Learning scattering waves via coupling physics-informed neural networks and their convergence analysis

Abstract

Time harmonic acoustic wave scattering problems modeled by the unbounded Helmholtz equations are widely used in medical and military fields. A fast and flexible solver for the Helmholtz equation is expected in engineering applications. Fortunately, data-driven neural network algorithms show extraordinary promise in scientific computing and artificial intelligence. We solve the Helmholtz equation by decomposing the equation into a coupling equation system that explicitly includes the real and imaginary parts of the solution. The real and imaginary parts are approximated by the different neural networks, respectively. The unbounded domain is reduced to a problem on a bounded domain via the Dirichlet-to-Neumann operator. Thus, a coupling physics-informed neural network (cPINN) is constructed. On the other hand, in exploring the convergence analysis for PINNs, we obtain the convergence results for cPINNs through the universal approximation and Sobolev space theory instead of using the neural tangent kernel theory. We prove that the cPINNs converge to the solution of the Helmholtz equation in H01 space as the number of neurons approaches infinity. Finally, we propose some numerical results.