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  1. Modeling fluid flow and transport in heterogeneous systems is often challenged by unknown parameters that vary in space. In inverse modeling, measurement data are used to estimate these parameters. Due to the spatial variability of these unknown parameters in heterogeneous systems (e.g., permeability or diffusivity), the inverse problem is ill-posed and infinite solutions are possible. Physics-informed neural networks (PINN) have become a popular approach for solving inverse problems. However, in inverse problems in heterogeneous systems, PINN can be sensitive to hyperparameters and can produce unrealistic patterns. Motivated by the concept of ensemble learning and variance reduction in machine learning, we propose an ensemble PINN (ePINN) approach where an ensemble of parallel neural networks is used and each sub-network is initialized with a meaningful pattern of the unknown parameter. Subsequently, these parallel networks provide a basis that is fed into a main neural network that is trained using PINN. It is shown that an appropriately selected set of patterns can guide PINN in producing more realistic results that are relevant to the problem of interest. To assess the accuracy of this approach, inverse transport problems involving unknown heat conductivity, porous media permeability, and velocity vector fields were studied. The proposed ePINN approach was shown to increase the accuracy in inverse problems and mitigate the challenges associated with non-uniqueness. 
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    Free, publicly-accessible full text available May 1, 2024
  2. The problem of nonlinear natural convection in a fluid saturated porous layer heated from below is reviewed focusing on the specific result of a collapse of the wave function. When the conditions for the onset of convection are met, a wave function is obtained as the solution of the linearized equations expressed in terms of a Fourier expansion. Only one mode of this expansion survives at the onset of convection, a result that can be seen as the “collapse of the wave function” in a very similar fashion as in quantum mechanics, although the explanations of the latter are very distinct from the ones in quantum mechanics. The reasons behind the “collapse of the wave function” result in natural convection are discussed and the analysis is extended into the nonlinear domain of convection, by using a weak nonlinear analysis. 
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