Recent years have witnessed a growth in mathematics for deep learning—which seeks a deeper understanding of the concepts of deep learning with mathematics, and explores how to make it more robust—and deep learning for mathematics, where deep learning algorithms are used to solve problems in mathematics. The latter has popularised the field of scientific machine learning where deep learning is applied to problems in scientific computing. Specifically, more and more neural network architectures have been developed to solve specific classes of partial differential equations (PDEs). Such methods exploit properties that are inherent to PDEs and thus solve the PDEs better than classical feed-forward neural networks, recurrent neural networks, and convolutional neural networks. This has had a great impact in the area of mathematical modeling where parametric PDEs are widely used to model most natural and physical processes arising in science and engineering, In this work, we review such methods and extend them for parametric studies as well as for solving the related inverse problems. We equally proceed to show their relevance in some industrial applications

We consider the inverse source problem for a Cauchy wave equation with passive cross-correlation data. We propose to consider the cross-correlation as a wave equation itself and reconstruct the cross-correlation in the support of the source for the original Cauchy wave equation. Having access to the cross-correlation in the support of the source, we use a Fourier method to reconstruct the source of the original Cauchy wave equation. We show the inverse source problem is ill-posed and suffer from non-uniqueness when the mean of the source is zero, and provide a uniqueness result and stability estimate in case of non-zero mean.

We analyse an inverse problem for water waves proposed by Richard Feynman in the BBC documentary "Fun to imagine". We show how the presence of both gravity and capillary waves makes water an excellent medium for the propagation of information.

We consider the problem of estimation of solutions to systems of coupled stochastic differential equations, as well as underlying system parameters, based on discrete-time measurements. We concentrate on the case where transition densities are not available in closed form, and focus on the technique of the Laplace approximation for integrating out unobserved state variables in a Bayesian setting. We demonstrate that the direct approach of inserting sufficiently many time points with unobserved states performs well, when the noise is additive. A pitfall arises when the noise intensity in the state equation depends on state variables, i.e., when the noise is not additive: In this case, maximizing the posterior density over unobserved states does not lead to useful state estimates (i.e., they are not consistent in the fine-time limit). This problem can be overcome by focusing in stead on the minimum-energy realization of the noise process which is consistent with data, which provides a connection to calculus of variations. We demonstrate the theory with numerical examples.