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