Physics-informed Neural Networks

A Combination of Physics-informed Neural Networks with the Fixed-stress Splitting Iteration for Solving Biot’s Model

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KAUST

Abstract In this study, we aim to solve Biot’s consolidation models by employing physics-informed neural networks. Based on the fixed-stress splitting method, loss functions are designed for the displacement variable and the pressure variables separately. Such a strategy enables us to train two independent and smaller neural networks. The coupling of different variables is realized via the fixed-stress iterative algorithm. Numerical experiments are provided to show that the fixed-stress iterative algorithm can improve the performance of neural network approximations of the solution of Biot’s

Physics-informed Neural Networks for Learning the Homogenized Coefficients of Multiscale Elliptic Equations

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KAUST

Abstract Multiscale elliptic equations with scale separation are often approximated by the corresponding homogenized equations with slowly varying homogenized coefficients (the G-limit). The traditional homogenization techniques typically rely on the periodicity of the multiscale coefficients, thus finding the G-limits often requires sophisticated techniques in more general settings even when multiscale coefficient is known, if possible. Alternatively, we propose a simple approach to estimate the G-limits from (noisy-free or noisy) multiscale solution data, either from the existing forward