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First published on Wednesday, Jun 3, 2026 and last modified on Wednesday, Jun 3, 2026 by François Chaplais.
College of Electronic Science and Technology, National University of Defense Technology, Changsha, China. This work was supported by the National Natural Science Foundation of China under Grant 62471476 and 62231026. Email
Microwave inverse scattering imaging, physics-informed neural network, alternating optimization, conjugate gradient method, batched zero-padded 2D-FFT
Abstract
Microwave inverse scattering imaging (MISI) is a crucial computational technique in microwave nondestructive evaluation and near-field microwave sensing systems. However, quantitative reconstruction of high-contrast targets remains a formidable challenge due to severe multiple scattering effects and the inherent ill-posedness of electromagnetic inverse problems. To overcome this fundamental bottleneck in computational microwave imaging, this paper proposes an alternating optimization framework based on cross-correlated physics-informed neural network (Alt-CC-PINN). This architecture deeply decouples the evolution of the microwave physical field from the neural-network-based dielectric parameter inference, replacing traditional joint optimization with a hybrid alternating engine. Specifically, the method employs an analytical Polak-Ribière conjugate gradient (PR-CG) algorithm driven by a cross-correlated loss to optimally update the contrast sources, and deploys batched zero-padded 2D-FFT to ensure high computational efficiency. Subsequently, a deep learning optimizer is utilized to update the continuous neural representation. Extensive validations based on simulated and measured data demonstrate that Alt-CC-PINN effectively overcomes the local minima problem in high-contrast and low-signal-to-noise-ratio (SNR) environments. It exhibits superior reconstruction fidelity and robustness under the frequency-hopping probing strategy, providing a powerful and reliable computational electromagnetic solver for practical microwave imaging systems.
1 Introduction
Microwave inverse scattering imaging (MISI) aims to quantitatively reconstruct the spatial distribution of the complex permittivity (relative permittivity and conductivity) of unknown targets within a probing domain using external scattered field measurements in the microwave frequency band. As a critical branch of microwave theory and technology, MISI has demonstrated irreplaceable value in engineering applications such as microwave nondestructive evaluation (NDE) [1, 2], near-field microwave sensing [3], and microwave medical imaging systems (e.g., early-stage microwave breast cancer detection [4, 5]). However, the microwave inverse scattering process governed by the Lippmann-Schwinger integral equation is fundamentally affected by the complex internal interactions between high-frequency microwaves and inhomogeneous media (i.e., multiple scattering effects), rendering it a highly nonlinear and ill-posed computational electromagnetics problem. To tackle this challenge, traditional iterative optimization algorithms, such as the Contrast Source Inversion (CSI) method [6, 7] and the Distorted Born Iterative Method (DBIM) [8, 9], rely on constructing sophisticated cost functions and utilizing analytical gradients for iterative searches. These deterministic methods have achieved promising inversion results in numerous microwave imaging systems [10, 11]. Nevertheless, when confronted with strong microwave scatterers of high contrast, the extreme non-convexity of the loss function landscape exposes these algorithms to a severe risk of falling into local minima [12].
In recent years, the deep integration of computational microwave techniques and deep learning has introduced physics-informed neural networks (PINNs) [13] to the field of MISI, offering an attractive new pathway to break the inversion bottlenecks associated with complex microwave targets [14, 13, 15]. PINNs utilize neural networks to implicitly represent continuous spatial medium distributions and embed the physical partial differential/integral equations governing microwave scattering as residual terms in the loss function. This facilitates unsupervised quantitative inversion, drastically reducing the reliance on massive paired microwave training datasets. In our previous research, a cross-correlated PINN (CC-PINN) architecture was proposed [13]. CC-PINN creatively introduces a cross-correlated term into the loss function, forcefully coupling the predicted microwave contrast and the contrast source to the far-field observations. Although CC-PINN significantly enhances the ability of the inversion algorithm to escape local minima, its convergence stability encounters bottlenecks when processing complex microwave targets with high dielectric contrasts. A deeper analysis of the underlying architecture of CC-PINN reveals that this performance ceiling stems from an optimization strategy mismatch. In CC-PINN and its derived traditional PINN architectures, spatial contrast (parameterized by neural network weights) and internal microwave contrast sources (treated as explicit leaf tensors) are bluntly concatenated and subjected to joint, end-to-end gradient updates by deep learning optimizers (e.g., Adam). However, the physical evolution of microwave contrast sources possesses an extremely rigorous algebraic structure. Indiscriminate joint optimization using pure first-order gradient descent not only severs the analytical connections within the electromagnetic equations but also leads to an extremely rugged non-convex surface in the parameter space. As the contrast of the target increases, the gradients among the data term, state term, and cross-correlated term are highly prone to severe mutual interference and directional conflicts, resulting in violent training oscillations or even complete collapse in the microwave field domain.
To resolve this computational bottleneck faced by microwave imaging systems during high-contrast inversion, this paper proposes an alternating optimization PINN framework combined with a cross-correlated cost function, termed Alt-CC-PINN. The core innovation of Alt-CC-PINN lies in abandoning the end-to-end joint optimization of contrast sources and network parameters, replacing it with a hybrid alternating optimization engine combining “analytical conjugate gradients + deep neural networks”. This architecture not only profoundly aligns with the physical nature of microwave field evolution but also significantly improves the robustness of high-contrast microwave inverse imaging.
The main contributions of this paper are summarized as follows:
- Alternating Microwave Evolution Architecture: A two-stage iterative framework that decouples internal microwave field evolution and dielectric parameter inference is constructed. This resolves the fundamental optimization mechanism issue of gradient interference and cancellation among multiple physical loss terms in high-contrast scenarios.
- PR-CG Physical Optimization Engine: During the microwave contrast source update phase, the conventional black-box neural network optimizer is bypassed. An optimal Polak-Ribière conjugate gradient (PR-CG) analytical update rule based on cross-correlated loss is derived, utilizing an analytical search step to ensure absolute descent for strongly nonlinear microwave field quantities.
- Extreme Optimization of Computational Architecture: A continuous neural representation incorporating restricted dynamic ranges and Weight Normalization is adopted. Additionally, batched zero-padded 2D-FFT acceleration is comprehensively deployed in the forward Green’s integration. This technique not only maintains computational complexity at \( \mathcal{O}(N^2 \log N)\) but also significantly boosts the computational efficiency via its batched parallel strategy, providing possibilities for real-time or near-real-time processing in microwave imaging systems.
- Comprehensive Validation of Microwave Imaging Robustness: Under the alternating update architecture, the significant advantage of the cross-correlated cost function in enhancing algorithmic robustness is further confirmed. Whether targeting single-frequency, broadband frequency-hopping, or simultaneous multi-frequency processing strategies, the proposed architecture demonstrates excellent robustness and universality.
The remainder of this paper is organized as follows: Section II formulates the mathematical modeling of the microwave inverse scattering problem and details the fundamental algorithmic theory and alternating evolution workflow of Alt-CC-PINN. Section III comprehensively validates the effectiveness and computational efficiency of the proposed algorithm under different SNRs and microwave frequency processing strategies, based on full-wave simulations and measured data from a microwave anechoic chamber. The final section concludes the paper.
2 Algorithm Architecture and Implementation
2.1 Inverse Scattering Problem Modeling
Consider a standard two-dimensional transverse magnetic (TM) polarized multi-frequency inverse scattering probing scenario, with a time dependence of \( \exp(\text{j}\omega t)\) , where \( \text{j}^2=-1\) . The unknown scatterers are located within a bounded domain of interest (DOI), denoted as \( \mathcal{D}\) . The probing system utilizes a multi-frequency broadband detection approach, defined by the frequency set \( \mathcal{F}_s=\{f_1, f_2, \dots, f_I\}\) and corresponding angular frequencies \( \omega_f = 2\pi f\) . The complex contrast of the target at spatial position \( \mathbf{r} \in \mathcal{D}\) is defined as:
(1)
where \( \varepsilon_r(\mathbf{r})\) and \( \sigma(\mathbf{r})\) represent the relative permittivity and conductivity to be reconstructed, respectively.
The system deploys \( N_{tx}\) transmitting antennas and corresponding receiving antennas uniformly distributed over \( S_{obs}\) outside domain \( \mathcal{D}\) . At frequency \( f\) , the \( p\) -th transmitting antenna generates a background incident field \( \mathbf{E}_{\text{inc}, f}^p(\mathbf{r})\) within region \( \mathcal{D}\) . By introducing the equivalent contrast source variable \( \mathbf{J}_f^p(\mathbf{r}) = \chi_f(\mathbf{r}) \odot \mathbf{E}_{\text{tot}, f}^p(\mathbf{r})\) , the forward physical process of inverse scattering is governed by the following two sets of Lippmann-Schwinger integral equations (expressed in operator matrix form to encompass all transceiver pairs):
-
State Equation: Describes the conservation relationship of the internal physical field
\[ \begin{equation} \mathbf{E}_{\text{tot}, f} = \mathbf{E}_{\text{inc}, f} + \mathcal{G}_{\mathcal{D}, f} \mathbf{J}_f \end{equation} \](2)
where \( \mathcal{G}_{\mathcal{D}, f}\) is the domain Green’s function integral operator.
-
Data Equation: Describes the mapping relationship of the external observed scattered field
\[ \begin{equation} \mathbf{E}_{\text{meas},f} = \mathcal{G}_{\text{S}, f} \mathbf{J}_f \end{equation} \](3)
where \( \mathcal{G}_{\text{S}, f}\) is the far-field Green’s integral operator mapping from region \( \mathcal{D}\) to the observation domain \( \text{S}_\text{obs}\) .
Combining the above two equations, the objective of the inverse problem is: given multi-frequency scattered measurements \( \mathbf{E}_{\text{meas},f}\) , iteratively solve for physically consistent contrast \( \chi_f(\mathbf{r})\) and internal contrast sources \( \mathbf{J}_f\) by minimizing physical residuals.
2.2 Definition of Cross-Correlated Loss Function
For multi-frequency joint inversion and frequency-hopping strategies, the training process is divided into multiple stages. For the active frequency set \( \mathcal{F}_s\) in Stage \( s\) , the total loss function for the current stage consists of the superposition of the data term, state term, and cross-correlated term across all active frequency points:
(4)
The cross-correlated error term acts similarly to a global regularizer in the objective function. While it smooths extrema, it may slightly sacrifice a minor portion of inversion accuracy in most cases. The introduction of an exponential decay factor \( \beta_{\text{Lc},s}^{(k)}\) balances the algorithmic robustness provided by the cross-correlated residual with the inversion precision typical of traditional cost functions. Under the frequency-hopping strategy, to ensure the cross-correlated term exerts a progressive annealing effect during each independent frequency set training, we define \( \beta_{\text{Lc},s}^{(k)}\) at iteration step \( k\) as:
(5)
where \( K_s\) represents the total number of Epochs allocated to Stage \( s\) . This implies that during the training process for each fixed frequency set, \( \beta_{\text{Lc},s}^{(k)}\) undergoes an exponential decay from \( 0.5^{(s-1)}\) approaching 0. The term \( 0.5^{(s-1)}\) is introduced based on the following consideration: as higher frequency components are sequentially incorporated, the training process gradually stabilizes along the correct optimization direction. Moderately decreasing the proportion of the cross-correlated term effectively boosts convergence speed and inversion accuracy. The specific definitions of the respective physical residual terms are as follows:
(6)
(7)
(8)
2.3 Alternating Optimization Engine
The core idea of Alt-CC-PINN is to decompose the originally unified and massive non-convex optimization problem into two well-posed subproblems that are executed alternately. Its overall workflow comprises two alternating phases: Phase 1 utilizes an improved conjugate gradient method (PR-CG) to analytically batch-update multi-frequency contrast sources based on the current dielectric parameters; Phase 2 utilizes a deep learning optimizer (Adam) to update the network parameters to approximate the updated physical fields.
2.3.1 Phase 1: Batched PR-CG for Contrast Sources Update
In this phase, the neural network parameters \( \theta\) are frozen (i.e., the contrast distribution \( \boldsymbol{\chi}_f\) is held as a known constant from the previous generation). Since the aforementioned loss function is strictly quadratic with respect to the contrast sources \( \mathbf{J}_f\) at each frequency point, PyTorch’s autograd.grad is employed to accurately obtain its complex gradient \( \mathbf{g}_{\mathbf{J}}\) . Subsequently, the Polak-Ribière conjugate gradient method is used to analytically determine the descent step. For iteration step \( k\) , the update procedure is as follows:
- Compute the batched complex gradient: \( \mathbf{g}_{\mathbf{J}}^{(k)} = \nabla_{\mathbf{J}_f} L_\text{total}(\mathbf{J}^{(k-1)}, \theta)\) . To prevent gradient explosion caused by singularities in the ill-posed equations, absolute magnitude clipping (e.g., threshold set to \( 100.0\) ) is applied to gradients whose modulus exceeds the limit.
-
Compute the PR-CG conjugate coefficient \( \gamma\) :
\[ \begin{equation} \gamma^{(k)} = \max \left( 0, \frac{\Re\left\{ \langle \mathbf{g}_{\mathbf{J}}^{(k)}, \mathbf{g}_{\mathbf{J}}^{(k)} - \mathbf{g}_{\mathbf{J}}^{(k-1)} \rangle \right\}}{\left\|\mathbf{g}_{\mathbf{J}}^{(k-1)}\right\|_2^2} \right) \end{equation} \](9)
-
Determine the conjugate descent direction \( \mathbf{v}^{(k)}\) :
\[ \begin{equation} \mathbf{v}^{(k)} = -\mathbf{g}_{\mathbf{J}}^{(k)} + \gamma^{(k)} \mathbf{v}^{(k-1)} \end{equation} \](10)
If \( \Re\{\langle \mathbf{g}_{\mathbf{J}}^{(k)}, \mathbf{v}^{(k)} \rangle\} \ge 0\) (i.e., deviation from the descent direction), the conjugate direction is reset to the steepest descent direction \( \mathbf{v}^{(k)} = -\mathbf{g}_{\mathbf{J}}^{(k)}\) .
-
Calculate the optimal step size \( \alpha\) via analytical line search: Substituting the update formula \( \mathbf{J}_f^{(k)} = \mathbf{J}_f^{(k-1)} + \alpha \mathbf{v}^{(k)}\) into the denominators of all loss terms (where \( \mathcal{G}_D \mathbf{v}\) is accelerated by 2D-FFT). With \( \boldsymbol{\chi}\) fixed, the objective function becomes a standard upward-opening quadratic function regarding the real number \( \alpha\) . Setting the derivative to zero yields the analytical step size:
\[ \begin{equation} \alpha = \frac{-\Re\left\{ \langle \mathbf{g}_{\mathbf{J}}^{(k)}, \mathbf{v}^{(k)} \rangle \right\}}{2 \cdot \text{Denom}(\mathbf{v}^{(k)}, \boldsymbol{\chi})} \end{equation} \](11)
where the denominator term \( \text{Denom}\) is composed of the second-order variation induced by \( \mathbf{v}\) across the three loss terms. Thus, the analytical update of the multi-frequency contrast sources is completed.
2.3.2 Phase 2: Neural Network Update
After accomplishing the precise analytical advancement of the contrast sources, \( \mathbf{J}_f\) is treated as a constant tensor (via Detach operation). The neural network \( \theta\) is unfrozen, and the algorithm enters the inner loop. At this point, utilizing the known and fixed internal total field \( \mathbf{E}_{\text{tot},f} = \mathbf{E}_{\text{inc},f} + \mathcal{G}_D \mathbf{J}_f^{(k)}\) , and considering that the data residual \( L_\text{data}\) is independent of \( \theta\) , the target loss function for network optimization degenerates to:
(12)
The Adam optimizer is employed to minimize this degenerated loss, propagating errors backward to update \( \theta\) , compelling the network-output contrast \( \boldsymbol{\chi}_f(\theta)\) to accurately match the current optimal physical field distribution. Before the update, a maximum norm clipping (e.g., restricted to \( 1.0\) ) is applied to \( \nabla_{\theta} L_\text{NN}\) to further ensure stable network convergence:
(13)
Finally, it should be noted that the updating speed of contrast sources is typically faster than that of contrast itself. Therefore, within the alternating optimization training framework, the number of updates \( N_{\text{inner}}\) in Phase 2 is set to 2. That is, for every single update of the contrast sources, the contrast is updated twice, achieving an optimal balance between computational efficiency and inversion accuracy.
2.4 Fourier Feature MLP Architecture Based on Weight Normalization
To implicitly represent the contrast, Fourier Feature Mapping is utilized to map the spatial coordinates \( (x,y)\) into a high-dimensional periodic space. Building upon this, the main body of the network (MaterialNet INR) adopts a multi-layer perceptron (MLP) architecture combined with Weight Normalization and SiLU activation functions. Unlike LayerNorm, which frequently generates high-frequency granular artifacts by dividing by the local variance in the spatial dimension, Weight Normalization exclusively decouples the direction and magnitude of the weight matrices. This modification not only significantly enhances the robustness of initialization but also perfectly recovers the physical spatial smoothness of the inverted medium. To prevent the network output from becoming abnormally large in the early stages of optimization—which could lead to severe background oscillation—the final layer of the network is passed through a Sigmoid activation, strictly restricting \( \varepsilon_r\) within a physically reasonable interval (e.g., a maximum limit of 80.0, adjustable depending on specific applications).
The proposed inversion architecture can be flexibly paired with various network models to realize the implicit neural representation of inverse scattering operators. During the specific research and experimental processes, the introduction of a Fourier Neural Operator (FNO) [13] was also attempted. Although FNO can marginally accelerate the convergence speed, it introduces two significant drawbacks: first, FNO involves extensive convolution operations that severely drag down the computational efficiency of the algorithm; second, FNO inevitably faces the frequency selection problem of frequency truncation, which induces Gibbs oscillations and causes accuracy loss in inversion. Given these two reasons, this study ultimately decided to retain the Fourier Feature MLP architecture based on Weight Normalization. It provides high computational efficiency and high inversion accuracy, while the proposed alternating update architecture guarantees the necessary convergence speed during inversion network training. Fig. 3 illustrates the block diagram of the Fourier Feature MLP architecture based on Weight Normalization. Algorithm 1 provides the precise flowchart of the Fast Batched Alternating Optimization for Alt-CC-PINN with Frequency Hopping.
2.5 Batched Zero-Padded 2D-FFT Operator Acceleration
To completely eliminate the \( \mathcal{O}(N_g^2)\) computational bottleneck introduced by the domain Green’s function integration \( \mathcal{G}_D \mathbf{J}\) during forward inference, this architecture integrates a fast Fourier transform technique based on zero-padding with a multiplier of 4 [16]. Whenever calculating \( \mathcal{G}_D \mathbf{J}\) or computing the PR-CG search direction \( \mathcal{G}_D \mathbf{v}\) , the algorithm automatically deploys a parallel batch process across the frequency and transmitter dimensions, executing 2D-FFT via the cuFFT library. This implementation guarantees that the circular convolution of FFT is strictly equivalent to the linear convolution in spatial physics, while the batch processing strategy dramatically amplifies the computational efficiency of the algorithm when handling multi-frequency data.
Algorithm 1 Fast Batched Alternating Optimization for Alt-CC-PINN with Frequency Hopping
3 Experimental Validation and Performance Analysis
To comprehensively and objectively evaluate the inversion performance of the proposed Alt-CC-PINN architecture, this section presents detailed validations based on both full-wave simulation data and measured data from a microwave anechoic chamber. To ensure absolute fairness in comparative experiments, identical network topologies (Fourier feature networks with Weight Normalization) and initialization boundary conditions are adopted across the board, whether using traditional physical-driven losses without cross-correlation or the proposed alternating optimization architecture.
Considering that the randomness in parameter distribution during the early training stages of neural networks might affect highly nonlinear inverse scattering outcomes, all validation cases in this section were subjected to 11 independent Monte Carlo runs. Subsequently, rigorous quantitative analysis of the convergence accuracy and robustness of Alt-CC-PINN was conducted from a statistical perspective, using average Peak Signal-to-Noise Ratio (PSNR) evolution curves and boxplot distributions, thereby effectively eliminating statistical bias caused by solitary accidental convergences. In this paper, both simultaneous multi-frequency inversion and frequency-hopping progression processing strategies are investigated. The results shown are selected from the median of the final PSNR of the 11 independent runs. Dashed lines mark the true contours of the target.
3.1 Simulation Experiment Analysis
3.1.1 Simulation Environment Setup and Evaluation Configurations
3.1.1.1 Simulation of Electromagnetic Scattering Measurement System
The generation environment for synthetic data simulates a classic two-dimensional Fresnel experimental configuration (its topology is illustrated in Fig. 4). Transmitting antennas (Tx) and receiving antennas (Rx) are uniformly deployed on a concentric observation circle with a radius of \( R = 3\) m. To highly replicate physical blind zones in authentic measurements, the received data within a \( 30^\circ\) sectorial region centered on the active transmitter is excluded during each excitation. In the remaining \( 300^\circ\) observation arc, receivers are densely sampled at an angular resolution of \( 3^\circ\) . Therefore, for each frequency point, the effective measurement matrix dimension in TM polarization mode is fixed at \( 12 (\text{Tx}) \times 101 (\text{Rx})\) .
In the forward physics solver, the TM polarized incident wave is excited by ideal line sources along the \( z\) -axis. To guarantee numerical discretization accuracy, the forward simulation employs a fine grid partition of \( 5 \text{mm} \times 5 \text{mm}\) , which strictly complies with the empirical convergence criterion \( \Delta \leq \lambda_0/(15\sqrt{\varepsilon_\text{r}})\) [13] (\( \lambda_0\) being the free-space wavelength). To truncate unbounded free space, perfectly matched layers (PML) are set on the outer boundaries of the \( x\) and \( y\) axes, while periodic boundary conditions (PBC) are applied along the \( z\) -axis to maintain two-dimensional electromagnetic characteristics. Ultimately, the pure scattered field data is rigorously extracted by subtracting the background incident field from the total field.
3.1.1.2 Benchmark Model and Inversion Domain Setup
The “Austria” profile, a classically challenging benchmark model in inverse scattering, was selected [17, 18, 19, 20]. This target is renowned for its complex disconnected topology and high contrast, consisting of two independent disks and one ring. In the established 2D Cartesian coordinate system, the two disks both have a radius of 0.1 m, with geometric centers located at \( (0.3, -0.15)\) m and \( (0.3, 0.15)\) m, respectively; the inner and outer radii of the ring are 0.15 m and 0.3 m, anchored at \( (-0.1, 0)\) m. The physical dimension of the DOI is defined as \( [-0.5, 0.5] \text{m} \times [-0.5, 0.5] \text{m}\) . To reduce the ill-posedness of the problem while matching the output resolution of the network, this region is discretized into a coarse grid of \( 64 \times 64\) during the inversion calculation process.
3.1.1.3 Multi-Frequency Strategies and Noise Resistance Testing Scheme
Three frequency points, 0.3 GHz, 0.4 GHz, and 0.5 GHz, were selected for the simulated broadband test. When evaluating Alt-CC-PINN with the frequency-hopping strategy, the total number of global training Epochs is set to 25,000, dynamically divided into three progressive stages: Stage 1 (only 0.3 GHz active) and Stage 2 (0.3 + 0.4 GHz active) each account for 20% of the total iterations (i.e., 3,000 epochs each), while the final frequency-fusion Stage 3 (all band 0.3 + 0.4 + 0.5 GHz active) occupies the remaining 60%.
Furthermore, to rigorously verify the algorithm’s noise resistance under complex electromagnetic interference, complex Gaussian white noise was artificially injected into the extracted total field data in the observation domain. The corrupted data vector \( \boldsymbol{y}'_{p,f}\) follows the degradation model:
(14)
where \( N\) is the dimension of the vector \( \boldsymbol{y}_{p,f}\) ; the real and imaginary noise components \( n_{\text{real}}\) and \( n_{\text{imag}}\) both conform to the standard normal distribution \( \mathcal{N}(0, 1)\) . To prevent statistical bias in algorithmic comparison caused by randomized noise instantiations, noisy dataset files with three fixed SNRs (SNR \( = 20\) dB, 10 dB, 0 dB) were pre-generated, and all compared baselines operated strictly on the exact same dataset files.
3.1.1.4 Quantitative Evaluation Metrics
The quantitative fidelity of the reconstructed images is evaluated by the Peak Signal-to-Noise Ratio (PSNR), defined as:
(15)
where \( \text{peakval}\) represents the peak value of the true distribution profile for the target’s permittivity (or conductivity), and MSE denotes the mean squared error between the network-inferred distribution matrix and the true profile matrix. A higher PSNR indicates superior quantitative analytical capability and better artifact suppression of the inversion algorithm.
3.1.2 Performance Analysis with Different Contrast Values
An increase in the dielectric target’s contrast sharply elevates the non-linearity and multiple scattering effects of the inverse scattering problem, easily leading gradient-based optimization algorithms into local minima. To verify the capability of the proposed algorithm in handling highly nonlinear problems, the “Austria” dielectric targets with relative permittivity \( \varepsilon_\text{r} = 6, 7, 8, 9\) were investigated under SNR\( =20\) dB utilizing the frequency-hopping strategy. The statistical PSNR convergence curves and final inversion results are displayed in Fig. 5 and Fig. 10, respectively.
When the target contrast is moderate (\( \varepsilon_\text{r} = 6\) ), as shown in Fig. 5(a) and 10(a, e, i), all three algorithms successfully reconstruct the target’s topology. However, significant differences appear in convergence dynamics: the PSNR curve of Simul-CC-PINN consistently remains the lowest with large variance, indicating that updating contrast sources and network parameters simultaneously in a single optimizer causes gradient conflict, which severely retards convergence speed and reduces final precision. In comparison, Alt-PINN, without the cross-correlated term, exhibits a slightly higher absolute accuracy than Alt-CC-PINN. This is because the cross-correlated term functions similarly to global regularization in the objective function; although it smooths extrema, it sacrifices minute edge sharpness at lower contrasts.
As target contrast increases further, the ill-posedness of the inverse scattering problem becomes remarkably prominent. At \( \varepsilon_\text{r} = 7\) , Simul-CC-PINN is the first to fail; not only does its PSNR cease growing, but its reconstructed image degrades into meaningless background noise. Concurrently, the convergence curve of Alt-PINN starts to oscillate violently (the error band in Fig. 5(b) broadens noticeably), indicating that the optimization trajectory is struggling near local minima. When the relative permittivity reaches extreme levels of \( \varepsilon_\text{r} = 8\) and \( 9\) (Fig. 5(c-d) and the last two columns of Fig. 10), the traditional Simul-CC-PINN and Alt-PINN architectures collapse entirely, losing all inversion capabilities. Exclusively, the proposed Alt-CC-PINN maintains a remarkably stable convergence trajectory, successfully and clearly inverting the contours of the high-contrast ring and cylinders. This strongly demonstrates that when confronting strong non-linearities dominated by high-frequency multiple scattering, mere alternating optimization (Alt-PINN) is insufficient to overcome the ruggedness of the loss landscape. The deep integration of the cross-correlated term and the alternating engine effectively guides the network across local minima, endowing Alt-CC-PINN with outstanding algorithmic robustness.
3.1.3 Performance Analysis under Different SNRs
To further test the algorithm’s anti-interference capabilities in harsh electromagnetic environments, the evaluation is extended to low-SNR scenarios. Under the frequency-hopping strategy, strong noise levels with SNR \( = 10\) dB and \( 0\) dB were injected into the forward data for \( \varepsilon_\text{r} = 6\) and \( \varepsilon_\text{r} = 7\) , with results documented in Fig. 23 and Fig. 28.
Under SNR \( = 10\) dB conditions, severely corrupted measurement data directly destroys the matching conditions for physical loss without the cross-correlation term. As shown in Fig. 23(a-b), the performance of Alt-PINN sharply declines when the contrast increases to 7, while Simul-CC-PINN fails completely. Conversely, because the cross-correlated term establishes a self-consistent validation mechanism based on physical constraints between the data equation and state equation, Alt-CC-PINN’s intrinsic noise-filtering effect substantially suppresses the misguidance of gradient updates by high-frequency noise, thus maintaining an extremely high reconstruction quality. When environmental conditions deteriorate to the extreme limit of SNR \( = 0\) dB (where noise power equals signal power), the observation data for \( \varepsilon_\text{r} = 7\) experiences severe distortion. As depicted in Fig. 23(c-d), Alt-PINN exhibits continuous instability, and Simul-CC-PINN predictably breaks down. However, Alt-CC-PINN remarkably manages to robustly delineate the core topological features and location information of the “Austria” target. This further validates the significant superiority of Alt-CC-PINN under extremely ill-posed conditions, demonstrating exceptional tolerance towards measurement noise. The corresponding inversion reconstructions are provided in Fig. 28.
3.1.4 Simultaneous Multi-Frequency Processing Strategy
Beyond the classic frequency-hopping strategy, the feeding format of multi-frequency broadband data remains a decisive factor influencing PINN performance. In this section, under SNR \( = 20\) dB conditions, the data from three frequency points (\( 0.3, 0.4, 0.5\) GHz) are simultaneously fed into the network (Simultaneous Multi-Frequency Processing) to test targets with \( \varepsilon_\text{r} = 6, 7, 8\) . The statistical inversion results are presented in Fig. 41.
The experimental outcomes reveal a crucial conclusion: simultaneous multi-frequency processing significantly amplifies the high-dimensional complexity of the loss space. For \( \varepsilon_\text{r} = 6\) and \( 7\) , Alt-PINN—which was functional under the hopping strategy—fails completely in the simultaneous mode, as does Simul-CC-PINN. Under these challenging conditions, only Alt-CC-PINN, relying on its robust adaptive optimization capabilities, marginally overcomes the non-convexity induced by high-frequency phase wrapping, succeeding in inverting the target’s permittivity distribution.
However, when the dielectric constant reaches \( \varepsilon_\text{r} = 8\) , facing the dual pressures of extreme contrast and simultaneous multi-frequency mapping, all three algorithms (including Alt-CC-PINN) fail universally. This phenomenon is entirely physically coherent: the frequency-hopping strategy intrinsically adheres to the asymptotic physical logic of the Born approximation—first utilizing low-frequency data to establish a smooth global convex hull (localizing the target and determining low spatial frequency components), and then leveraging high-frequency data to refine topological boundaries. Conversely, simultaneous multi-frequency processing demands that the network simultaneously fit violent high-frequency oscillating phases during the initialization stage, readily causing gradient vanishing or explosion. Therefore, for electromagnetic inversion architectures driven by PINNs, the frequency-hopping strategy possesses an unparalleled scientific advantage in reducing dimensionality, decoupling features, and raising the upper limit for solving nonlinearities compared to simultaneous processing.
3.1.5 Testing and Analysis on Lossy Targets
To verify the decoupling and optimization capabilities of the proposed Alt-CC-PINN architecture in joint dual-parameter inversion, two sets of tests were conducted on the “Austria” lossy target involving complex permittivity under SNR \( = 20\) dB and utilizing the frequency-hopping strategy.
In the first set of experiments, the target parameters are configured as follows: small cylinders \( \varepsilon_\text{r} = 6, \sigma = 0.05\) S/m; large ring \( \varepsilon_\text{r} = 9, \sigma = 0.03\) S/m (see Fig. 48(a)). Fig. 45 and Fig. 48 showcase the statistical convergence curves and final reconstruction results, respectively. Under this configuration, the traditional end-to-end synchronous optimization architecture Simul-CC-PINN exposes severe flaws. As shown in Fig. 45(a)(b), its PSNR curves for both \( \varepsilon_\text{r}\) and \( \sigma\) remain at extremely low levels with massive variance. In stark contrast, Alt-PINN and Alt-CC-PINN—employing the alternating evolution engine—both successfully achieve high-precision separation and reconstruction of permittivity and conductivity. Particularly noteworthy is that in the previous tests involving purely lossless media, Alt-PINN failed completely at \( \varepsilon_\text{r} = 8\) ; however, when facing the ring up to \( \varepsilon_\text{r} = 9\) here, Alt-PINN converges excellently. This phenomenon is explained by the fact that dielectric loss effectively smooths the non-convex landscape of the loss function, enabling even a pure alternating optimization algorithm to smoothly slide into the global optimal solution region.
To explore the limits of the algorithm’s performance, in the second set of experiments, the dielectric constant of the small cylinders was pushed to \( \varepsilon_\text{r} = 12\) , and the conductivity of the large ring was elevated to \( \sigma = 0.10\) S/m (see Fig. 59(a)). Higher contrasts often mandate more training epochs; hence, epochs were set to 50,000 for this trial. For a purely lossless target, standard inversion algorithms would collapse under these parameters. Nonetheless, benefiting from the suppression of multiple scattering caused by high dissipation (\( \sigma = 0.10\) S/m), Alt-PINN still demonstrates decent reconstruction capabilities (Fig. 59(c)(f)), reaffirming the microwave inverse scattering principle that loss mitigates nonlinearity. In spite of this, observing the statistical boxplots on the right side of Fig. 56 clearly reveals that the final PSNR distribution of Alt-PINN is broader, presenting a degree of instability. The proposed Alt-CC-PINN, however, exhibits the highest absolute precision and an extremely compact convergence variance in the reconstruction of both permittivity and conductivity. As shown in Fig. 59(b)(e), Alt-CC-PINN characterizes the quantitative values of the high-contrast regions (\( \varepsilon_\text{r} = 12\) and \( \sigma = 0.10\) S/m) far more precisely. This suggests that although medium loss decreases the problem’s ill-posedness, the stringent data-physics consistency validation provided by the cross-correlated term remains indispensable. By offering stable gradient guidance to the alternating engine during the initial training stages, it endows Alt-CC-PINN with preeminent robustness for inverse scattering problems.
3.2 Validation with Experimental Data
3.2.1 Description of Experimental Data and Preprocessing Configurations
To further validate the generalization ability and anti-interference robustness of the proposed algorithm in authentic physical environments, this section uses the publicly available microwave anechoic chamber experimental dataset provided by Institut Fresnel for benchmark testing. The experimental scenario for inversion imaging is illustrated in Fig. 67. Specifically, the highly topologically complex TM-polarized target FoamTwinDielTM was selected. As displayed in Fig. 70, this target consists of a large, low-contrast foam cylinder (relative permittivity \( \varepsilon_r \approx 1.45\) , diameter 80 mm) nested with two high-contrast small dielectric cylinders (\( \varepsilon_r \approx 3.0\) , diameter 31 mm). This scenario incorporates not only disconnected domains but also the coexistence of strong and weak scatterers, imposing extremely high demands on the spatial resolution and quantitative accuracy of inversion algorithms [21].
The original measurement system deployed 18 equally spaced transmitting antennas on an observation circle with a radius of 1.67 m, and complex scattered fields were received by 241 probes. To decrease data redundancy and accelerate computation, the spatial sampling interval on the receiving end was downsampled to \( 5^\circ\) , yielding a compact measurement matrix of size \( 18 (\text{Tx}) \times 49 (\text{Rx})\) . The physical DOI for inversion is defined as \( [-0.1, 0.1] \text{m} \times [-0.1, 0.1] \text{m}\) , partitioned into a discrete \( 64 \times 64\) grid.
To comprehensively investigate the impact of frequency strategies on algorithm convergence, broadband data spanning from 2 GHz to 10 GHz were partitioned into two subsets featuring progressive nonlinear traits: a smooth low-frequency dataset FoamTwinDielTM_345 (comprising 3, 4, 5 GHz) and a highly ill-posed high-frequency dataset FoamTwinDielTM_678 (comprising 6, 7, 8 GHz). During quantitative evaluation, owing to minute positional shifts in the placement of the measured target, the actual geometric centers of the cylinders were extracted via the optimally reconstructed images to construct the reference ground-truth distribution with nominal \( \varepsilon_r = 1.45\) and \( 3.0\) , which was subsequently used to calculate PSNR for each independent test. Global training Epochs were set to 15,000.
3.2.2 Analysis of Experimental Data Inversion Under the Frequency-Hopping Strategy
In the frequency-hopping evolution mode, the network divides training into three stages from low to high frequency (with Epoch allocations of 20%, 20%, and 60%). The results of 11 independent Monte Carlo runs for both low-frequency and high-frequency subsets are presented in Fig. 71 and Fig. 74.
When processing the low-frequency dataset FoamTwinDielTM_345 (Fig. 71(a) and Fig. 74(a-c)), the physical scattering process is relatively mild. Both the proposed Alt-CC-PINN and the Alt-PINN lacking the cross-correlated term display exceptional reconstruction abilities, explicitly distinguishing the outer foam contour from the two inner high-permittivity cylinders. Notably, the average final PSNR of Alt-CC-PINN approaches 27 dB, outperforming Alt-PINN. Conversely, the baseline algorithm Simul-CC-PINN, executing synchronous optimization, crashes, producing only blurred artifact masses. This reveals that under discontinuous physical field jumps induced by the hopping mechanism, forcing the contrast source and network parameters to bind within a single optimizer easily precipitates gradient conflicts. In contrast, the analytical update mechanism of the Alternating Engine seamlessly adapts to such frequency band transitions.
When confronting the FoamTwinDielTM_678 dataset, which is replete with rich high-frequency multiple scattering (Fig. 71(b) and Fig. 74(d-f)), the ill-posedness of the inverse problem dramatically escalates due to increased electrical size and the introduction of experimental measurement noise. At this juncture, Alt-PINN—which previously performed well at low frequencies—fails completely (severe ring artifacts manifest in Fig. 74(e), with PSNR dropping below 18 dB). Simul-CC-PINN similarly remains trapped in local minima. Against this stark contrast, Alt-CC-PINN demonstrates an overwhelming robustness advantage. As seen in Fig. 74(d), even amidst high-frequency experimental noise interference, Alt-CC-PINN accurately locks onto and reconstructs the topological structure and quantitative values of the twin cylinders, with PSNR stabilizing above 28 dB. This conclusively proves that the “bridge” effect established by the cross-correlated term between the data equation and state equation exerts potent robust correction against high-frequency phase wrapping and measurement noise, serving as the essential catalyst for bypassing high-frequency nonlinear traps.
3.2.3 Analysis of Experimental Data Inversion Under Simultaneous Multi-Frequency Processing Strategy
To furnish a comprehensive algorithmic evaluation perspective, testing under a training strategy where multi-frequency data is simultaneously fed into the network (Simultaneous Multi-Frequency) was conducted without altering the total Epochs. The outcomes are provided in Fig. 81 and Fig. 84.
In the low-frequency subset FoamTwinDielTM_345 (Fig. 81(a)), an interesting observation arises: Simul-CC-PINN, which performed poorly under the hopping mode, achieves the highest final accuracy in this mode (PSNR \( \sim 27.6\) dB), followed closely by Alt-CC-PINN (\( \sim 27.2\) dB). Because the phases of low-frequency signals are relatively gentle, processing multiple frequency points simultaneously provides a consistent data augmentation effect for the neural network, allowing pure gradient descent algorithms to locate favorable extrema under well-posed conditions. However, once entering the highly ill-posed high-frequency subset FoamTwinDielTM_678, as presented in Fig. 81(b) and Fig. 84(e, f), the superimposition of complex interference fringes induced by high-frequency waves alongside measurement noise wildly twists the network’s loss landscape. Faced with this challenge, Alt-PINN without cross-correlation constraints collapses, its convergence curves stalling entirely in an ineffective low-precision domain of 16 dB, completely stripping it of its imaging capability. In stark contrast, Alt-CC-PINN’s PSNR climbs steadily, eventually exceeding 29 dB (Fig. 81(b), red curve), which is slightly higher than Simul-CC-PINN (\( \sim 28.5\) dB). From Fig. 84(d), the quantitative magnitudes of the internal high-permittivity areas are seen to be perfectly decoupled from the outer background.
Synthesizing the exhaustive tests across all scenarios above, the proposed Alt-CC-PINN algorithm, supported by the profound coupling of its “alternating analytical update engine” and “cross-correlated physical constraints” effectively transcends the fragile convergence performance typical of traditional deep learning models processing highly nonlinear measured inverse scattering data.
3.3 Computational Time Comparison
| Model |
Experiment Type |
Epochs |
Hopping Strategy Time (s) |
Simultaneous Multi-Freq. Processing Time (s) |
| Alt-CC-PINN | Simulation | 25000 | 322 | 333 |
| Alt-PINN | Simulation | 25000 | 281 | 283 |
| Simul-CC-PINN | Simulation | 25000 | 131 | 134 |
| Alt-CC-PINN | Fresnel Data | 15000 | 200 | 202 |
| Alt-PINN | Fresnel Data | 15000 | 171 | 177 |
| Simul-CC-PINN | Fresnel Data | 15000 | 79 | 80 |
This section provides the execution times of Alt-CC-PINN, Alt-PINN, and Simul-CC-PINN during simulation and measured data testing. The algorithmic codes were constructed based on the PyTorch framework and executed on an NVIDIA A6000 GPU with CUDA acceleration, employing batched deployment acceleration. Table 1 catalogs the execution times for the three algorithms. An examination of the temporal comparison in Table 1 indicates that Alt-CC-PINN requires the longest processing time—approximately 2.5 times that of Simul-CC-PINN. As Alt-PINN is exempt from calculating the cross-correlated cost, its execution time falls intermediately. Although Alt-CC-PINN consumes the most time per epoch, its convergence rate significantly outpaces Simul-CC-PINN. Consequently, even when inverting simple targets, Alt-CC-PINN mandates noticeably fewer training iterations compared to Simul-CC-PINN. Under comprehensive evaluation, Alt-CC-PINN retains a distinct competitive edge in terms of computational efficiency.
4 Conclusion
This paper proposes Alt-CC-PINN, an alternating optimization inversion architecture based on a cross-correlated cost function. By thoroughly analyzing the fundamental flaw of gradient-coupling collision triggered by the joint optimization of physical variables and network parameters in the original CC-PINN, an alternating optimization framework is innovatively introduced. At the physical evolution level, the combination of a \( 4N \times 4N\) scale fully parallel FFT operator and the PR-CG analytical descent method guarantees robust optimization within the contrast source domain. This alternating decoupling mechanism successfully flattens the rugged landscape characteristic of non-convex optimization, dramatically minimizing the risk of entrapment in local minima. Simulation experiments and comparative analyses confirm that Alt-CC-PINN not only sustains extreme robustness for traditional weak-scattering targets but also exhibits reconstruction precision and resilience far surpassing those of conventional physical baselines and jointly-trained networks when addressing high-contrast targets. This proposes a highly promising novel technological paradigm for the practical deployment of neural-network-based solvers in microwave probing and quantitative inverse problems.
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