Wavefield solutions from machine learned functions

2021/06/02 by Tariq Alkhalifah, Alkhalifah, Tariq, Chao Song +5 · 4 citations
Earth and Planetary Sciences · Engineering · #Computational Physics (physics.comp-ph) #FOS: Physical sciences #Geophysics (physics.geo-ph) #Hydraulic Fracturing and Reservoir Analysis #Seismic Imaging and Inversion Techniques #Seismic Waves and Analysis

paper · pdf · doi:10.48550/arxiv.2106.01433

openalex publication_date 2021/06/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Solving the wave equation is one of the most (if not the most) fundamental problems we face as we try to illuminate the Earth using recorded seismic data. The Helmholtz equation provides wavefield solutions that are dimensionally reduced, per frequency, compared to the time domain, which is useful for many applications, like full waveform inversion (FWI). However, our ability to attain such wavefield solutions depends often on the size of the model and the complexity of the wave equation. Thus, we use here a recently introduced framework based on neural networks to predict functional solutions through setting the underlying physical equation as a loss function to optimize the neural network parameters. For an input given by a location in the model space, the network learns to predict the wavefield value at that location, and its partial derivatives using a concept referred to as automatic differentiation, to fit, in our case, a form of the Helmholtz equation. We specifically seek the solution of the scattered wavefield considering a simple homogeneous background model that allows for analytical solutions of the background wavefield. Providing the neural network (NN) a reasonable number of random points from the model space will ultimately train a fully connected deep NN to predict the scattered wavefield function. The size of the network depends mainly on the complexity of the desired wavefield, with such complexity increasing with increasing frequency and increasing model complexity. However, smaller networks can provide smoother wavefields that might be useful for inversion applications. Preliminary tests on a two-box-shaped scatterer model with a source in the middle, as well as, the Marmousi model with a source on the surface demonstrate the potential of the NN for this application. Additional tests on a 3D model demonstrate the potential versatility of the approach.

Citations

• On the limited memory BFGS method for large scale optimization
• Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations
• Approximating the solution to wave propagation using deep neural networks
• Regularized elastic full-waveform inversion using deep learning
• Wave physics as an analog recurrent neural network

Cited by

• Inspired by machine learning optimization: can gradient-based optimizers solve cycle skipping in full waveform inversion given sufficient iterations?
• Physics-informed waveform inversion using pretrained wavefield neural operators
• Physics-Informed Neural Network for Elastic Wave-Mode Separation
• Prediction of acoustic field in 1-D uniform duct with varying mean flow and temperature using neural networks

Related

• An effective physics-informed neural operator framework for predicting wavefields
• Solving the wave equation with physics-informed deep learning
• Learned frequency-domain scattered wavefield solutions using neural operators
• Robust data driven discovery of a seismic wave equation
• Seismic wave propagation and inversion with Neural Operators