Source-linked AI summary
SwitchNet: a neural network model for forward and inverse scattering problems
Yuehaw Khoo, Lexing Ying
TL;DR
The paper addresses forward and inverse scattering maps whose global coupling makes local convolutional networks inadequate and fully connected alternatives expensive. It proposes SwitchNet, using low-rank structure and sparse switching connections, and reports numerical experiments demonstrating promising map-learning accuracy with substantially fewer parameters than a fully connected network. The method’s scope includes fixed-frequency settings and the small-scatterer regime used for its perturbative formulation.
Problem
Scattering maps are globally coupled, so local convolutional networks are insufficient while fully connected parameterizations can be costly.
Method
SwitchNet uses a low-complexity switching layer and low-rank factorization to connect inputs and outputs globally with sparse parameterization.
Results
2900K parameters were used for the reported model problem, while far-field SwitchNet models used 3100K and 4200K parameters versus at least 40960K for a fully connected network.
Takeaways & Limitations
SwitchNet provides an efficient neural-network approach for approximating forward and inverse maps in wave-equation scattering problems.
Abstract
from arXiv · showhide
We propose a novel neural network architecture, SwitchNet, for solving the wave equation based inverse scattering problems via providing maps between the scatterers and the scattered field (and vice versa). The main difficulty of using a neural network for this problem is that a scatterer has a global impact on the scattered wave field, rendering typical convolutional neural network with local connections inapplicable. While it is possible to deal with such a problem using a fully connected network, the number of parameters grows quadratically with the size of the input and output data. By leveraging the inherent low-rank structure of the scattering problems and introducing a novel switching layer with sparse connections, the SwitchNet architecture uses much fewer parameters and facilitates the training process. Numerical experiments show promising accuracy in learning the forward and inverse maps between the scatterers and the scattered wave field.
1. Introduction
The paper studies neural-network approximations of forward and inverse acoustic scattering maps in two-dimensional, time-harmonic settings. Because scattering is globally coupled, it proposes SwitchNet, which uses a low-complexity switching layer instead of purely local connectivity.
- Problem setting: The study focuses on forward and inverse scattering for the two-dimensional time-harmonic acoustic wave equation at fixed frequency.The inhomogeneous medium is modeled with a Helmholtz operator.
- Problem setting: The inverse problem recovers an unknown compactly supported scatterer η from observation data derived from the Helmholtz Green’s function.The scatterer is represented relative to a known background velocity field.
- Motivation: Efficient forward maps can replace expensive Helmholtz PDE solvers, while inverse maps can recover scatterers without the usual iterative process.These are the stated numerical motivations for learning η → d and d → η.
- Challenge: Unlike image-processing maps with local dependencies, scattering maps allow each scatterer location to influence every point of the scattered field.The input and output may also be defined on different domains.
- Proposed approach: SwitchNet addresses this non-locality with a low-complexity switch layer that sends information between all pairs of sites effectively.The architecture is introduced to learn both forward and inverse maps.
- Scope: The paper applies SwitchNet to far-field patterns and to seismic imaging with sources and receivers at finite but well-separated distances.The two settings are treated in separate sections.
2. Preliminary
The preliminary formulation discretizes the Helmholtz scattering problem on a Cartesian grid and represents data acquisition through sources, propagation, and receivers. A background Green’s function yields a perturbative expansion of the scattered field when the scatterer is sufficiently small.
- Geometric and numerical setting: The analysis is two-dimensional, with a compactly supported scatterer in a bounded domain and a Helmholtz operator posed on R2 with a radiative boundary condition.The computational domain is truncated to the scatterer-supporting region.
- Geometric and numerical setting: The domain is discretized on a Cartesian grid at a few points per wavelength, making the grid resolution depend on the frequency.The resulting scatterer is a vector and the discretized Helmholtz operator is a matrix.
- Perturbative formulation: Using a background operator L0 and perturbation E, the Green’s function admits the expansion G ≈ G0 + G0EG0 + G0EG0EG0 + ···.The expansion terms are defined successively as G1, G2, and so on.
- Perturbative formulation: The perturbative expansion is valid when η is sufficiently small, and data collection commonly focuses on G−G0 because G0 is computable from the background velocity.This isolates the scatterer-dependent contribution.
- Data acquisition: Data acquisition applies sources, solves for the scattering field, and gathers measurements with receivers at specified locations or directions.Source- and receiver-dependent operators represent the first and third steps.
- Data acquisition: The observation operator is d = ΠR(G−G0)ΠS, where G−G0 captures the difference between the perturbed and background Green’s functions.The source and receiver operators surround the Green’s-function difference.
- Problem settings: The paper studies far-field and seismic settings, differing mainly in whether sources and receivers are treated as being at infinity or at finite separated distances.The far-field case motivates the SwitchNet construction before the seismic case is considered.
3. SwitchNet for far field pattern
For far-field scattering, SwitchNet approximates both scatterer-to-data and data-to-scatterer maps by exploiting low-rank structure and sparse global switching. This reduces parameterization complexity while retaining convolutional nonlinear processing for the inverse map.
- 3.1. Problem setup.: The far-field setup uses plane-wave sources and directional receivers, with data sampled over receiver and source directions.The scatterer is compactly supported, and the source and receiver direction sets are uniformly distributed with size M typically scaling linearly with frequency.
- 3.4. Neural networks.: When the scatterer is small, the inverse map can be approximated by back-projection with A* followed by the regularized deconvolution filter (A*A + ϵI)^−1.The filter is translation-invariant because A*A is a translation-invariant convolution operator.
- 3.2. Low-rank property.: Partitioning spatial and angular domains into squares of side length 1/√ω yields low-rank submatrices underlying the factorization of the scattering map.The spatial grid X and observation grid R × S are partitioned into PX and PD subgroups, respectively.
- 3.3. Matrix factorization.: The resulting factorization reduces approximate storage from O(N^4) to O(N^3) when M and N have comparable sizes.The factorization stores tP(M^2 + P + N^2) entries with P approximately max(M, N).
- 3.4. Neural networks.: SwitchNet uses a trainable Switch layer to send local information globally, while Vect and Square reshape data and convolution layers provide nonlinear inverse-map capacity.The Switch layer mimics A* ≈ VΣ*U* with tunable coefficients; replacing linear deconvolution by convolution layers supports nonlinear inversion.
- 3.4. Neural networks.: The Switch layer has 2tPn^2 parameters for n × n input and output, while the reported inverse and forward architectures use 3,100K and 4,200K parameters, respectively.The reported settings use t = 3 for the inverse network and t = 4 for the forward network, with three convolution layers in each.
4. SwitchNet for seismic imaging
For seismic imaging, SwitchNet exploits low-rank structure in the linearized scattering operator to represent both inverse and forward maps between observation data and scatterers. The model uses sparse switching connections, pointwise multiplication for non-translation-invariant effects, and achieves low prediction errors in the reported experiment.
- 4.1. Problem setup.: Sources and receivers are uniformly sampled along a near-surface horizontal line, while the scatterer is supported below and well-separated from them.The setup uses a 64 × 64 grid for η and 80 × 80 observation data in the numerical experiments.
- 4.2. Low-rank property.: Under smooth background velocity and source–receiver separation, the linearized operator A has numerically low-rank blocks and a low-complexity factorization.The factorization is written as A ≈ UΣV* and A* ≈ VΣ*U*.
- 4.3. Neural networks.: SwitchNet uses the low-rank factorization to implement the inverse map from observations d to scatterers η through a Switch layer.The inverse architecture also includes pointwise multiplication because the seismic operator is not translation-invariant.
- 4.3. Neural networks.: The forward map reverses the inverse architecture and retains an extra pointwise multiplication layer to capture the loss of translation invariance.The inverse network includes convolutional layers, vectorization, switching, reshaping, and pointwise multiplication operations.
- 4.4. Numerical results.: With 2,900K parameters, the seismic imaging networks were evaluated on 64 × 64 scatterers and 80 × 80 observations, with predictions compared against ground truth.Figure 7 provides a visual comparison for one typical test sample, while Table 2 reports prediction errors for both maps.
5. Discussion
SwitchNet approximates forward and inverse maps for time-harmonic wave equations whose local inputs have global effects on outputs. Its sparse global connections exploit low-rank structure to reduce complexity while representing oscillatory scattering fields.
- SwitchNet targets forward and inverse maps arising from the time-harmonic wave equation.
- Because local input information globally affects the output, fully connected networks are generally needed, but SwitchNet replaces them with sparse connections based on low-rank linearized operators.The architecture connects input and output globally rather than using only local convolutional filters.
- Global input–output connections enable SwitchNet to represent highly oscillatory scattered wave fields and solve associated inverse problems.