Neural network representation of the probability density function of diffusion processes.

Physics-informed neural networks are developed to characterize the state of dynamical systems in a random environment. The neural network approximates the probability density function (pdf) or the characteristic function (chf) of the state of these systems, which satisfy the Fokker-Planck equation or an integro-differential equation under Gaussian and/or Poisson white noises. We examine analytically and numerically the advantages and disadvantages of solving each type of differential equation to characterize the state. It is also demonstrated how prior information of the dynamical system can be exploited to design and simplify the neural network architecture. Numerical examples show that (1) the neural network solution can approximate the target solution even for partial integro-differential equations and a system of partial differential equations describing the time evolution of the pdf/chf, (2) solving either the Fokker-Planck equation or the chf differential equation using neural networks yields similar pdfs of the state, and (3) the solution to these differential equations can be used to study the behavior of the state for different types of random forcings.

Stochastic reduced order models for inverse problems under uncertainty.

This work presents a novel methodology for solving inverse problems under uncertainty using stochastic reduced order models (SROMs). Given statistical information about an observed state variable in a system, unknown parameters are estimated probabilistically through the solution of a model-constrained, stochastic optimization problem. The point of departure and crux of the proposed framework is the representation of a random quantity using a SROM - a low dimensional, discrete approximation to a continuous random element that permits e cient and non-intrusive stochastic computations. Characterizing the uncertainties with SROMs transforms the stochastic optimization problem into a deterministic one. The non-intrusive nature of SROMs facilitates e cient gradient computations for random vector unknowns and relies entirely on calls to existing deterministic solvers. Furthermore, the method is naturally extended to handle multiple sources of uncertainty in cases where state variable data, system parameters, and boundary conditions are all considered random. The new and widely-applicable SROM framework is formulated for a general stochastic optimization problem in terms of an abstract objective function and constraining model. For demonstration purposes, however, we study its performance in the specific case of inverse identification of random material parameters in elastodynamics. We demonstrate the ability to efficiently recover random shear moduli given material displacement statistics as input data. We also show that the approach remains effective for the case where the loading in the problem is random as well.