Single Reference Frequency Loss for Multifrequency Wavefield Representation Using Physics-Informed Neural Networks

Abstract

Physics-informed neural networks (PINNs) can offer approximate multidimensional functional solutions to the Helmholtz equation that are flexible, require low memory, and have no limitations on the shape of the solution space. However, the neural network (NN) training can be costly and the cost dramatically increases as we train for multi-frequency wavefields by adding frequency as an additional input to the NN multidimensional function. In this case, the often large variation of the wavefield features (specifically wavelength) with frequency adds more complexity to the NN training. Thus, we propose a new loss function for the NN multidimensional input training that allows us to seamlessly include frequency as a dimension. We specifically utilize the linear relation between frequency and wavenumber (the wavefield space representation) to incorporate a reference frequency scaling to the loss function. As a result, the effective wavenumber of the wavefield solution as a function of frequency remains almost stationary, which reduces the learning burden on the NN function. We demonstrate the effectiveness of this modified loss function on a layered model.

Source location using physics-informed neural networks with hard constraints

“…Recall that if we do the source imaging in the frequency domain, then the wavefield belonging to multiple frequencies is needed for the transformation to the time domain wavefield. As shown in Huang and Alkhalifah [2022b], direct use of the PINN with the loss (Equation 4) would decrease the accuracy and convergence speed. Similar to their implementation, we modify the loss function with a single reference frequency loss by replacing ω to αω ref and U (x, z, ω) to D(x, ω) + αzΦ(x, z, ω, θ), yielding…”

“…Accordingly, the dimensions of the velocity v is also scaled, by α, to v(αx, αz). For details, we refer the reader to [Huang and Alkhalifah, 2022b].…”

M‐PINN: A mesh‐based physics‐informed neural network for linear elastic problems in solid mechanics

“…At present, there are few reports on the PINN solution under unknown boundary conditions. 27 To address the issues mentioned above, a mesh-based PINN (M-PINN) approach is proposed in this paper. This approach is inspired by the idea of FEM, where the solution domain is partitioned by mesh.…”

“…For traditional numerical methods, the lack of boundary conditions makes it impossible to obtain reliable computational results. At present, there are few reports on the PINN solution under unknown boundary conditions 27 …”

High-Dimensional Wavefield Solutions Using Physics-Informed Neural Networks with Frequency-Extension

“…The proposed method allows the neural operators to simultaneously handle the modeling for various velocities, frequencies, and source locations. Moreover, to improve generalization for larger-domain velocities beyond the training sample scope, we propose incorporating the single reference frequency [11] to enhance the generalization ability. We demonstrate the effectiveness of the proposed method on the OpenFWI dataset [12] and more realistic models, e.g., extracted from Overthrust models.…”