Results demonstrate effective label-scarce estimation in this case, while extension to other inverse problems requires an affordable differentiable forward model, sufficiently accurate coarse localization, and structured mismatch that can be learned from scarce measurements.
Abstract
Inverse problems in computational physics often face dual scarcity: few real measurements and no ground-truth labels during deployment. When an imperfect forward model exhibits structured simulation-to-measurement mismatch and the target parameter is coupled with nuisance factors, neither model-based fitting nor supervised learning alone is sufficient. This paper presents a modular framework: (i) Measurement-Guided Data Augmentation (MDGA) converts each measurement into a dense local simulation bank via coarse fitting and Latin Hypercube Sampling; (ii) a Physics-Informed Dual-Stem Network (PI-DSN) processes real and simulated inputs through separate feature extractors and predicts a residual correction using label-free physics-consistency losses; and (iii) a label-free checkpoint selection protocol ranks candidates using forward-model consistency metrics while isolating ground-truth labels for final evaluation only. Evaluation is conducted in a Fraunhofer diffraction filament-metrology case where off-axis acquisition limits each setup to 5–10 patterns, the target diameter couples to nuisance parameters, and the forward model introduces structured mismatch. Results demonstrate effective label-scarce estimation in this case, while extension to other inverse problems requires an affordable differentiable forward model, sufficiently accurate coarse localization, and structured mismatch that can be learned from scarce measurements.
Physics-informed neural networks (PINNs) have emerged as a powerful paradigm in scientific machine learning by embedding governing physical laws into neural network training through loss functions. They have demonstrated remarkable success in solving various forward and inverse problems governed by partial differential equations (PDEs). However, in practical applications, purely physics-constrained PINNs that rely solely on PDE residuals often suffer from slow or non-convergence and limited prediction accuracy, particularly when modeling high-order dynamical systems (e.g., second-order and above). Moreover, conventional PINNs struggle to effectively capture high-frequency components in complex physical fields, which further limits their generalization and representational capability.
To address these challenges, this study proposes a data-guided physics-informed neural network with Fourier feature enhancement. In the proposed framework, a small amount of high-fidelity measurement or simulation data is incorporated to guide the training process, providing explicit guidance that complement the physics-based constraints. Meanwhile, Fourier feature embeddings are introduced into the input layer of the network to enhance its ability to represent high-frequency variations and multi-scale solution structures. This synergistic integration of data guidance and Fourier-enhanced representations accelerates convergence, improves robustness, and enhances the accuracy of PDE solutions.
The effectiveness of the proposed PINN model is validated through numerical and simulation studies on Euler-Bernoulli beam vibration problems, which serve as representative examples of high-order mechanical systems. The results demonstrate that PINN achieves more accurate and stable full-field vibration reconstructions than conventional PINNs, particularly under conditions involving high-frequency modes. These findings highlight the potential of hybrid data-physics neural frameworks as an efficient and reliable approach for solving complex PDE-governed dynamical systems.
Hailong Liu, S. Hedayatrasa, Yunpeng Zhu et al.· e-Journal of Nondestructive...· 0 citations
Inverse physics-informed neural networks (PINNs) can reconstruct a field accurately while returning an incorrect physical parameter. We introduce a two-axis post-training diagnosis that separates finite-sample resolution under a specified observation-and-estimation protocol from the signed parameter preference encoded by the final learned field and residual metric. The first axis repeatedly fits noisy observations with a matched forward estimator. At known synthetic truth, the second freezes the field and residual view and computes a local score displacement toward a nearby residual-profile minimum. Endpoint consistency then tests whether joint training delivers that preference under the same final view. Across three synthetic one-dimensional, scalar-parameter PDEs, matched-forward mean absolute relative error ranges from 2.34 percent to 17.46 percent. The displacement tracks frozen-profile minima across locked seeds, architectures, and fresh-noise retraining (r from .945 to .982), and it tracks delivered signed log-error in 240 fresh-noise RBA runs (r = .994; 237/240 correct directions). A coupled two-parameter Darcy check validates the full matrix calculation. The axes are complementary diagnostic coordinates, not additive error components or a deployable oracle-free estimator. Together, they route follow-up work toward observations, residual evidence, or endpoint delivery.
Electromagnetic inverse scattering is a nonlinear and ill-posed problem, where accurate reconstruction is challenging due to measurement limitations, noise, and high computational costs, especially for 3-D imaging. Although physics-driven neural networks (PDNNs) reduce the dependence on labeled training data, existing accelerated PDNN frameworks often rely on preliminary reconstruction-based region selection, which may introduce instability when the selected region is inaccurate. In this paper, a coordinate-residual physics-driven neural network (CRPDNN) is proposed for 3-D electromagnetic inverse scattering. The proposed solver directly reconstructs the unknown contrast distribution using normalized spatial coordinates and a residual convolutional network, without requiring a preliminary reconstruction. For the reported noise-free 3-D synthetic cases, CRPDNN achieves an average relative error of 2.10\%, compared with 7.97\% for CSI and 3.99\% for $L_{2/3}$-FBE-WCIE, while providing approximately 5.5- and 12.1-fold speedups over the two baselines, respectively. Supplementary 2-D comparisons further confirm its stability and computational efficiency relative to existing PDNN frameworks. CRPDNN also maintains reliable reconstruction performance under noisy measurements, and the 3-D Fresnel experiments further indicate its potential for practical imaging applications. The related code is available at https://github.com/Physics-driven-methods.
Physics-informed neural networks (PINNs) encounter ill-posed optimization, loss competition, and parameter compensation in partial differential equation (PDE) inverse problems. Transfer learning can reuse representations from source tasks, but direct fine-tuning may introduce negative transfer when dominant physical mechanisms, governing parameters, or observation noise differ between source and target domains: the model achieves low field error yet recovers incorrect target physical parameters. To mitigate, we propose Target-Guided Selective Reweighting PINN (TGSR-PINN), a target-evidence-driven representation correction method for PINN inverse transfer learning. TGSR-PINN transfers only the weights and biases from the source PINN, while target physical parameters are independently initialized; after a short target-adaptation phase, the method computes neuron target scores using first-order Taylor sensitivity and pre-activation variance on fixed scoring batches, and converts evidence associated with low-scoring neurons into continuous weak-adaptation signals via a Gaussian mixture model (GMM) with rank fallback. TGSR-PINN then applies selective soft decay to input weight rows and biases of low-scoring neurons instead of hard pruning or random resetting. In experiments, TGSR-PINN improves target parameter recovery while maintaining comparable field accuracy in the high-P\'{e}clet 2D advection-diffusion task and in the Allen--Cahn to Burgers cross-PDE-family transfer task; a 5%-noise reaction--diffusion case provides supplementary evidence under milder source-target mismatch. Ablation studies suggest that neuron target scoring, weak-adaptation signal estimation, layer protection, and selective soft decay jointly contribute to the benefits.
Non-intrusive optical measurement techniques are widely used to obtain high-resolution pressure, temperature, and velocity fields, but they often suffer from random data loss caused by geometric obstruction, surface reflection, or illumination non-uniformity. Conventional reconstruction methods usually depend on high-fidelity CFD priors or large paired datasets, which limits their flexibility for arbitrary missing patterns and scarce experimental samples. This study proposes a Physics-Informed Multi-Scale Resampled Denoising Diffusion Probabilistic Model (MSR-DDPM) for missing-data reconstruction in optical flow measurements. The method shifts the reconstruction paradigm from deterministic mapping to probabilistic distribution modeling. Three features are introduced: a multi-scale hierarchical reconstruction strategy that reduces computational cost by 56% and improves stability; embedded physics-informed constraints, including divergence and gradient-continuity terms, to enhance physical consistency and reduce dependence on large training datasets; and an optional conditional diffusion module that incorporates auxiliary low-fidelity data for extreme missing-data scenarios. The framework is validated for incompressible velocity-field imputation using turbine cascade passage data, demonstrating its ability to recover complex unsteady flow structures. For missing ratios of 10%-45%, MSR-DDPM achieves high-accuracy reconstruction with Symmetric Mean Absolute Percentage Error (SMAPE) below 6% without auxiliary data. Under more severe missing ratios of 45%-65%, auxiliary guidance substantially recovers flow details and reduces reconstruction errors by approximately 60%. These results indicate that MSR-DDPM offers a flexible, data-efficient, and physically consistent solution for missing-data imputation in complex experimental flow measurements.
Bo Yu, Pingting Chen, JunKui Mao· Journal of turbomachinery· 0 citations
Due to the scarcity of industrial fault data, deep learning-based fault diagnosis models often struggle to fully learn fault features under few-shot conditions, leading to degraded diagnostic performance. To address this issue, this article, for the first time, proposes a physics-informed Fourier prior-guided denoising diffusion probabilistic model (FPG-DDPM) for fault sample generation. The proposed model introduces theoretical fault characteristic frequencies and fault labels as dual-condition inputs, enabling the generated samples to exhibit clear class discriminability and fault-related frequency characteristics. In terms of model structure, a Mamba module is introduced to enhance the capture of global features and long-range contextual dependencies in time–frequency representations. Meanwhile, Fourier prior guidance is introduced during the reverse denoising process to dynamically constrain and correct the noise predicted by the model, thereby improving the accuracy and physical plausibility of the generated samples. To evaluate the effectiveness and physical plausibility of the generated data in downstream diagnosis tasks, layer-wise relevance propagation (LRP) is introduced. By comparing the classification decision basis of real and generated samples in the CNN diagnostic model, LRP provides interpretable validation for the evaluation of generated data. Experimental results on the Case Western Reserve University (CWRU) and Southeast University (SEU) datasets show that fault diagnosis models trained with augmented datasets achieve accuracies of 99.96% and 98.82%, respectively, demonstrating that the proposed method can improve fault diagnosis performance under few-shot learning conditions.
Jing Yang, Ming Lv, Jie Zhang· IEEE Sensors Journal· 0 citations