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Physics-informed neural networks are developed to characterize the state ofdynamical systems in a random environment. The neural network approximates theprobability density function (pdf) or the characteristic function (chf) of thestate of these systems which satisfy the Fokker-Planck equation or anintegro-differential equation under Gaussian and/or Poisson white noises. Weexamine analytically and numerically the advantages and disadvantages ofsolving each type of differential equation to characterize the state. It isalso demonstrated how prior information of the dynamical system can beexploited to design and simplify the neural network architecture. Numericalexamples show that: 1) the neural network solution can approximate the targetsolution even for partial integro-differential equations and system of PDEsdescribing the time evolution of the pdf/chf, 2) solving either theFokker-Planck equation or the chf differential equation using neural networksyields similar pdfs of the state, and 3) the solution to these differentialequations can be used to study the behavior of the state for different types ofrandom forcings.

Neural network representation of the probability density function of diffusion processes