We present a solver for the 2D high-frequency Helmholtz equation in heterogeneous acoustic media, with online parallel complexity that scales optimally as O( N L ), where N is the number of volume unknowns, and L is the number of processors, as long as L grows at most like a small fractional power of N . The solver decomposes the domain into layers, and uses transmission conditions in boundary integral form to explicitly define "polarized traces", i.e., up-and down-going waves sampled at interfaces. Local direct solvers are used in each layer to precompute traces of local Green's functions in an embarrassingly parallel way (the offline part), and incomplete Green's formulas are used to propagate interface data in a sweeping fashion, as a preconditioner inside a GMRES loop (the online part). Adaptive low-rank partitioning of the integral kernels is used to speed up their application to interface data. The method uses secondorder finite differences. The complexity scalings are empirical but motivated by an analysis of ranks of off-diagonal blocks of oscillatory integrals. They continue to hold in the context of standard geophysical community models such as BP and Marmousi 2, where convergence occurs in 5 to 10 GMRES iterations. While the parallelism in this paper stems from decomposing the domain, we do not explore the alternative of parallelizing the systems solves with distributed linear algebra routines.

In this work we introduce a new multiscale artificial neural network based on the structure of H-matrices. This network generalizes the latter to the nonlinear case by introducing a local deep neural network at each spatial scale. Numerical results indicate that the network is able to efficiently approximate discrete nonlinear maps obtained from discretized nonlinear partial differential equations, such as those arising from nonlinear Schrödinger equations and the Kohn-Sham density functional theory.

We present a ray-based finite element method for the high-frequency Helmholtz equation in smooth media, whose basis is learned adaptively from the medium and source. The method requires a fixed number of grid points per wavelength to represent the wave field; moreover, it achieves an asymptotic convergence rate of O(ωwhere ω is the frequency parameter in the Helmholtz equation. The local basis is motivated by the geometric optics ansatz and is composed of polynomials modulated by plane waves propagating in a few dominant ray directions. The ray directions are learned by processing a low-frequency wave field that probes the medium with the same source. Once the local ray directions are extracted, they are incorporated into the local basis to solve the high-frequency Helmholtz equation. This process can be continued to further improve the approximations for both local ray directions and high-frequency wave fields iteratively. Finally, a fast solver is developed for solving the resulting linear system with an empirical complexity O(ω d ) up to a poly-logarithmic factor. Numerical examples in 2D are presented to corroborate the claims.

In recent years, deep learning has led to impressive results in many fields. In this paper, we introduce a multiscale artificial neural network for high-dimensional nonlinear maps based on the idea of hierarchical nested bases in the fast multipole method and the H 2 -matrices. This approach allows us to efficiently approximate discretized nonlinear maps arising from partial differential equations or integral equations. It also naturally extends our recent work based on the generalization of hierarchical matrices [Fan et al. arXiv:1807.01883] but with a reduced number of parameters. In particular, the number of parameters of the neural network grows linearly with the dimension of the parameter space of the discretized PDE. We demonstrate the properties of the architecture by approximating the solution maps of nonlinear Schrödinger equation, the radiative transfer equation, and the Kohn-Sham map.which can be viewed as a nonlinear generalization of pseudo-differential operators. This type of maps may arise from parameterized and discretized partial differential equations (PDE) or integral equations (IE), with u being the quantity of interest and v the parameter that serves to identify a particular configuration of the system.We propose a neural network architecture based on the idea of hierarchical nested bases used in the fast multipole method (FMM) [19] and the H 2 -matrix [22] to represent nonlinear maps arising in computational physics, motivated by the favorable complexity of the FMM / H 2 -matrices in the linear setting. The proposed neural network, which we call MNN-H 2 , is able to efficiently represent the nonlinear maps benchmarked in the sequel, in such cases the number of parameters required to approximate the maps can grow linearly with respect to N , the dimension of the parameter space of the discretized PDE. Our presentation will mostly follow the notation of the H 2 -matrix framework due to its algebraic nature.The proposed architecture, MNN-H 2 , is a direct extension of the framework used to build a multiscale neural networks based on H-matrices (MNN-H) [17] to H 2 -matrices. We demonstrate the capabilities of MNN-H 2 with three classical yet challenging examples in computational physics: the nonlinear Schrödinger equation [2,43], the radiative transfer equation [29,30,40,44], and the Kohn-Sham map [25,31]. We find that MNN-H 2 can yield comparable results to those obtained from MNN-H, but with a reduced number of parameters, thanks to the use of hierarchical nested bases.The outline of the paper is as follows. Section 2 reviews the H 2 -matrices and interprets them within the framework of neural networks. Section 3 extends the neural network representation of H 2 -matrices to the nonlinear case. Section 4 discusses the implementation details and demonstrates the accuracy of the architecture in representing nonlinear maps, followed by the conclusion and future directions in Section 5. Neural network architecture for H -matricesIn this section, we reinterpret the matrix-vector multiplication of H...

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