Contrast-Source-Based Physics-Driven Neural Network for Inverse Scattering Problems

Deep neural networks (DNNs) have recently been applied to inverse scattering problems (ISPs) due to their strong nonlinear mapping capabilities. However, supervised DNN solvers require large-scale datasets, which limits their generalization in practical applications. Untrained neural networks (UNNs) address this issue by updating weights from measured electric fields and prior physical knowledge, but existing UNN solvers suffer from long inference time. To overcome these limitations, this paper proposes a contrast-source-based physics-driven neural network (CSPDNN), which predicts the induced current distribution to improve efficiency and incorporates an adaptive total variation loss for robust reconstruction under varying contrast and noise conditions. The improved imaging performance is validated through comprehensive numerical simulations and experimental data.

💡 Research Summary

The paper addresses the longstanding challenges of electromagnetic inverse scattering problems (ISPs), namely the severe non‑linearity caused by multiple scattering and the ill‑posedness due to limited measurement data. While supervised deep neural networks (DNNs) have shown promise, they require large labeled datasets that are rarely available in practice, leading to poor generalization. Untrained neural networks (UNNs) mitigate the data‑dependency by updating network weights directly from measured fields and physical constraints, yet existing UNN‑based solvers suffer from long inference times because they still rely on explicit matrix inversions or repeated forward simulations.

To overcome these limitations, the authors propose a Contrast‑Source‑Based Physics‑Driven Neural Network (CSPDNN). Instead of directly estimating the relative permittivity distribution ε_r, CSPDNN predicts the induced current (contrast source) J, which is the product of the contrast χ = ε_r − 1 and the total field. By formulating the ISP state equation as E_tot = E_inc + G_D J and the data equation as E_sca = G_S J, the network only needs to output an updated J. The permittivity is then recovered in a closed‑form post‑processing step (χ = J/E_tot, ε_r = 1 + χ), thereby avoiding the costly inversion of the discretized Green‑operator G_D that dominates the computational load of prior methods such as PDNN.

The network architecture is a hybrid of three convolutional layers (16, 32, 64 channels) each followed by a residual block and LeakyReLU activation, then flattened and processed by two fully‑connected layers with dropout. Input consists of the initial current J(0) and an initial permittivity estimate ε_r(0) (obtained by a fast back‑propagation ISP solver) concatenated as four real‑valued channels (real/imag parts of both quantities). Output is the updated current J_θ represented by two channels (real/imag). This design enables hierarchical feature extraction while preserving the spatial resolution required for accurate field reconstruction.

Training employs the Adam optimizer (initial LR = 1e‑3, halved every 1000 epochs) for up to 1500 epochs on a workstation equipped with an RTX 4090 GPU. The loss function combines four terms:

1. State consistency (L_State) – normalized L2 distance between the predicted current and the product χ·E_tot, enforcing that the network’s output respects the physical definition of the contrast source.
2. Data fidelity (L_Data) – normalized L2 distance between simulated scattered fields (computed from the predicted current) and the measured scattered fields, ensuring agreement with observations.
3. Bound constraint (L_Bound) – a ReLU‑based penalty that prevents the real part of ε_r from falling below unity, reflecting the physical requirement that relative permittivity be ≥ 1 for passive media.
4. Total‑variation regularization (L_TV) – a smoothness prior applied separately to the real and imaginary parts of χ. Crucially, the TV weight β is made adaptive: β(u) = β₀ · M(u), where u is a measure of the current contrast level. This allows the regularizer to strengthen in low‑contrast or high‑noise regimes and relax when the contrast is strong, improving robustness across a wide range of scenarios.

Numerical experiments are conducted on a 0.15 m × 0.15 m domain discretized into a 64 × 64 grid, illuminated by 36 transmitters and recorded by 36 receivers placed on a circle of radius 20 λ at 4 GHz. Six benchmark cases are examined, covering complex geometries (overlapping circles, concentric structures, Austria‑shaped objects), varying contrast levels, and lossy (conductive) materials. CSPDNN consistently yields sharper boundaries, more accurate contrast values, and fewer artifacts compared with three baselines: SOM (a physics‑informed unrolled optimizer), uSOM (its untrained variant), and PDNN (a physics‑driven network that updates ε_r directly).

A key advantage is computational speed. While SOM and uSOM require roughly 80 seconds per reconstruction, PDNN varies between 70 s and 140 s depending on the case, CSPDNN completes every test in about 28 seconds, delivering a 3–4× speed‑up without sacrificing image quality.

Robustness to measurement noise is also demonstrated. With additive white Gaussian noise at SNR = 20, 10, 5, and 1 dB, CSPDNN maintains recognizable shapes and accurate contrast even at 1 dB, whereas PDNN deteriorates markedly, producing background artifacts and contrast over‑estimation. The adaptive TV weighting is identified as the primary factor enabling this resilience.

Finally, experimental validation uses the “FoamDielExt” dataset from the Fresnel Institute. In this real‑world measurement, PDNN underestimates a strong scatterer, while CSPDNN recovers its permittivity more faithfully, confirming that the method generalizes beyond synthetic data.

In conclusion, CSPDNN introduces three pivotal innovations: (i) contrast‑source prediction that eliminates explicit matrix inversion, (ii) an adaptive total‑variation regularizer that self‑tunes to contrast and noise levels, and (iii) a lightweight CNN‑FC architecture that achieves near‑real‑time inference. These contributions collectively address the data‑scarcity and computational‑burden issues that have limited the practical deployment of deep‑learning‑based ISP solvers, paving the way for reliable, fast, and physics‑consistent imaging in nondestructive testing, security screening, and subsurface exploration.