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Physics-informed neural networks (PINNs) commonly use the mean squared error (MSE) as the loss function. However, this MSE is sensitive to high-residual regions and noise, often causing nonconvergence, overfitting, and loss imbalance during training. To address these challenges, we propose a Huber+ that combines the robustness of the Huber loss with a residual-driven weighting mechanism. The Huber loss transitions smoothly from the MSE for small residuals to the mean absolute error for large residuals, enhancing robustness and accuracy. Furthermore, the dynamic weighting mechanism adaptively adjusts loss weights on the basis of residual variations at each training point, effectively mitigating loss imbalance and enabling PINNs to focus on high-residual regions. To validate the effectiveness of the proposed method, we conduct comparative experiments, ablation studies, and noise sensitivity tests on the Allen–Cahn equation, the Burgers equation, and the Helmholtz equation. The experimental results show that the proposed strategy improves both accuracy and convergence speed.

Despite its rapid development, Physics-Informed Neural Network (PINN)-based computational solid mechanics is still in its infancy. In PINN, the loss function plays a critical role that significantly influences the performance of the predictions. In this paper, by using the Least Squares Weighted Residual (LSWR) method, we proposed a modified loss function, namely the LSWR loss function, which is tailored to a dimensionless form with only one manually determined parameter. Based on the LSWR loss function, an advanced PINN technique is developed for computational 2D and 3D solid mechanics. The performance of the proposed PINN technique with the LSWR loss function is tested through 2D and 3D (geometrically nonlinear) problems. Thoroughly studies and comparisons are conducted between the two existing loss functions, the energy-based loss function and the collocation loss function, and the proposed LSWR loss function. Through numerical experiments, we show that the PINN based on the LSWR loss function is effective, robust, and accurate for predicting both the displacement and stress fields. The source codes for the numerical examples in this work are available at https://github.com/JinshuaiBai/LSWR_loss_function_PINN/ .

Summary Significant advances in both mathematical and molecular approaches in ecology offer unprecedented opportunities to describe and understand ecosystem functioning. Ecological networks describe interactions between species, the underlying structure of communities and the function and stability of ecosystems. They provide the ability to assess the robustness of complex ecological communities to species loss, as well as a novel way of guiding restoration. However, empirically quantifying the interactions between entire communities remains a significant challenge. Concomitantly, advances in DNA sequencing technologies are resolving previously intractable questions in functional and taxonomic biodiversity and provide enormous potential to determine hitherto difficult to observe species interactions. Combining DNA metabarcoding approaches with ecological network analysis presents important new opportunities for understanding large‐scale ecological and evolutionary processes, as well as providing powerful tools for building ecosystems that are resilient to environmental change. We propose a novel ‘nested tagging’ metabarcoding approach for the rapid construction of large, phylogenetically structured species‐interaction networks. Taking tree–insect–parasitoid ecological networks as an illustration, we show how measures of network robustness, constructed using DNA metabarcoding, can be used to determine the consequences of tree species loss within forests, and forest habitat loss within wider landscapes. By determining which species and habitats are important to network integrity, we propose new directions for forest management. Merging metabarcoding with ecological network analysis provides a revolutionary opportunity to construct some of the largest, phylogenetically structured species‐interaction networks to date, providing new ways to: (i) monitor biodiversity and ecosystem functioning; (ii) assess the robustness of interacting communities to species loss; and (iii) build ecosystems that are more resilient to environmental change. A lay summary is available for this article. Lay Summary

The advent of technology including big data has allowed machine learning technology to strengthen its place in solving different science and engineering complex problems. Conventional deep machine learning algorithms work as a black box while dealing with various complex physics-driven problems. This problem can be reduced by integrating the physical laws with machine learning algorithms to ensure the developed models are complied with the physics and are potentially more explainable. This physics-informed machine learning (PIML) approach allows the integration of physical laws in the form of PDEs into the loss function of the neural network, hence, constraining the training of the complex problems based on both the physical, experimental, and mathematical boundaries. This, hence, allows the development of a more general predictive model for different science, engineering, and optimization tasks. Considering such advancements in the machine learning domain, this review presents the systematic progress in the development of integrating physics into the neural networks and recent applications in solving various forward and inverse problems in science and engineering. This paper can serve as a reference for the researchers, developers, and users to get all information they need before developing, implementing, and deploying AI models and smart systems that are equipped with the PIML methodology. It highlights the benefits and points out its limitations and recommendations for further development. The review also compares the traditional data-driven machine learning and PIML approach in dealing with the physics of complex problems. In general, the PIML has been found to provide consistent results with the exact solutions and physical nature of the system. However, similar to other AI system development, a more robust and complex AI algorithm requires more computational power which is also the case in PIML development and implementation. It should be noted that different terminologies such as physics-informed neural networks (PINN), science-informed neural networks, physics-inspired neural networks, and physics-constrained neural networks have been used in the literature that describes the very similar concept of integrating physical laws with machine intelligence. For consistency, we use the PIML term throughout this paper which covers all listed terminologies in this regard.

Additive coefficient models generalize linear regression models by assuming that the relationship between the response and some covariates is linear, while their regression coefficients are additive functions. Because of its advantages in dealing with the “curse of dimensionality”, additive coefficient models gain a lot of attention. The commonly used estimation methods for additive coefficient models are not robust against high leverage points. To circumvent this difficulty, we develop a robust variable selection procedure based on the exponential squared loss function and group penalty for the additive coefficient models, which can tackle outliers in the response and covariates simultaneously. Under some regularity conditions, we show that the oracle estimator is a local solution of the proposed method. Furthermore, we apply the local linear approximation and minorization-maximization algorithm for the implementation of the proposed estimator. Meanwhile, we propose a data-driven procedure to select the tuning parameters. Simulation studies and an application to a plasma beta-carotene level data set illustrate that the proposed method can offer more reliable results than other existing methods in contamination schemes.

A large body of research shows that biodiversity loss can reduce ecosystem functioning. However, much of the evidence for this relationship is drawn from biodiversity–ecosystem functioning experiments in which biodiversity loss is simulated by randomly assembling communities of varying species diversity, and ecosystem functions are measured. This random assembly has led some ecologists to question the relevance of biodiversity experiments to real-world ecosystems, where community assembly or disassembly may be non-random and influenced by external drivers, such as climate, soil conditions or land use. Here, we compare data from real-world grassland plant communities with data from two of the largest and longest-running grassland biodiversity experiments (the Jena Experiment in Germany and BioDIV in the United States) in terms of their taxonomic, functional and phylogenetic diversity and functional-trait composition. We found that plant communities of biodiversity experiments cover almost all of the multivariate variation of the real-world communities, while also containing community types that are not currently observed in the real world. Moreover, they have greater variance in their compositional features than their real-world counterparts. We then re-analysed a subset of experimental data that included only ecologically realistic communities (that is, those comparable to real-world communities). For 10 out of 12 biodiversity–ecosystem functioning relationships, biodiversity effects did not differ significantly between the full dataset of biodiversity experiments and the ecologically realistic subset of experimental communities. Although we do not provide direct evidence for strong or consistent biodiversity–ecosystem functioning relationships in real-world communities, our results demonstrate that the results of biodiversity experiments are largely insensitive to the exclusion of unrealistic communities and that the conclusions drawn from biodiversity experiments are generally robust. By comparing data from real-world grassland communities with data from two of the longest-running grassland biodiversity–ecosystem functioning experiments, the authors show that conclusions derived from experimental systems are robust to the removal of unrealistic experimental communities.

Label noise has been broadly observed in real-world datasets. To mitigate the negative impact of overfitting to label noise for deep models, effective strategies (e.g., re-weighting, or loss rectification) have been broadly applied in prevailing approaches, which have been generally learned under the meta-learning scenario. Despite the robustness of noise achieved by the probabilistic meta-learning models, they usually suffer from model collapse that degenerates generalization performance. In this paper, we propose variational rectification inference (VRI) to formulate the adaptive rectification for loss functions as an amortized variational inference problem and derive the evidence lower bound under the meta-learning framework. Specifically, VRI is constructed as a hierarchical Bayes by treating the rectifying vector as a latent variable, which can rectify the loss of the noisy sample with the extra randomness regularization and is, therefore, more robust to label noise. To achieve the inference of the rectifying vector, we approximate its conditional posterior with an amortization meta-network. By introducing the variational term in VRI, the conditional posterior is estimated accurately and avoids collapsing to a Dirac delta function, which can significantly improve the generalization performance. The elaborated meta-network and prior network adhere to the smoothness assumption, enabling the generation of reliable rectification vectors. Given a set of clean meta-data, VRI can be efficiently meta-learned within the bi-level optimization programming. Besides, theoretical analysis guarantees that the meta-network can be efficiently learned with our algorithm. Comprehensive comparison experiments and analyses validate its effectiveness for robust learning with noisy labels, particularly in the presence of open-set noise.

Although circadian oscillators in diverse eukaryotes all depend on interlinked transcriptional feedback loops, specific components are not conserved across higher taxa. Moreover, the circadian network in the model plant Arabidopsis thaliana is notably more complex than those found in animals and fungi. Here, we combine mathematical modeling and experimental approaches to investigate the functions of two classes of Myb-like transcription factors that antagonistically regulate common target genes. Both CCA1/LHY- and RVE8-clade factors bind directly to the same cis-element, but the former proteins act primarily as repressors, while the latter act primarily as activators of gene expression. We find that simulation of either type of loss-of-function mutant recapitulates clock phenotypes previously reported in mutant plants, while simulated simultaneous loss of both type of factors largely rescues circadian phase at the expense of rhythmic amplitude. In accord with this prediction, we find that plants mutant for both activator- and repressor-type Mybs have near-normal circadian phase and period but reduced rhythmic amplitude. Although these mutants exhibit robust rhythms when grown at mild temperatures, they are largely arrhythmic at physiologically relevant but nonoptimal temperatures. LHY- and RVE8-type Mybs are found in separate clades across the land plant lineage and even in some unicellular green algae, suggesting that they both may have functioned in even the earliest arising plant circadian oscillators. Our data suggest that the complexity of the plant circadian network may have arisen to provide rhythmic robustness across the range of environmental extremes to which plants, as sessile organisms, are regularly subjected.

Robustness is a desirable property for many statistical techniques. As an important measure of robustness, the breakdown point has been widely used for regression problems and many other settings. Despite the existing development, we observe that the standard breakdown point criterion is not directly applicable for many classification problems. In this paper, we propose a new breakdown point criterion, namely angular breakdown point, to better quantify the robustness of different classification methods. Using this new breakdown point criterion, we study the robustness of binary large margin classification techniques, although the idea is applicable to general classification methods. Both bounded and unbounded loss functions with linear and kernel learning are considered. These studies provide useful insights on the robustness of different classification methods. Numerical results further confirm our theoretical findings.

The purpose of the subspace clustering approach is to discover the similarity between samples by learning a self-representation matrix, and it has been widely employed in machine learning and pattern recognition. Most existing subspace clustering techniques discover subspace structures from raw data and simply adopt L2 loss to characterize the reconstruction error. To break through these limitations, a novel robust model named Feature extraction and Cauchy loss function-based Subspace Clustering (FCSC) is proposed. FCSC performs low dimensional and low-rank feature extraction at the same time, as well as processing large noise in the data to generate a more ideal similarity matrix. Furthermore, we provide an efficient iterative strategy to solve the resultant problem. Extensive experiments on benchmark datasets confirm its superiority in the robustness of some advanced subspace clustering algorithms.