schemes can be derived by discretizing these differential equations.It is numerically verified that in certain cases, by selecting appropriatesystems and applying suitable numerical schemes, the memory cost required inthe i-QuAPI method can be significantly reduced.

Differential Equation Based Path Integral for System-Bath Dynamics