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Reliable methods for obtaining time-dependent solutions of Langevin equationsare in high demand in the field of non-equilibrium theory. In this paper, wepresent a new method based on variational superposed Gaussian approximation(VSGA) and Pad\'e approximant. The VSGA obtains time-dependent probabilitydensity functions as a superposition of multiple Gaussian distributions.However, a limitation of the VSGA is that the expectation of the drift termwith respect to the Gaussian distribution should be calculated analytically,which is typically satisfied when the drift term is a polynomial function. Whenthis condition is not met, the VSGA must rely on the numerical integration ofthe expectation at each step, resulting in huge computational cost. We proposean augmented VSGA (A-VSGA) method that effectively overcomes the limitation ofthe VSGA by approximating non-linear functions with the Pad\'e approximant. Weapply the A-VSGA to two systems driven by chaotic input signals, a stochasticgenetic regulatory system and a soft bistable system, whose drift terms are arational polynomial function and a hyperbolic tangent function, respectively.The numerical calculations show that the proposed method can provide accurateresults with less temporal cost than that required for Monte Carlo simulation.

Augmented Variational Superposed Gaussian Approximation for Langevin Equations with Rational Polynomial Functions