Deep neural networks for waves assisted by the Wiener–Hopf method

Abstract

<jats:p>In this work, the classical Wiener–Hopf method is incorporated into the emerging deep neural networks for the study of certain wave problems. The essential idea is to use the first-principle-based analytical method to efficiently produce a large volume of datasets that would supervise the learning of data-hungry deep neural networks, and to further explain the working mechanisms on underneath. To demonstrate such a combinational research strategy, a deep feed-forward network is first used to approximate the forward propagation model of a duct acoustic problem, which can find important aerospace applications in aeroengine noise tests. Next, a convolutional type U-net is developed to learn spatial derivatives in wave equations, which could help to promote computational paradigm in mathematical physics and engineering applications. A couple of extensions of the U-net architecture are proposed to further impose possible physical constraints. Finally, after giving the implementation details, the performance of the neural networks are studied by comparing with analytical solutions from the Wiener–Hopf method. Overall, the Wiener–Hopf method is used here from a totally new perspective and such a combinational research strategy shall represent the key achievement of this work.</jats:p>

Solving Partial Differential Equations Using Deep Learning and Physical Constraints

“…Nowadays, there has been a growing number of researchers using deep learning methods to study partial differential equations [24][25][26][27]. For example, Huang [28] combines deep neural networks and the Wiener-Hopf method to study some wave problems. This combinational research strategy has achieved excellent experimental results in solving two particular problems.…”

Reinvigorating the Wiener-Hopf technique in the pursuit of understanding processes and materials

“…The importance of the WH method in bridging and advancing seemingly unrelated fields is illustrated in its application to financially inspired Levi processes or discretely monitored path-dependent option pricing. Other promising directions include the development of hybrid methods, combining analytical modelling, artificial intelligence and experimental testing [ 7 ]. Potential benefits include the rapid production of high-fidelity ground truth data for the training of machine learning models and the development of forward models required by inverse testing methods applied in acoustic and ultrasound imaging.…”