Dongxiao Zhang

Jian Li, Dongxiao Zhang, Tianhao He et al.

Uncertainty quantification (UQ) of subsurface two-phase flow usually requires numerous executions of forward simulations under varying conditions. In this work, a novel coupled theory-guided neural network (TgNN) based surrogate model is built to facilitate computation efficiency under the premise of satisfactory accuracy. The core notion of this proposed method is to bridge two separate blocks on top of an overall network. They underlie the TgNN model in a coupled form, which reflects the coupling nature of pressure and water saturation in the two-phase flow equation. The TgNN model not only relies on labeled data, but also incorporates underlying scientific theory and experiential rules (e.g., governing equations, stochastic parameter fields, boundary and initial conditions, well conditions, and expert knowledge) as additional components into the loss function. The performance of the TgNN-based surrogate model for two-phase flow problems is tested by different numbers of labeled data and collocation points, as well as the existence of data noise. The proposed TgNN-based surrogate model offers an effective way to solve the coupled nonlinear two-phase flow problem and demonstrates good accuracy and strong robustness when compared with the purely data-driven surrogate model. By combining the accurate TgNN-based surrogate model with the Monte Carlo method, UQ tasks can be performed at a minimum cost to evaluate statistical quantities. Since the heterogeneity of the random fields strongly impacts the results of the surrogate model, corresponding variance and correlation length are added to the input of the neural network to maintain its predictive capacity. The results show that the TgNN-based surrogate model achieves satisfactory accuracy, stability, and efficiency in UQ problems of subsurface two-phase flow.

Hao Xu, Junsheng Zeng, Dongxiao Zhang

Data-driven discovery of PDEs has made tremendous progress recently, and many canonical PDEs have been discovered successfully for proof-of-concept. However, determining the most proper PDE without prior references remains challenging in terms of practical applications. In this work, a physics-informed information criterion (PIC) is proposed to measure the parsimony and precision of the discovered PDE synthetically. The proposed PIC achieves state-of-the-art robustness to highly noisy and sparse data on seven canonical PDEs from different physical scenes, which confirms its ability to handle difficult situations. The PIC is also employed to discover unrevealed macroscale governing equations from microscopic simulation data in an actual physical scene. The results show that the discovered macroscale PDE is precise and parsimonious, and satisfies underlying symmetries, which facilitates understanding and simulation of the physical process. The proposition of PIC enables practical applications of PDE discovery in discovering unrevealed governing equations in broader physical scenes.

Qiang Zheng, Xiaoguang Yin, Dongxiao Zhang

The Li-ion battery is a complex physicochemical system that generally takes applied current as input and terminal voltage as output. The mappings from current to voltage can be described by several kinds of models, such as accurate but inefficient physics-based models, and efficient but sometimes inaccurate equivalent circuit and black-box models. To realize accuracy and efficiency simultaneously in battery modeling, we propose to build a data-driven surrogate for a battery system while incorporating the underlying physics as constraints. In this work, we innovatively treat the functional mapping from current curve to terminal voltage as a composite of operators, which is approximated by the powerful deep operator network (DeepONet). Its learning capability is firstly verified through a predictive test for Li-ion concentration at two electrodes. In this experiment, the physics-informed DeepONet is found to be more robust than the purely data-driven DeepONet, especially in temporal extrapolation scenarios. A composite surrogate is then constructed for mapping current curve and solid diffusivity to terminal voltage with three operator networks, in which two parallel physics-informed DeepONets are firstly used to predict Li-ion concentration at two electrodes, and then based on their surface values, a DeepONet is built to give terminal voltage predictions. Since the surrogate is differentiable anywhere, it is endowed with the ability to learn from data directly, which was validated by using terminal voltage measurements to estimate input parameters. The proposed surrogate built upon operator networks possesses great potential to be applied in on-board scenarios, such as battery management system, since it integrates efficiency and accuracy by incorporating underlying physics, and also leaves an interface for model refinement through a totally differentiable model structure.

Hao Xu, Jinglong Lin, Dongxiao Zhang et al.

A new research framework is proposed to incorporate machine learning techniques into the field of experimental chemistry to facilitate chromatographic enantioseparation. A documentary dataset of chiral molecular retention times (CMRT dataset) in high-performance liquid chromatography is established to handle the challenge of data acquisition. Based on the CMRT dataset, a quantile geometry-enhanced graph neural network is proposed to learn the molecular structure-retention time relationship, which shows a satisfactory predictive ability for enantiomers. The domain knowledge of chromatography is incorporated into the machine learning model to achieve multi-column prediction, which paves the way for chromatographic enantioseparation prediction by calculating the separation probability. Experiments confirm that the proposed research framework works well in retention time prediction and chromatographic enantioseparation facilitation, which sheds light on the application of machine learning techniques to the experimental scene and improves the efficiency of experimenters to speed up scientific discovery.

Nanzhe Wang, Haibin Chang, Xiangzhao Kong et al.

To maximize the economic benefits of geothermal energy production, it is essential to optimize geothermal reservoir management strategies, in which geologic uncertainty should be considered. In this work, we propose a closed-loop optimization framework, based on deep learning surrogates, for the well control optimization of geothermal reservoirs. In this framework, we construct a hybrid convolution-recurrent neural network surrogate, which combines the convolution neural network (CNN) and long short-term memory (LSTM) recurrent network. The convolution structure can extract spatial information of geologic parameter fields and the recurrent structure can approximate sequence-to-sequence mapping. The trained model can predict time-varying production responses (rate, temperature, etc.) for cases with different permeability fields and well control sequences. In the closed-loop optimization framework, production optimization based on the differential evolution (DE) algorithm, and data assimilation based on the iterative ensemble smoother (IES), are performed alternately to achieve real-time well control optimization and geologic parameter estimation as the production proceeds. In addition, the averaged objective function over the ensemble of geologic parameter estimations is adopted to consider geologic uncertainty in the optimization process. Several geothermal reservoir development cases are designed to test the performance of the proposed production optimization framework. The results show that the proposed framework can achieve efficient and effective real-time optimization and data assimilation in the geothermal reservoir production process.

Tianhao He, Haibin Chang, Dongxiao Zhang

Identification of unknown physical processes and parameters of groundwater contaminant sources is a challenging task due to their ill-posed and non-unique nature. Numerous works have focused on determining nonlinear physical processes through model selection methods. However, identifying corresponding nonlinear systems for different physical phenomena using numerical methods can be computationally prohibitive. With the advent of machine learning (ML) algorithms, more efficient surrogate models based on neural networks (NNs) have been developed in various disciplines. In this work, a theory-guided U-net (TgU-net) framework is proposed for surrogate modeling of three-dimensional (3D) groundwater contaminant problems in order to efficiently elucidate their involved processes and unknown parameters. In TgU-net, the underlying governing equations are embedded into the loss function of U-net as soft constraints. For the considered groundwater contaminant problem, sorption is considered to be a potential process of an uncertain type, and three equilibrium sorption isotherm types (i.e., linear, Freundlich, and Langmuir) are considered. Different from traditional approaches in which one model corresponds to one equation, these three sorption types are modeled through only one TgU-net surrogate. The three mentioned sorption terms are integrated into one equation by assigning indicators. Accurate predictions illustrate the satisfactory generalizability and extrapolability of the constructed TgU-net. Furthermore, based on the constructed TgU-net surrogate, a data assimilation method is employed to identify the physical process and parameters simultaneously. This work shows the possibility of governing equation discovery of physical problems that contain multiple and even uncertain processes by using deep learning and data assimilation methods.

Qian Li, Yuxiao Hu, Ye Liu et al.

Classical adversarial attacks for Face Recognition (FR) models typically generate discrete examples for target identity with a single state image. However, such paradigm of point-wise attack exhibits poor generalization against numerous unknown states of identity and can be easily defended. In this paper, by rethinking the inherent relationship between the face of target identity and its variants, we introduce a new pipeline of Generalized Manifold Adversarial Attack (GMAA) to achieve a better attack performance by expanding the attack range. Specifically, this expansion lies on two aspects - GMAA not only expands the target to be attacked from one to many to encourage a good generalization ability for the generated adversarial examples, but it also expands the latter from discrete points to manifold by leveraging the domain knowledge that face expression change can be continuous, which enhances the attack effect as a data augmentation mechanism did. Moreover, we further design a dual supervision with local and global constraints as a minor contribution to improve the visual quality of the generated adversarial examples. We demonstrate the effectiveness of our method based on extensive experiments, and reveal that GMAA promises a semantic continuous adversarial space with a higher generalization ability and visual quality

Mengge Du, Yuntian Chen, Longfeng Nie et al.

Unveiling the underlying governing equations of nonlinear dynamic systems remains a significant challenge. Insufficient prior knowledge hinders the determination of an accurate candidate library, while noisy observations lead to imprecise evaluations, which in turn result in redundant function terms or erroneous equations. This study proposes a framework to robustly uncover open-form partial differential equations (PDEs) from limited and noisy data. The framework operates through two alternating update processes: discovering and embedding. The discovering phase employs symbolic representation and a novel reinforcement learning (RL)-guided hybrid PDE generator to efficiently produce diverse open-form PDEs with tree structures. A neural network-based predictive model fits the system response and serves as the reward evaluator for the generated PDEs. PDEs with higher rewards are utilized to iteratively optimize the generator via the RL strategy and the best-performing PDE is selected by a parameter-free stability metric. The embedding phase integrates the initially identified PDE from the discovering process as a physical constraint into the predictive model for robust training. The traversal of PDE trees automates the construction of the computational graph and the embedding process without human intervention. Numerical experiments demonstrate our framework's capability to uncover governing equations from nonlinear dynamic systems with limited and highly noisy data and outperform other physics-informed neural network-based discovery methods. This work opens new potential for exploring real-world systems with limited understanding.

Mengge Du, Yuntian Chen, Dongxiao Zhang

Imposing physical constraints on neural networks as a method of knowledge embedding has achieved great progress in solving physical problems described by governing equations. However, for many engineering problems, governing equations often have complex forms, including complex partial derivatives or stochastic physical fields, which results in significant inconveniences from the perspective of implementation. In this paper, a scientific machine learning framework, called AutoKE, is proposed, and a reservoir flow problem is taken as an instance to demonstrate that this framework can effectively automate the process of embedding physical knowledge. In AutoKE, an emulator comprised of deep neural networks (DNNs) is built for predicting the physical variables of interest. An arbitrarily complex equation can be parsed and automatically converted into a computational graph through the equation parser module, and the fitness of the emulator to the governing equation is evaluated via automatic differentiation. Furthermore, the fixed weights in the loss function are substituted with adaptive weights by incorporating the Lagrangian dual method. Neural architecture search (NAS) is also introduced into the AutoKE to select an optimal network architecture of the emulator according to the specific problem. Finally, we apply transfer learning to enhance the scalability of the emulator. In experiments, the framework is verified by a series of physical problems in which it can automatically embed physical knowledge into an emulator without heavy hand-coding. The results demonstrate that the emulator can not only make accurate predictions, but also be applied to similar problems with high efficiency via transfer learning.

Hao Xu, Wei Fan, Ambrose C. Taylor et al.

Computational solid mechanics has become an indispensable approach in engineering, and numerical investigation of fracture in composites is essential as composites are widely used in structural applications. Crack evolution in composites is the bridge to elucidate the relationship between the microstructure and fracture performance, but crack-based finite element methods are computationally expensive and time-consuming, limiting their application in computation-intensive scenarios. Here we propose a deep learning framework called Crack-Net, which incorporates the relationship between crack evolution and stress response to predict the fracture process in composites. Trained on a high-precision fracture development dataset generated using the phase field method, Crack-Net demonstrates a remarkable capability to accurately forecast the long-term evolution of crack growth patterns and the stress-strain curve for a given composite design. The Crack-Net captures the essential principle of crack growth, which enables it to handle more complex microstructures such as binary co-continuous structures. Moreover, transfer learning is adopted to further improve the generalization ability of Crack-Net for composite materials with reinforcements of different strengths. The proposed Crack-Net holds great promise for practical applications in engineering and materials science, in which accurate and efficient fracture prediction is crucial for optimizing material performance and microstructural design.

Yuntian Chen, Dongxiao Zhang, Qun Zhao et al.

An algorithm named InterOpt for optimizing operational parameters is proposed based on interpretable machine learning, and is demonstrated via optimization of shale gas development. InterOpt consists of three parts: a neural network is used to construct an emulator of the actual drilling and hydraulic fracturing process in the vector space (i.e., virtual environment); the Sharpley value method in interpretable machine learning is applied to analyzing the impact of geological and operational parameters in each well (i.e., single well feature impact analysis); and ensemble randomized maximum likelihood (EnRML) is conducted to optimize the operational parameters to comprehensively improve the efficiency of shale gas development and reduce the average cost. In the experiment, InterOpt provides different drilling and fracturing plans for each well according to its specific geological conditions, and finally achieved an average cost reduction of 9.7% for a case study with 104 wells.

Hao Xu, Yuntian Chen, Zhenzhong Zeng et al.

Despite the remarkable strides made by AI-driven models in modern precipitation forecasting, these black-box models cannot inherently deepen the comprehension of underlying mechanisms. To address this limitation, we propose an AI-driven knowledge discovery framework known as genetic algorithm-geographic weighted regression (GA-GWR). Our approach seeks to unveil the explicit equations that govern the intricate relationship between precipitation patterns and terrain characteristics in regions marked by complex terrain. Through this AI-driven knowledge discovery, we uncover previously undisclosed explicit equations that shed light on the connection between terrain features and precipitation patterns. These equations demonstrate remarkable accuracy when applied to precipitation data, outperforming conventional empirical models. Notably, our research reveals that the parameters within these equations are dynamic, adapting to evolving climate patterns. Ultimately, the unveiled equations have practical applications, particularly in fine-scale downscaling for precipitation predictions using low-resolution future climate data. This capability offers invaluable insights into the anticipated changes in precipitation patterns across diverse terrains under future climate scenarios, which enhances our ability to address the challenges posed by contemporary climate science.

Jiaxin Gao, Wenbo Hu, Dongxiao Zhang et al.

Electrical energy is essential in today's society. Accurate electrical load forecasting is beneficial for better scheduling of electricity generation and saving electrical energy. In this paper, we propose theory-guided deep-learning load forecasting 2.0 (TgDLF2.0) to solve this issue, which is an improved version of the theory-guided deep-learning framework for load forecasting via ensemble long short-term memory (TgDLF). TgDLF2.0 introduces the deep-learning model Transformer and transfer learning on the basis of dividing the electrical load into dimensionless trends and local fluctuations, which realizes the utilization of domain knowledge, captures the long-term dependency of the load series, and is more appropriate for realistic scenarios with scarce samples. Cross-validation experiments on different districts show that TgDLF2.0 is approximately 16% more accurate than TgDLF and saves more than half of the training time. TgDLF2.0 with 50% weather noise has the same accuracy as TgDLF without noise, which proves its robustness. We also preliminarily mine the interpretability of Transformer in TgDLF2.0, which may provide future potential for better theory guidance. Furthermore, experiments demonstrate that transfer learning can accelerate convergence of the model in half the number of training epochs and achieve better performance.

Hao Xu, Yuntian Chen, Dongxiao Zhang

The interpretability of deep neural networks has attracted increasing attention in recent years, and several methods have been created to interpret the "black box" model. Fundamental limitations remain, however, that impede the pace of understanding the networks, especially the extraction of understandable semantic space. In this work, we introduce the framework of semantic explainable AI (S-XAI), which utilizes row-centered principal component analysis to obtain the common traits from the best combination of superpixels discovered by a genetic algorithm, and extracts understandable semantic spaces on the basis of discovered semantically sensitive neurons and visualization techniques. Statistical interpretation of the semantic space is also provided, and the concept of semantic probability is proposed for the first time. Our experimental results demonstrate that S-XAI is effective in providing a semantic interpretation for the CNN, and offers broad usage, including trustworthiness assessment and semantic sample searching.

Mengge Du, Yuntian Chen, Dongxiao Zhang

The working mechanisms of complex natural systems tend to abide by concise and profound partial differential equations (PDEs). Methods that directly mine equations from data are called PDE discovery, which reveals consistent physical laws and facilitates our adaptive interaction with the natural world. In this paper, an enhanced deep reinforcement-learning framework is proposed to uncover symbolically concise open-form PDEs with little prior knowledge. Particularly, based on a symbol library of basic operators and operands, a structure-aware recurrent neural network agent is designed and seamlessly combined with the sparse regression method to generate concise and open-form PDE expressions. All of the generated PDEs are evaluated by a meticulously designed reward function by balancing fitness to data and parsimony, and updated by the model-based reinforcement learning in an efficient way. Customized constraints and regulations are formulated to guarantee the rationality of PDEs in terms of physics and mathematics. The experiments demonstrate that our framework is capable of mining open-form governing equations of several dynamic systems, even with compound equation terms, fractional structure, and high-order derivatives, with excellent efficiency. Without the need for prior knowledge, this method shows great potential for knowledge discovery in more complicated circumstances with exceptional efficiency and scalability.

Hao Xu, Jinshen Sun, Yuntian Chen et al.

Soil hydraulic functions are fundamental to modelling water flow and transport in vadose-zone hydrology and are central to a wide range of hydrological and geoscientific applications. Yet in practice, these functions are still predominantly specified through expert-designed empirical formulations, such as the Mualem-van Genuchten (MvG) model. Although such models have proved highly influential, their derivation relies on predefined functional assumptions that make it difficult to simultaneously achieve accuracy, compactness, and robustness across diverse soil textures. Here we present a graph-based automated model discovery framework for discovering explicit soil hydraulic functions directly from experimental data. Applied to the original datasets used in the development of the MvG model, the method identifies a concise soil water retention function and its associated unsaturated hydraulic conductivity function whose mathematical structure differs fundamentally from classical empirical forms. Across 249 real soil samples spanning diverse textural classes, the discovered functions achieve more accurate predictions of unsaturated hydraulic conductivity than the MvG model. The fitted parameters also exhibit correlations with soil physical properties. This work demonstrates that data-driven model discovery can move beyond traditional empirical derivation and provide a promising route for developing accurate and explicit constitutive models.

Hao Xu, Yuntian Chen, Dongxiao Zhang

Constitutive models are fundamental to solid mechanics and materials science, underpinning the quantitative description and prediction of material responses under diverse loading conditions. Traditional phenomenological models, which are derived through empirical fitting, often lack generalizability and rely heavily on expert intuition and predefined functional forms. In this work, we propose a graph-based equation discovery framework for the automated discovery of constitutive laws directly from multisource experimental data. This framework expresses equations as directed graphs, where nodes represent operators and variables, edges denote computational relations, and edge features encode parametric dependencies. This enables the generation and optimization of free-form symbolic expressions with undetermined material-specific parameters. Through the proposed framework, we have discovered new constitutive models for strain-rate effects in alloy steel materials and the deformation behavior of lithium metal. Compared with conventional empirical models, these new models exhibit compact analytical structures and achieve higher accuracy. The proposed graph-based equation discovery framework provides a generalizable and interpretable approach for data-driven scientific modelling, particularly in contexts where traditional empirical formulations are inadequate for representing complex physical phenomena.

Dayin Chen, Xiaodan Shi, Mingkun Jiang et al.

Photovoltaic power forecasting (PVPF) is a critical area in time series forecasting (TSF), enabling the efficient utilization of solar energy. With advancements in machine learning and deep learning, various models have been applied to PVPF tasks. However, constructing an optimal predictive architecture for specific PVPF tasks remains challenging, as it requires cross-domain knowledge and significant labor costs. To address this challenge, we introduce AutoPV, a novel framework for the automated search and construction of PVPF models based on neural architecture search (NAS) technology. We develop a brand new NAS search space that incorporates various data processing techniques from state-of-the-art (SOTA) TSF models and typical PVPF deep learning models. The effectiveness of AutoPV is evaluated on diverse PVPF tasks using a dataset from the Daqing Photovoltaic Station in China. Experimental results demonstrate that AutoPV can complete the predictive architecture construction process in a relatively short time, and the newly constructed architecture is superior to SOTA predefined models. This work bridges the gap in applying NAS to TSF problems, assisting non-experts and industries in automatically designing effective PVPF models.

Hao Xu, Yuntian Chen, Dongxiao Zhang

Knowledge constitutes the accumulated understanding and experience that humans use to gain insight into the world. In deep learning, prior knowledge is essential for mitigating shortcomings of data-driven models, such as data dependence, generalization ability, and compliance with constraints. To enable efficient evaluation of the worth of knowledge, we present a framework inspired by interpretable machine learning. Through quantitative experiments, we assess the influence of data volume and estimation range on the worth of knowledge. Our findings elucidate the complex relationship between data and knowledge, including dependence, synergistic, and substitution effects. Our model-agnostic framework can be applied to a variety of common network architectures, providing a comprehensive understanding of the role of prior knowledge in deep learning models. It can also be used to improve the performance of informed machine learning, as well as distinguish improper prior knowledge.

Hao Xu, Yuntian Chen, Chongqing Kang et al.

The global shift towards renewable energy necessitates the development of ultrahigh-voltage (UHV) AC transmission to bridge the gap between remote energy sources and urban demand. While UHV grids offer superior capacity and efficiency, their implementation is often hindered by corona-induced audible noise (AN) and radio interference (RI). Since these emissions must meet strict environmental compliance standards, accurate prediction is vital for the large-scale deployment of UHV infrastructure. Existing engineering practices often rely on empirical laws, in which fixed log-linear structures limit accuracy and extrapolation. Herein, we present a monotonicity-constrained graph symbolic discovery framework, Mono-GraphMD, which uncovers compact, interpretable laws for corona-induced AN and RI. The framework provides mechanistic insight into how nonlinear interactions among the surface gradient, bundle number and diameter govern high-field emissions and enables accurate predictions for both corona-cage data and multicountry real UHV lines with up to 16-bundle conductors. Unlike black-box models, the discovered closed-form laws are highly portable and interpretable, allowing for rapid predictions when applied to various scenarios, thereby facilitating the engineering design process.

Yuxiao Hu, Qian Li, Dongxiao Zhang et al.

Recently, leveraging pre-trained Large Language Models (LLMs) for time series (TS) tasks has gained increasing attention, which involves activating and enhancing LLMs' capabilities. Many methods aim to activate LLMs' capabilities based on token-level alignment but overlook LLMs' inherent strength on natural language processing -- their deep understanding of linguistic logic and structure rather than superficial embedding processing. We propose Context-Alignment, a new paradigm that aligns TS with a linguistic component in the language environments familiar to LLMs to enable LLMs to contextualize and comprehend TS data, thereby activating their capabilities. Specifically, such context-level alignment comprises structural alignment and logical alignment, which is achieved by a Dual-Scale Context-Alignment GNNs (DSCA-GNNs) applied to TS-language multimodal inputs. Structural alignment utilizes dual-scale nodes to describe hierarchical structure in TS-language, enabling LLMs treat long TS data as a whole linguistic component while preserving intrinsic token features. Logical alignment uses directed edges to guide logical relationships, ensuring coherence in the contextual semantics. Demonstration examples prompt are employed to construct Demonstration Examples based Context-Alignment (DECA) following DSCA-GNNs framework. DECA can be flexibly and repeatedly integrated into various layers of pre-trained LLMs to improve awareness of logic and structure, thereby enhancing performance. Extensive experiments show the effectiveness of DECA and the importance of Context-Alignment across tasks, particularly in few-shot and zero-shot forecasting, confirming that Context-Alignment provide powerful prior knowledge on context.

Mengge Du, Yuntian Chen, Zhongzheng Wang et al.

Equation discovery is aimed at directly extracting physical laws from data and has emerged as a pivotal research domain. Previous methods based on symbolic mathematics have achieved substantial advancements, but often require the design of implementation of complex algorithms. In this paper, we introduce a new framework that utilizes natural language-based prompts to guide large language models (LLMs) in automatically mining governing equations from data. Specifically, we first utilize the generation capability of LLMs to generate diverse equations in string form, and then evaluate the generated equations based on observations. In the optimization phase, we propose two alternately iterated strategies to optimize generated equations collaboratively. The first strategy is to take LLMs as a black-box optimizer and achieve equation self-improvement based on historical samples and their performance. The second strategy is to instruct LLMs to perform evolutionary operators for global search. Experiments are extensively conducted on both partial differential equations and ordinary differential equations. Results demonstrate that our framework can discover effective equations to reveal the underlying physical laws under various nonlinear dynamic systems. Further comparisons are made with state-of-the-art models, demonstrating good stability and usability. Our framework substantially lowers the barriers to learning and applying equation discovery techniques, demonstrating the application potential of LLMs in the field of knowledge discovery.

Qian Li, Yuxiao Hu, Yinpeng Dong et al.

Adversarial training is often formulated as a min-max problem, however, concentrating only on the worst adversarial examples causes alternating repetitive confusion of the model, i.e., previously defended or correctly classified samples are not defensible or accurately classifiable in subsequent adversarial training. We characterize such non-ignorable samples as "hiders", which reveal the hidden high-risk regions within the secure area obtained through adversarial training and prevent the model from finding the real worst cases. We demand the model to prevent hiders when defending against adversarial examples for improving accuracy and robustness simultaneously. By rethinking and redefining the min-max optimization problem for adversarial training, we propose a generalized adversarial training algorithm called Hider-Focused Adversarial Training (HFAT). HFAT introduces the iterative evolution optimization strategy to simplify the optimization problem and employs an auxiliary model to reveal hiders, effectively combining the optimization directions of standard adversarial training and prevention hiders. Furthermore, we introduce an adaptive weighting mechanism that facilitates the model in adaptively adjusting its focus between adversarial examples and hiders during different training periods. We demonstrate the effectiveness of our method based on extensive experiments, and ensure that HFAT can provide higher robustness and accuracy.

Dayin Chen, Xiaodan Shi, Haoran Zhang et al.

Enhancing the energy efficiency of buildings significantly relies on monitoring indoor ambient temperature. The potential limitations of conventional temperature measurement techniques, together with the omnipresence of smartphones, have redirected researchers'attention towards the exploration of phone-based ambient temperature estimation methods. However, existing phone-based methods face challenges such as insufficient privacy protection, difficulty in adapting models to various phones, and hurdles in obtaining enough labeled training data. In this study, we propose a distributed phone-based ambient temperature estimation system which enables collaboration among multiple phones to accurately measure the ambient temperature in different areas of an indoor space. This system also provides an efficient, cost-effective approach with a few-shot meta-learning module and an automated label generation module. It shows that with just 5 new training data points, the temperature estimation model can adapt to a new phone and reach a good performance. Moreover, the system uses crowdsourcing to generate accurate labels for all newly collected training data, significantly reducing costs. Additionally, we highlight the potential of incorporating federated learning into our system to enhance privacy protection. We believe this study can advance the practical application of phone-based ambient temperature measurement, facilitating energy-saving efforts in buildings.

Yongzheng Liu, Yiming Wang, Po Xu et al.

Due to the extensive availability of operation data, data-driven methods show strong capabilities in predicting building energy loads. Buildings with similar features often share energy patterns, reflected by spatial dependencies in their operational data, which conventional prediction methods struggle to capture. To overcome this, we propose a multi-building prediction approach using spatio-temporal graph neural networks, comprising graph representation, graph learning, and interpretation. First, a graph is built based on building characteristics and environmental factors. Next, a multi-level graph convolutional architecture with attention is developed for energy prediction. Lastly, a method interpreting the optimized graph structure is introduced. Experiments on the Building Data Genome Project 2 dataset confirm superior performance over baselines such as XGBoost, SVR, FCNN, GRU, and Naive, highlighting the method's robustness, generalization, and interpretability in capturing meaningful building similarities and spatial relationships.

Xiangnan Yu, Hao Xu, Zhiping Mao et al.

In complex physical systems, conventional differential equations often fall short in capturing non-local and memory effects, as they are limited to local dynamics and integer-order interactions. This study introduces a stepwise data-driven framework for discovering fractional differential equations (FDEs) directly from data. FDEs, known for their capacity to model non-local dynamics with fewer parameters than integer-order derivatives, can represent complex systems with long-range interactions. Our framework applies deep neural networks as surrogate models for denoising and reconstructing sparse and noisy observations while using Gaussian-Jacobi quadrature to handle the challenges posed by singularities in fractional derivatives. To optimize both the sparse coefficients and fractional order, we employ an alternating optimization approach that combines sparse regression with global optimization techniques. We validate the framework across various datasets, including synthetic anomalous diffusion data, experimental data on the creep behavior of frozen soils, and single-particle trajectories modeled by Lévy motion. Results demonstrate the framework's robustness in identifying the structure of FDEs across diverse noise levels and its capacity to capture integer-order dynamics, offering a flexible approach for modeling memory effects in complex systems.

Qinglong Cao, Yuntian Chen, Lu Lu et al.

Large-scale Vision-Language Models (VLMs) have demonstrated exceptional performance in natural vision tasks, motivating researchers across domains to explore domain-specific VLMs. However, the construction of powerful domain-specific VLMs demands vast amounts of annotated data, substantial electrical energy, and computing resources, primarily accessible to industry, yet hindering VLM research in academia. To address this challenge and foster sustainable and equitable VLM research, we present the Generalized Domain Prompt Learning (GDPL) framework. GDPL facilitates the transfer of VLMs' robust recognition capabilities from natural vision to specialized domains, without the need for extensive data or resources. By leveraging small-scale domain-specific foundation models and minimal prompt samples, GDPL empowers the language branch with domain knowledge through quaternion networks, uncovering cross-modal relationships between domain-specific vision features and natural vision-based contextual embeddings. Simultaneously, GDPL guides the vision branch into specific domains through hierarchical propagation of generated vision prompt features, grounded in well-matched vision-language relations. Furthermore, to fully harness the domain adaptation potential of VLMs, we introduce a novel low-rank adaptation approach. Extensive experiments across diverse domains like remote sensing, medical imaging, geology, Synthetic Aperture Radar, and fluid dynamics, validate the efficacy of GDPL, demonstrating its ability to achieve state-of-the-art domain recognition performance in a prompt learning paradigm. Our framework paves the way for sustainable and inclusive VLM research, transcending the barriers between academia and industry.

Ni Tang, Xiaotong Luo, Zihan Cheng et al.

Diffusion models have revealed powerful potential in all-in-one image restoration (AiOIR), which is talented in generating abundant texture details. The existing AiOIR methods either retrain a diffusion model or fine-tune the pretrained diffusion model with extra conditional guidance. However, they often suffer from high inference costs and limited adaptability to diverse degradation types. In this paper, we propose an efficient AiOIR method, Diffusion Once and Done (DOD), which aims to achieve superior restoration performance with only one-step sampling of Stable Diffusion (SD) models. Specifically, multi-degradation feature modulation is first introduced to capture different degradation prompts with a pretrained diffusion model. Then, parameter-efficient conditional low-rank adaptation integrates the prompts to enable the fine-tuning of the SD model for adapting to different degradation types. Besides, a high-fidelity detail enhancement module is integrated into the decoder of SD to improve structural and textural details. Experiments demonstrate that our method outperforms existing diffusion-based restoration approaches in both visual quality and inference efficiency.

Xinyue Liu, Xiao Peng, Shuyue Yan et al.

Observed records of climate extremes provide an incomplete picture of risk, missing "unseen" extremes that exceed historical bounds. In parallel, neglecting spatial dependence undervalues the risk of synchronized hazards that amplify impacts. To address these challenges, we develop DeepX-GAN (Dependence-Enhanced Embedding for Physical eXtremes - Generative Adversarial Network), a knowledge-informed deep generative model designed to better capture the spatial structure of rare extremes. The zero-shot generalizability of DeepX-GAN enables simulation of unseen extremes that fall outside historical experience yet remain statistically plausible. We define two types of unseen extremes: "checkmate" extremes that directly hit targets, and "stalemate" extremes that narrowly miss. These unrealized scenarios expose latent risks in fragile systems and may reinforce a false sense of resilience if overlooked. Near misses, in particular, can prompt either proactive adaptation or dangerous complacency, depending on how they are interpreted. Applying DeepX-GAN to the Middle East and North Africa (MENA), we find that these unseen extremes disproportionately affect regions with high vulnerability and low socioeconomic readiness, but differ in urgency and interpretation. Future warming could expand and redistribute these unseen extremes, with emerging exposure hotspots in Indo-Pakistan and Central Africa. This distributional shift highlights critical blind spots in conventional hazard planning and underscores the need to develop spatially adaptive policies that anticipate emergent risk hotspots rather than simply extrapolating from historical patterns.

Hao Xu, Yuntian Chen, Rui Cao et al.

Data driven discovery of partial differential equations (PDEs) is a promising approach for uncovering the underlying laws governing complex systems. However, purely data driven techniques face the dilemma of balancing search space with optimization efficiency. This study introduces a knowledge guided approach that incorporates existing PDEs documented in a mathematical handbook to facilitate the discovery process. These PDEs are encoded as sentence like structures composed of operators and basic terms, and used to train a generative model, called EqGPT, which enables the generation of free form PDEs. A loop of generation evaluation optimization is constructed to autonomously identify the most suitable PDE. Experimental results demonstrate that this framework can recover a variety of PDE forms with high accuracy and computational efficiency, particularly in cases involving complex temporal derivatives or intricate spatial terms, which are often beyond the reach of conventional methods. The approach also exhibits generalizability to irregular spatial domains and higher dimensional settings. Notably, it succeeds in discovering a previously unreported PDE governing strongly nonlinear surface gravity waves propagating toward breaking, based on real world experimental data, highlighting its applicability to practical scenarios and its potential to support scientific discovery.

Longfei Ma, Nan Cheng, Xiucheng Wang et al.

The development of Digital Twins (DTs) represents a transformative advance for simulating and optimizing complex systems in a controlled digital space. Despite their potential, the challenge of constructing DTs that accurately replicate and predict the dynamics of real-world systems remains substantial. This paper introduces an intelligent framework for the construction and evaluation of DTs, specifically designed to enhance the accuracy and utility of DTs in testing algorithmic performance. We propose a novel construction methodology that integrates deep learning-based policy gradient techniques to dynamically tune the DT parameters, ensuring high fidelity in the digital replication of physical systems. Moreover, the Mean STate Error (MSTE) is proposed as a robust metric for evaluating the performance of algorithms within these digital space. The efficacy of our framework is demonstrated through extensive simulations that show our DT not only accurately mirrors the physical reality but also provides a reliable platform for algorithm evaluation. This work lays a foundation for future research into DT technologies, highlighting pathways for both theoretical enhancements and practical implementations in various industries.

Lei Xu, Yulong Chen, Yuntian Chen et al.

Machine learning models offer the capability to forecast future energy production or consumption and infer essential unknown variables from existing data. However, legal and policy constraints within specific energy sectors render the data sensitive, presenting technical hurdles in utilizing data from diverse sources. Therefore, we propose adopting a Swarm Learning (SL) scheme, which replaces the centralized server with a blockchain-based distributed network to address the security and privacy issues inherent in Federated Learning (FL)'s centralized architecture. Within this distributed Collaborative Learning framework, each participating organization governs nodes for inter-organizational communication. Devices from various organizations utilize smart contracts for parameter uploading and retrieval. Consensus mechanism ensures distributed consistency throughout the learning process, guarantees the transparent trustworthiness and immutability of parameters on-chain. The efficacy of the proposed framework is substantiated across three real-world energy series modeling scenarios with superior performance compared to Local Learning approaches, simultaneously emphasizing enhanced data security and privacy over Centralized Learning and FL method. Notably, as the number of data volume and the count of local epochs increases within a threshold, there is an improvement in model performance accompanied by a reduction in the variance of performance errors. Consequently, this leads to an increased stability and reliability in the outcomes produced by the model.

Zhongzheng Wang, Yuntian Chen, Guodong Chen et al.

Maximizing storage performance in geological carbon storage (GCS) is crucial for commercial deployment, but traditional optimization demands resource-intensive simulations, posing computational challenges. This study introduces the multimodal latent dynamic (MLD) model, a deep learning framework for fast flow prediction and well control optimization in GCS. The MLD model includes a representation module for compressed latent representations, a transition module for system state evolution, and a prediction module for flow responses. A novel training strategy combining regression loss and joint-embedding consistency loss enhances temporal consistency and multi-step prediction accuracy. Unlike existing models, the MLD supports diverse input modalities, allowing comprehensive data interactions. The MLD model, resembling a Markov decision process (MDP), can train deep reinforcement learning agents, specifically using the soft actor-critic (SAC) algorithm, to maximize net present value (NPV) through continuous interactions. The approach outperforms traditional methods, achieving the highest NPV while reducing computational resources by over 60%. It also demonstrates strong generalization performance, providing improved decisions for new scenarios based on knowledge from previous ones.

Yongan Zhang, Junfeng Zhao, Jian Li et al.

The prediction of formation resistivity plays a crucial role in the evaluation of oil and gas reservoirs, identification and assessment of geothermal energy resources, groundwater detection and monitoring, and carbon capture and storage. However, traditional well logging techniques fail to measure accurate resistivity in cased boreholes, and the transient electromagnetic method for cased borehole resistivity logging encounters challenges of high-frequency disaster (the problem of inadequate learning by neural networks in high-frequency features) and noise interference, badly affecting accuracy. To address these challenges, frequency-aware framework and temporal anti-noise block are proposed to build frequency aware LSTM (FAL). The frequency-aware framework implements a dual-stream structure through wavelet transformation, allowing the neural network to simultaneously handle high-frequency and low-frequency flows of time-series data, thus avoiding high-frequency disaster. The temporal anti-noise block integrates multiple attention mechanisms and soft-threshold attention mechanisms, enabling the model to better distinguish noise from redundant features. Ablation experiments demonstrate that the frequency-aware framework and temporal anti-noise block contribute significantly to performance improvement. FAL achieves a 24.3% improvement in R2 over LSTM, reaching the highest value of 0.91 among all models. In robustness experiments, the impact of noise on FAL is approximately 1/8 of the baseline, confirming the noise resistance of FAL. The proposed FAL effectively reduces noise interference in predicting formation resistivity from cased transient electromagnetic well logging curves, better learns high-frequency features, and thereby enhances the prediction accuracy and noise resistance of the neural network model.

Jiaxin Gao, Qinglong Cao, Yuntian Chen et al.

Photovoltaic (PV) power forecasting plays a crucial role in optimizing the operation and planning of PV systems, thereby enabling efficient energy management and grid integration. However, un certainties caused by fluctuating weather conditions and complex interactions between different variables pose significant challenges to accurate PV power forecasting. In this study, we propose PV-Client (Cross-variable Linear Integrated ENhanced Transformer for Photovoltaic power forecasting) to address these challenges and enhance PV power forecasting accuracy. PV-Client employs an ENhanced Transformer module to capture complex interactions of various features in PV systems, and utilizes a linear module to learn trend information in PV power. Diverging from conventional time series-based Transformer models that use cross-time Attention to learn dependencies between different time steps, the Enhanced Transformer module integrates cross-variable Attention to capture dependencies between PV power and weather factors. Furthermore, PV-Client streamlines the embedding and position encoding layers by replacing the Decoder module with a projection layer. Experimental results on three real-world PV power datasets affirm PV-Client's state-of-the-art (SOTA) performance in PV power forecasting. Specifically, PV-Client surpasses the second-best model GRU by 5.3% in MSE metrics and 0.9% in accuracy metrics at the Jingang Station. Similarly, PV-Client outperforms the second-best model SVR by 10.1% in MSE metrics and 0.2% in accuracy metrics at the Xinqingnian Station, and PV-Client exhibits superior performance compared to the second-best model SVR with enhancements of 3.4% in MSE metrics and 0.9% in accuracy metrics at the Hongxing Station.

Wenchao Wu, Hao Xu, Dongxiao Zhang et al.

We present an innovative of artificial intelligence with column chromatography, aiming to resolve inefficiencies and standardize data collection in chemical separation and purification domain. By developing an automated platform for precise data acquisition and employing advanced machine learning algorithms, we constructed predictive models to forecast key separation parameters, thereby enhancing the efficiency and quality of chromatographic processes. The application of transfer learning allows the model to adapt across various column specifications, broadening its utility. A novel metric, separation probability ($S_p$), quantifies the likelihood of effective compound separation, validated through experimental verification. This study signifies a significant step forward int the application of AI in chemical research, offering a scalable solution to traditional chromatography challenges and providing a foundation for future technological advancements in chemical analysis and purification.

Yuntian Chen, Dongxiao Zhang

Scientific research's mandate is to comprehend and explore the world, as well as to improve it based on experience and knowledge. Knowledge embedding and knowledge discovery are two significant methods of integrating knowledge and data. Through knowledge embedding, the barriers between knowledge and data can be eliminated, and machine learning models with physical common sense can be established. Meanwhile, humans' understanding of the world is always limited, and knowledge discovery takes advantage of machine learning to extract new knowledge from observations. Knowledge discovery can not only assist researchers to better grasp the nature of physics, but it can also support them in conducting knowledge embedding research. A closed loop of knowledge generation and usage are formed by combining knowledge embedding with knowledge discovery, which can improve the robustness and accuracy of models and uncover previously unknown scientific principles. This study summarizes and analyzes extant literature, as well as identifies research gaps and future opportunities.

Hao Xu, Jinglong Lin, Qianyi Liu et al.

As an essential attribute of organic compounds, polarity has a profound influence on many molecular properties such as solubility and phase transition temperature. Thin layer chromatography (TLC) represents a commonly used technique for polarity measurement. However, current TLC analysis presents several problems, including the need for a large number of attempts to obtain suitable conditions, as well as irreproducibility due to non-standardization. Herein, we describe an automated experiment system for TLC analysis. This system is designed to conduct TLC analysis automatically, facilitating high-throughput experimentation by collecting large experimental data under standardized conditions. Using these datasets, machine learning (ML) methods are employed to construct surrogate models correlating organic compounds' structures and their polarity using retardation factor (Rf). The trained ML models are able to predict the Rf value curve of organic compounds with high accuracy. Furthermore, the constitutive relationship between the compound and its polarity can also be discovered through these modeling methods, and the underlying mechanism is rationalized through adsorption theories. The trained ML models not only reduce the need for empirical optimization currently required for TLC analysis, but also provide general guidelines for the selection of conditions, making TLC an easily accessible tool for the broad scientific community.

Nanzhe Wang, Qinzhuo Liao, Haibin Chang et al.

Large-scale or high-resolution geologic models usually comprise a huge number of grid blocks, which can be computationally demanding and time-consuming to solve with numerical simulators. Therefore, it is advantageous to upscale geologic models (e.g., hydraulic conductivity) from fine-scale (high-resolution grids) to coarse-scale systems. Numerical upscaling methods have been proven to be effective and robust for coarsening geologic models, but their efficiency remains to be improved. In this work, a deep-learning-based method is proposed to upscale the fine-scale geologic models, which can assist to improve upscaling efficiency significantly. In the deep learning method, a deep convolutional neural network (CNN) is trained to approximate the relationship between the coarse grid of hydraulic conductivity fields and the hydraulic heads, which can then be utilized to replace the numerical solvers while solving the flow equations for each coarse block. In addition, physical laws (e.g., governing equations and periodic boundary conditions) can also be incorporated into the training process of the deep CNN model, which is termed the theory-guided convolutional neural network (TgCNN). With the physical information considered, dependence on the data volume of training the deep learning models can be reduced greatly. Several subsurface flow cases are introduced to test the performance of the proposed deep-learning-based upscaling method, including 2D and 3D cases, and isotropic and anisotropic cases. The results show that the deep learning method can provide equivalent upscaling accuracy to the numerical method, and efficiency can be improved significantly compared to numerical upscaling.

Rui Xu, Dongxiao Zhang, Nanzhe Wang

We build surrogate models for dynamic 3D subsurface single-phase flow problems with multiple vertical producing wells. The surrogate model provides efficient pressure estimation of the entire formation at any timestep given a stochastic permeability field, arbitrary well locations and penetration lengths, and a timestep matrix as inputs. The well production rate or bottom hole pressure can then be determined based on Peaceman's formula. The original surrogate modeling task is transformed into an image-to-image regression problem using a convolutional encoder-decoder neural network architecture. The residual of the governing flow equation in its discretized form is incorporated into the loss function to impose theoretical guidance on the model training process. As a result, the accuracy and generalization ability of the trained surrogate models are significantly improved compared to fully data-driven models. They are also shown to have flexible extrapolation ability to permeability fields with different statistics. The surrogate models are used to conduct uncertainty quantification considering a stochastic permeability field, as well as to infer unknown permeability information based on limited well production data and observation data of formation properties. Results are shown to be in good agreement with traditional numerical simulation tools, but computational efficiency is dramatically improved.

Nanzhe Wang, Haibin Chang, Dongxiao Zhang

The theory-guided convolutional neural network (TgCNN) framework, which can incorporate discretized governing equation residuals into the training of convolutional neural networks (CNNs), is extended to two-phase porous media flow problems in this work. The two principal variables of the considered problem, pressure and saturation, are approximated simultaneously with two CNNs, respectively. Pressure and saturation are coupled with each other in the governing equations, and thus the two networks are also mutually conditioned in the training process by the discretized governing equations, which also increases the difficulty of model training. The coupled and discretized equations can provide valuable information in the training process. With the assistance of theory-guidance, the TgCNN surrogates can achieve better accuracy than ordinary CNN surrogates in two-phase flow problems. Moreover, a piecewise training strategy is proposed for the scenario with varying well controls, in which the TgCNN surrogates are constructed for different segments on the time dimension and stacked together to predict solutions for the whole time-span. For scenarios with larger variance of the formation property field, the TgCNN surrogates can also achieve satisfactory performance. The constructed TgCNN surrogates are further used for inversion of permeability fields by combining them with the iterative ensemble smoother (IES) algorithm, and sufficient inversion accuracy is obtained with improved efficiency.

Xing Luo, Dongxiao Zhang

Accurate forecasts of photovoltaic power generation (PVPG) are essential to optimize operations between energy supply and demand. Recently, the propagation of sensors and smart meters has produced an enormous volume of data, which supports the development of data based PVPG forecasting. Although emerging deep learning (DL) models, such as the long short-term memory (LSTM) model, based on historical data, have provided effective solutions for PVPG forecasting with great successes, these models utilize offline learning. As a result, DL models cannot take advantage of the opportunity to learn from newly-arrived data, and are unable to handle concept drift caused by installing extra PV units and unforeseen PV unit failures. Consequently, to improve day-ahead PVPG forecasting accuracy, as well as eliminate the impacts of concept drift, this paper proposes an adaptive LSTM (AD-LSTM) model, which is a DL framework that can not only acquire general knowledge from historical data, but also dynamically learn specific knowledge from newly-arrived data. A two-phase adaptive learning strategy (TP-ALS) is integrated into AD-LSTM, and a sliding window (SDWIN) algorithm is proposed, to detect concept drift in PV systems. Multiple datasets from PV systems are utilized to assess the feasibility and effectiveness of the proposed approaches. The developed AD-LSTM model demonstrates greater forecasting capability than the offline LSTM model, particularly in the presence of concept drift. Additionally, the proposed AD-LSTM model also achieves superior performance in terms of day-ahead PVPG forecasting compared to other traditional machine learning models and statistical models in the literature.

Pengfei Tang, Junsheng Zeng, Dongxiao Zhang et al.

Particle settling in inclined channels is an important phenomenon that occurs during hydraulic fracturing of shale gas production. Generally, in order to accurately simulate the large-scale (field-scale) proppant transport process, constructing a fast and accurate sub-scale proppant settling model, or surrogate model, becomes a critical issue, since mapping between physical parameters and proppant settling velocity is complex. Previously, particle settling has usually been investigated via high-fidelity experiments and meso-scale numerical simulations, both of which are time-consuming. In this work, a new method is proposed and utilized, i.e., the multi-fidelity neural network (MFNN), to construct a settling surrogate model, which could utilize both high-fidelity and low-fidelity (thus, less expensive) data. The results demonstrate that constructing the settling surrogate with the MFNN can reduce the need for high-fidelity data and thus computational cost by 80%, while the accuracy lost is less than 5% compared to a high-fidelity surrogate. Moreover, the investigated particle settling surrogate is applied in macro-scale proppant transport simulation, which shows that the settling model is significant to proppant transport and yields accurate results. This opens novel pathways for rapidly predicting proppant settling velocity in reservoir applications.

Qiang Zheng, Dongxiao Zhang

Random reconstruction of three-dimensional (3D) digital rocks from two-dimensional (2D) slices is crucial for elucidating the microstructure of rocks and its effects on pore-scale flow in terms of numerical modeling, since massive samples are usually required to handle intrinsic uncertainties. Despite remarkable advances achieved by traditional process-based methods, statistical approaches and recently famous deep learning-based models, few works have focused on producing several kinds of rocks with one trained model and allowing the reconstructed samples to satisfy certain given properties, such as porosity. To fill this gap, we propose a new framework, named RockGPT, which is composed of VQ-VAE and conditional GPT, to synthesize 3D samples based on a single 2D slice from the perspective of video generation. The VQ-VAE is utilized to compress high-dimensional input video, i.e., the sequence of continuous rock slices, to discrete latent codes and reconstruct them. In order to obtain diverse reconstructions, the discrete latent codes are modeled using conditional GPT in an autoregressive manner, while incorporating conditional information from a given slice, rock type, and porosity. We conduct two experiments on five kinds of rocks, and the results demonstrate that RockGPT can produce different kinds of rocks with the same model, and the reconstructed samples can successfully meet certain specified porosities. In a broader sense, through leveraging the proposed conditioning scheme, RockGPT constitutes an effective way to build a general model to produce multiple kinds of rocks simultaneously that also satisfy user-defined properties.

Yuntian Chen, Yingtao Luo, Qiang Liu et al.

Partial differential equations (PDEs) are concise and understandable representations of domain knowledge, which are essential for deepening our understanding of physical processes and predicting future responses. However, the PDEs of many real-world problems are uncertain, which calls for PDE discovery. We propose the symbolic genetic algorithm (SGA-PDE) to discover open-form PDEs directly from data without prior knowledge about the equation structure. SGA-PDE focuses on the representation and optimization of PDE. Firstly, SGA-PDE uses symbolic mathematics to realize the flexible representation of any given PDE, transforms a PDE into a forest, and converts each function term into a binary tree. Secondly, SGA-PDE adopts a specially designed genetic algorithm to efficiently optimize the binary trees by iteratively updating the tree topology and node attributes. The SGA-PDE is gradient-free, which is a desirable characteristic in PDE discovery since it is difficult to obtain the gradient between the PDE loss and the PDE structure. In the experiment, SGA-PDE not only successfully discovered nonlinear Burgers' equation, Korteweg-de Vries (KdV) equation, and Chafee-Infante equation, but also handled PDEs with fractional structure and compound functions that cannot be solved by conventional PDE discovery methods.

Junsheng Zeng, Hao Xu, Yuntian Chen et al.

Although deep-learning has been successfully applied in a variety of science and engineering problems owing to its strong high-dimensional nonlinear mapping capability, it is of limited use in scientific knowledge discovery. In this work, we propose a deep-learning based framework to discover the macroscopic governing equation of viscous gravity current based on high-resolution microscopic simulation data without the need for prior knowledge of underlying terms. For two typical scenarios with different viscosity ratios, the deep-learning based equations exactly capture the same dominated terms as the theoretically derived equations for describing long-term asymptotic behaviors, which validates the proposed framework. Unknown macroscopic equations are then obtained for describing short-term behaviors, and additional deep-learned compensation terms are eventually discovered. Comparison of posterior tests shows that the deep-learning based PDEs actually perform better than the theoretically derived PDEs in predicting evolving viscous gravity currents for both long-term and short-term regimes. Moreover, the proposed framework is proven to be very robust against non-biased data noise for training, which is up to 20%. Consequently, the presented deep-learning framework shows considerable potential for discovering unrevealed intrinsic laws in scientific semantic space from raw experimental or simulation results in data space.

Hao Xu, Dongxiao Zhang

Data-driven discovery of partial differential equations (PDEs) has achieved considerable development in recent years. Several aspects of problems have been resolved by sparse regression-based and neural network-based methods. However, the performances of existing methods lack stability when dealing with complex situations, including sparse data with high noise, high-order derivatives and shock waves, which bring obstacles to calculating derivatives accurately. Therefore, a robust PDE discovery framework, called the robust deep learning-genetic algorithm (R-DLGA), that incorporates the physics-informed neural network (PINN), is proposed in this work. In the framework, a preliminary result of potential terms provided by the deep learning-genetic algorithm is added into the loss function of the PINN as physical constraints to improve the accuracy of derivative calculation. It assists to optimize the preliminary result and obtain the ultimately discovered PDE by eliminating the error compensation terms. The stability and accuracy of the proposed R-DLGA in several complex situations are examined for proof-and-concept, and the results prove that the proposed framework is able to calculate derivatives accurately with the optimization of PINN and possesses surprising robustness to complex situations, including sparse data with high noise, high-order derivatives, and shock waves.

Yuntian Chen, Dou Huang, Dongxiao Zhang et al.

Machine learning models have been successfully used in many scientific and engineering fields. However, it remains difficult for a model to simultaneously utilize domain knowledge and experimental observation data. The application of knowledge-based symbolic AI represented by an expert system is limited by the expressive ability of the model, and data-driven connectionism AI represented by neural networks is prone to produce predictions that violate physical mechanisms. In order to fully integrate domain knowledge with observations, and make full use of the prior information and the strong fitting ability of neural networks, this study proposes theory-guided hard constraint projection (HCP). This model converts physical constraints, such as governing equations, into a form that is easy to handle through discretization, and then implements hard constraint optimization through projection. Based on rigorous mathematical proofs, theory-guided HCP can ensure that model predictions strictly conform to physical mechanisms in the constraint patch. The performance of the theory-guided HCP is verified by experiments based on the heterogeneous subsurface flow problem. Due to the application of hard constraints, compared with fully connected neural networks and soft constraint models, such as theory-guided neural networks and physics-informed neural networks, theory-guided HCP requires fewer data, and achieves higher prediction accuracy and stronger robustness to noisy observations.

Qiang Zheng, Dongxiao Zhang

Uncertainty is ubiquitous with flow in subsurface rocks because of their inherent heterogeneity and lack of in-situ measurements. To complete uncertainty analysis in a multi-scale manner, it is a prerequisite to provide sufficient rock samples. Even though the advent of digital rock technology offers opportunities to reproduce rocks, it still cannot be utilized to provide massive samples due to its high cost, thus leading to the development of diversified mathematical methods. Among them, two-point statistics (TPS) and multi-point statistics (MPS) are commonly utilized, which feature incorporating low-order and high-order statistical information, respectively. Recently, generative adversarial networks (GANs) are becoming increasingly popular since they can reproduce training images with excellent visual and consequent geologic realism. However, standard GANs can only incorporate information from data, while leaving no interface for user-defined properties, and thus may limit the representativeness of reconstructed samples. In this study, we propose conditional GANs for digital rock reconstruction, aiming to reproduce samples not only similar to the real training data, but also satisfying user-specified properties. In fact, the proposed framework can realize the targets of MPS and TPS simultaneously by incorporating high-order information directly from rock images with the GANs scheme, while preserving low-order counterparts through conditioning. We conduct three reconstruction experiments, and the results demonstrate that rock type, rock porosity, and correlation length can be successfully conditioned to affect the reconstructed rock images. Furthermore, in contrast to existing GANs, the proposed conditioning enables learning of multiple rock types simultaneously, and thus invisibly saves computational cost.

Hao Xu, Dongxiao Zhang, Nanzhe Wang

Data-driven discovery of partial differential equations (PDEs) has attracted increasing attention in recent years. Although significant progress has been made, certain unresolved issues remain. For example, for PDEs with high-order derivatives, the performance of existing methods is unsatisfactory, especially when the data are sparse and noisy. It is also difficult to discover heterogeneous parametric PDEs where heterogeneous parameters are embedded in the partial differential operators. In this work, a new framework combining deep-learning and integral form is proposed to handle the above-mentioned problems simultaneously, and improve the accuracy and stability of PDE discovery. In the framework, a deep neural network is firstly trained with observation data to generate meta-data and calculate derivatives. Then, a unified integral form is defined, and the genetic algorithm is employed to discover the best structure. Finally, the value of parameters is calculated, and whether the parameters are constants or variables is identified. Numerical experiments proved that our proposed algorithm is more robust to noise and more accurate compared with existing methods due to the utilization of integral form. Our proposed algorithm is also able to discover PDEs with high-order derivatives or heterogeneous parameters accurately with sparse and noisy data.