Neural network augmented inverse problems for PDEs
This addresses inverse problems in PDEs for computational science and engineering, but it is incremental as it builds on existing methods with neural network augmentation.
The paper tackles the problem of estimating coefficients in inverse partial differential equations (PDEs) from noisy data by augmenting classical methods with neural networks as a prior, showing robustness across dimensions and noisy conditions.
In this paper we show how to augment classical methods for inverse problems with artificial neural networks. The neural network acts as a prior for the coefficient to be estimated from noisy data. Neural networks are global, smooth function approximators and as such they do not require explicit regularization of the error functional to recover smooth solutions and coefficients. We give detailed examples using the Poisson equation in 1, 2, and 3 space dimensions and show that the neural network augmentation is robust with respect to noisy and incomplete data, mesh, and geometry.