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This paper presents a novel physics‐informed neural network (PINN) approach for developing pedotransfer functions (PTFs) to predict continuous soil water retention curves (SWRCs) based on soil textural fractions, organic carbon content, and bulk density. In contrast to conventional parametric PTFs developed for specific SWRC models, the PINN learns a non‐specific form of the SWRC from both measurements and physical constraints imposed during the training process. This approach allows the estimated SWRC to maintain its physical integrity from saturation to oven‐dry conditions, even in scenarios with sparse data. The new approach is particularly effective for tackling the challenges encountered in developing PTFs on large SWRC data sets, which often have an imbalance toward the wet‐end (pF≤4.2$pF\\mathit{\\le }4.2$ ) and include numerous samples with limited and unevenly distributed measurements, many of which do not meet the requirements to fit traditional SWRC models. We compared the performance of the PINN with that of a conventional physics‐agnostic neural network using a data set of 4,200 soil samples. While both networks performed similarly at the wet‐end where data are abundant, with RMSE values of around 0.041 m3 m−3, the PINN excelled at the dry‐end (pF>4.2$pF > 4.2$ ) where data are sparse and unevenly distributed, achieving a normalized RMSE of 0.172 (RMSE = 0.0045 m3 m−3) compared to a normalized RMSE of 0.522 (RMSE = 0.0136 m3 m−3) for the conventional neural network. The SWRC derived from the PINN is differentiable with respect to matric potential, making it well‐suited for integration into models of water flow in the unsaturated zone. Key Points A novel physics‐informed machine learning method for developing continuous soil pedotransfer functions is introduced The method is suitable for data sets of SWRC with imbalanced and incomplete wet and dry measurements The SWRC derived from PINN is differentiable with respect to matric potential and can be integrated into numerical algorithms

Terahertz (THz) metamaterials with high‐figure‐of‐merit (high‐FoM) performance resonance are essential for advancing sensors, detectors, and imagers. Conventional designs focus on symmetric or low‐asymmetry geometric structures, leaving high‐asymmetry designs largely unexplored due to the inefficiency of trial‐and‐error‐based rational design. Recent deep learning techniques offer automation and acceleration but are constrained by the need for large datasets inherent to their data‐driven nature. Here, a novel prior knowledge‐guided generative model augmented by a physics‐constrained active learning mechanism to design high‐asymmetry metamaterials. An advanced diffusion model learns features from a small set of classical structures with high‐FoM THz resonance and generates new high‐asymmetry structures. To mitigate the limited number of classical structures, the generated high‐asymmetry structures are actively selected and integrated into the initial training dataset based on their physical characteristics. Experimental results demonstrate the superior resonance performance of the generated high‐asymmetry metamaterials over classical designs, exhibiting improvements exceeding 30% in key resonance metrics. Remarkably, this performance is attained using only 68 classical structures as the initial training dataset, significantly reducing the data requirements for deep learning‐based metamaterial design. The proposed scheme for generating high‐asymmetry structures provides a new effective and efficient solution for high‐FoM resonance, expanding applications in high‐sensitivity THz metadevices. A prior knowledge‐guided diffusion model augmented by physics‐constrained active learning is developed to design high‐asymmetry terahertz metamaterials. Trained on only a small set of classical structures, the model efficiently generates new high‐metrics designs. Experimental results confirm notable improvements and reveal multi‐resonance phenomena, highlighting the method's capability to overcome data limitations and discover novel metamaterial solutions.

PurposeAccurate fault diagnosis is essential for ensuring the safety and reliability of engineering systems. However, many artificial intelligence (AI)–based diagnostic models rely on purely data-driven black-box approaches that may produce predictions inconsistent with established physical behaviour, limiting confidence in safety-sensitive applications. This study proposes a unified framework that integrates predictive accuracy, physical consistency and interpretability within a single learning formulation.Design/methodology/approachA physics-informed and explainable neural network framework is developed for vibration-based fault severity estimation. Physically meaningful statistical features such as RMS energy, peak-to-peak amplitude and kurtosis, are used as model inputs. Physical knowledge is embedded through a monotonicity-based loss constraint that penalises violations of the expected relationship between vibration energy and fault severity. A combined objective balances prediction error and physics-based regularisation. Global permutation feature importance is applied to assess alignment between learned feature relevance and physical expectations. The framework is evaluated using physics-inspired synthetic vibration data under controlled conditions, supported by cross-validation and quantitative monotonicity assessment.FindingsCompared with an unconstrained neural network, the physics-informed model achieves lower prediction error, smoother convergence behaviour and substantially reduced monotonicity violations. The embedded constraint acts as a structural regulariser while preserving model flexibility. Explainability analysis indicates closer alignment between feature relevance and physically meaningful vibration indicators.Originality/valueThe study presents a reproducible framework that integrates loss-level physics constraints with global explainability analysis for fault severity diagnosis. By embedding physical admissibility directly into optimisation, the approach contributes toward the development of more transparent and physically grounded AI systems for engineering diagnostics.A conceptual flow diagram shows physics informed neural network processing vibration signals to produce interpretable fault.The conceptual flow diagram is arranged from left to right in three sections. The left section titled “PHYSICS INSPIRED VIBRATION SIGNALS AND FEATURES” contains three stacked rectangular panels. The top panel labeled “HEALTHY (Blue)” shows a low amplitude waveform, the middle panel labeled “INCIPIENT FAULT (Orange)” shows a moderate amplitude waveform, and the bottom panel labeled “SEVERE FAULT (Red)” shows a high amplitude waveform. To the right of these panels, three vertical bar charts labeled “R M S Energy”, “Peak to Peak Amplitude”, and “Kurtosis” show increasing bar heights from healthy to severe. A large downward arrow beside the panels indicates “INCREASING FAULT SEVERITY, SIGNAL AMPLITUDE and ENERGY”. A solid rightward arrow runs from this section to the central section titled “PHYSICS INFORMED NEURAL NETWORK (P I N N) FRAMEWORK”. In the central section, a box labeled “LOSS” with a downward arrow connects to “PHYSICS BASED REGULARIZATION (Monotonic Energy Severity Constraint)”. Below, a neural network diagram shows input nodes on the left connected through multiple hidden layers to output nodes on the right, with green highlighted nodes indicating the physics informed constraint. A crossed out black box icon with a red X below is labeled “AVOID UNPHYSICAL, UNINTERPRETABLE BLACK BOX MODEL”. A solid rightward arrow runs from this section to the right section titled “ROBUST and INTERPRETABLE DIAGNOSTIC OUTPUTS”. The top panel in this section labeled “FAULT SEVERITY SCORE” shows a horizontal gauge with segments labeled “Low (Healthy)”, “Medium (Incipient)”, and “High (Severe)”, with a marker positioned around 1.69 and a label “Status”. Below, a panel labeled “GLOBAL FEATURE IMPORTANCE (PHYSICALLY CONSISTENT)” displays horizontal bars for “R M S Energy”, “Peak to Peak Amplitude”, and “Kurtosis”, increasing from healthy to severe, with a note “Consistent Evolution with Fault Progression”.

This study addresses energy-response prediction in sheet metal forming under small-data conditions, where conventional simulation-based approaches are computationally expensive and data acquisition is limited. We propose an Energy-Informed Neural Network (EINN) framework that integrates energy consistency constraints and a Fenchel–Young duality-based loss to enforce physically consistent learning without relying on explicit governing equations. Using a dataset generated from 54 finite element simulations across 18 materials and three friction conditions, the proposed model demonstrates significant performance improvements. Specifically, EINN achieves an RMSE of 0.0096, MAE of 0.0065, and R2 of 0.9778, corresponding to approximately a 48% reduction in RMSE compared to the best baseline model. Compared to an energy-constrained neural network without the Fenchel–Young term, prediction error is reduced by approximately 50% with substantially improved stability. These results indicate that embedding energy-consistent dual structures enhances both prediction accuracy and robustness, providing a practical surrogate modeling approach for process optimization in sheet metal forming under limited data availability.

Inverse problems are fundamental in tasks like computer vision, where model parameters need to be estimated from observable data. We propose a novel approach that combines physics-constrained deep learning with automatic differentiation (AD) to tackle inverse problems in such as computer vision. Our method integrates variational approaches with deep learning-based algorithms by leveraging deep neural networks and AD. To handle nonconvex variational models, we employ the operator splitting technique, decomposing them into simpler sub-problems solvable using deep neural networks and AD. By combining physics-informed constraints, deep learning capabilities and operator splitting, our approach offers a promising framework for addressing inverse problems in computer vision. It bridges the gap between traditional variational methods and deep learning, providing effective solutions in the presence of noise. The integration of physics-based priors and deep learning enhances accuracy and robustness in estimating solutions, advancing the field of computer vision.

Recently, the development of deep learning has accomplished unbelievable success in many fields, especially in scientific computational fields. And almost all computational problems and physical phenomena can be described by partial differential equations. In this work, we proposed two potential high-order geometric flows. Motivation by the physical-information neural networks and the traditional level set method (LSM), we have integrated deep neural networks and LSM to make the proposed method more robust and efficient. Also, to test the sensitivity of the system to different input data, we set up three sets of initial conditions to test the model. Furthermore, numerical experiments on different input data are implemented to demonstrate the effectiveness and superiority of the proposed models compared to the state-of-the-art approach.

Accurate and physically consistent state prediction is essential for shipboard power systems (SPS) operating under dynamic conditions. However, purely data-driven models often exhibit degraded robustness and physically inconsistent outputs when exposed to transient disturbances or limited data coverage. To address these limitations, this paper proposes a physics-constrained hybrid prediction model that integrates a convolutional neural network–bidirectional long short-term memory (CNN–BiLSTM) architecture with wide residual connections (WRC) and a physics-constrained loss (PCL). The proposed modeling approach combines real operational measurement data with high-resolution simulation data to enhance data diversity and improve generalization capability. The CNN–BiLSTM structure captures nonlinear temporal dependencies, while the WRC preserves critical low-level transient electrical features during deep temporal modeling. In addition, multiple physical constraints, including power balance, voltage conversion relationships, and battery state-of-charge (SOC) dynamics, are incorporated into the training process to enforce physically consistent predictions. The model is validated using charging and discharging experiments on a laboratory-scale SPS under both steady-state and transient conditions. Comparative results demonstrate that the proposed approach achieves higher prediction accuracy, improved dynamic stability, and faster recovery following disturbances compared with conventional data-driven models. These results indicate that physics-constrained deep learning provides an effective and interpretable modeling framework for SPS state prediction, supporting digital twin-oriented monitoring and real-time prediction applications.

Two-dimensional transition metal carbides and nitrides, known collectively as MXenes, are attractive photocatalyst candidates because their surface chemistry and atomic composition can be tuned over a wide compositional window. A crucial design quantity is the electronic bandgap, which selects whether a given MXene couples with solar radiation and aligns with the redox levels of water splitting. High-fidelity bandgap calculations using the PBE0 hybrid functional are computationally expensive, which has motivated several machine learning surrogates. To the best of our knowledge, this is the first study to integrate a continuous-depth Neural Ordinary Differential Equation backbone with multi-fidelity Δ learning, distribution-free split-conformal calibration, and uncertainty-aware Pareto screening into a single mathematically grounded pipeline for MXene bandgap prediction. In this work, we develop a physics-constrained neural ordinary differential equation (PC-NODE) that predicts MXene bandgaps from a compact 34-dimensional descriptor set, without relying on the density of states. The model couples a classifier head for the metal/semiconductor decision with a regression head for the gap magnitude, and enforces three physically motivated properties: non-negativity of the predicted gap and monotonicity between the low-fidelity Perdew-Burke-Ernzerhof (PBE) and the high-fidelity PBE0 estimates are obtained exactly through a softplus-parameterised Δ learning construction, while a hurdle coupling that drives metal predictions towards zero is enforced via a quadratic penalty and verified empirically. In short, two of the three physical constraints are guaranteed by construction, and the third is approximately enforced and verified empirically; the same distinction is maintained consistently in the methodology, the constraint audit and the conclusion. Trained on the 4356-structure MXgap database, a ten-seed ensemble reaches a mean absolute error of 0.186 eV (per-seed 0.206±0.006 eV) and a coefficient of determination R2=0.880 on the semiconductor test subset, with a classifier accuracy of 0.856 and a Receiver Operating Characteristic Area Under the Curve (ROC-AUC) of 0.925. A split-conformal calibration step then delivers prediction intervals whose empirical coverage matches the 90% target within 0.5 percentage points. Finally, an uncertainty-aware Pareto screening step applies the trained surrogate to a held-out subset of 396 lanthanum-based MXenes and identifies 74 candidates inside the photocatalytic water splitting window [1.23, 3.10] eV. The framework offers a mathematically grounded, data-efficient alternative to feature-heavy pipelines and is reproducible from the open MXgap resource.

Almost sophisticated physical phenomena and computational problems arise as variational problems. Recently, the development of neural networks (NNs), which has accomplished unbelievable success in many fields, especially in scientific computational fields. And almost sophisticated computational problems of physical phenomena can be viewed as a variational or PDE problem. In this work, we proposed learning Laplace-Beltrami-operator with physics-constrained machine learning and automatic differentiation to discover the inverse variational problems from noisy data. Also, the hidden fields are approximated by neural networks. Meanwhile, the neural networks and traditional high-order inverse variational problems are integrated, to make the traditional variational problem gain stronger vitality once again. We propose a way for sensitivity analysis, utilizing the automatic differentiation mechanism embedded in the framework. We propose a neural network approach to approximate unknown functions based on curvature regularity to learn the high order inverse variational problems under noisy data. Also, theoretically, our framework is flexible to adapt to the different high-order curvature-based variational problems, then, the NNs are used to solve the problems to improve computational efficiency. Furthermore, several experiments with different initial data and different noise levels are implemented to demonstrate the effectiveness and superiority of the proposed models. Our experiments confirm this property.

The proliferation of sophisticated counterfeiting poses critical challenges to global security and commerce, with annual losses exceeding $2.2 trillion. This paper presents a novel physics-constrained deep learning framework for high-precision security ink colorimetry, integrating three key innovations: a physics-informed neural architecture achieving unprecedented color prediction accuracy (CIEDE2000 (ΔE00): 0.70 ± 0.08, p < 0.001), advanced attention mechanisms improving feature extraction efficiency by 58.3%, and a Bayesian optimization framework ensuring robust parameter tuning. Validated across 1500 industrial samples under varying conditions (±2 °C, 30–80% RH), this system demonstrates substantial improvements in production efficiency with a 50% reduction in rejections, a 35% decrease in calibration time, and 96.7% color gamut coverage. These achievements establish new benchmarks for security printing applications and provide scalable solutions for next-generation anti-counterfeiting technologies, offering a promising outlook for the future.