Fault Diagnosis (FD) on sequential data suffers from irregular sampling (with missing values), limited training data, and varying underlying environments. In response, this paper proposes FD by adjoint learning in continuous-time model space. Model-Space Learning employs well-fitted models that capture data's dynamics (i.e., changing information) as more stable and concise representations of the original data. The Continuous-Time Reservoir Computing Network (CT-Res) is first introduced, which embeds Ordinary Differential Equation (ODE) within the reservoir-based hidden layer to govern continuous-time hidden-state evolution, naturally handling irregular sampling without relying on fixed time steps and effectively capturing intrinsic data dynamics. By fitting each sequence via CT-Res and representing it with the fitted model, the original sequences are mapped from the data space into the continuous-time model space. We further develop an adjoint learning strategy by incorporating a discrete-time "adjoint Echo State Network (ESN)" that shares structure and parameters with CT-Res, thus enabling efficient training by bypassing the computationally intensive ODE solver, with joint optimization of fitting accuracy and class discrimination in the model space. Experiments on multiple FD benchmarks highlight the effectiveness and efficiency of our study, particularly with missing values and scarce training data.

Precise delineation of tumor target volumes in radiation therapy is a knowledge-intensive clinical task governed by three formally characterizable knowledge types: declarative geometric priors encoding anatomically plausible contour constraints, structural relational knowledge capturing rotational symmetry in polygon vertex sequences, and procedural tolerance knowledge defining clinically acceptable boundary deviations. Existing pixel-level segmentation methods discard this knowledge entirely, producing topologically inconsistent masks that require costly manual correction. We propose a knowledge-guided framework that explicitly encodes all three knowledge forms into both model architecture and training supervision. Specifically, (1) a Vision Transformer encoder captures global spatial knowledge via self-attention, transcending the locality limitations of convolutional kernels; (2) Rotary Positional Encoding embeds the rotational symmetry knowledge inherent in polygon vertex sequences, capturing a structural prior that standard absolute encodings fundamentally cannot represent; and (3) the proposed Convolutional Smoothing Loss operationalizes clinical positional tolerance knowledge by assigning spatially graded supervision signals, distinguishing catastrophic boundary violations from clinically acceptable minor deviations. Evaluated on the MSD Pancreas Tumour dataset and a proprietary cervical cancer CT dataset, the proposed model consistently outperforms CNN-based and Transformer-based baselines across all metrics. Ablation studies and robustness analysis under Gaussian noise confirm the independent contribution of each knowledge-encoding component.

Efficient anomaly detection of irregular sequences, especially those characterized by non-uniform sampling from discontinuous operations or unreliable sensors, presents challenges across various fields. In response, this paper introduces irregular-sequence classification in ''Ct-Echo Model Space''. A novel Continuous-time Echo Network (Ct-Echo) is proposed to fit irregular sequences, efficiently capturing their inherent dynamic characteristics. Ct-Echo utilizes the ''Echo'' mechanism, where history information influences the current state and diminishes over time, and employs Ordinary Differential Equation (ODE) to construct continuous-time transition of hidden states. Each sequence is individually fitted via Ct-Echo to derive a readout model. These fitted models, capturing the dynamic characteristics of the original data, serve as representations of the corresponding sequences, thus mapping the original data from the data space to the Ct-Echo model space. Anomaly detection is further performed in this model space, evaluating differences between models rather than directly on the original sequences. Our method enhances real-time processing and lessens reliance on the amount of labeled training data, as demonstrated by experimental studies.

Anomaly detection in sequential signals is gaining prominence, especially with limited training data and timeliness requirements. Fully extracting the data-inside changing information, we propose a novel Wavelet-Enhanced Reservoir Computing framework (WE-Res). Our framework uses Discrete Wavelet Transform (DWT) to decompose signals for multi-level detail and trend extraction recursively. Each level is fitted using an Echo State Network (ESN) respectively, extracting its inside dynamic features into fitted models. Integrating these dynamic features creates a "multi-level dynamic feature" that enhances signal representation, aiding in distinguishing between normal and anomalous patterns. We further introduce a dual-objective optimization to refine ESNs’ reservoirs, increasing the fitting accuracy and improving category discrimination among the captured features. Validation on real-world data confirms our method’s effectiveness, especially in data-limited scenarios and less training time compared to recent baselines.

Fault Diagnosis (FD) in time-varying systems faces challenges like limited training data, varying environments, and timeliness. Building upon the framework of model-space learning (MSL), we introduce co-objective learning in Dynamic-Integration network (DynInt) model space as a solution for FD. MSL involves utilizing well-fitted models that capture the dynamics within the data as more stable and parsimonious representations of the original data. DynInt integrates pooling and reservoir computing to adequately capture the data-inherent multi-scale dynamics. Representing the signal with the fitted DynInt model transforms the original signal from data space into the DynInt model space. A co-objective optimization is further introduced on DynInt, improving the fitting accuracy and category discrimination. Validation on real-world data confirms our method’s effectiveness, especially in data-limited scenarios.

Fault Diagnosis (FD) in time-varying data presents considerations such as limited training data, intra- and inter-dimensional correlations, and constraints of training time. In response, this paper introduces FD in the Reservoir-Embedded-Directional Network (REDNet) model space. Model-oriented methods utilize well-fitted networks or functions, denoted as "models" that capture data's changing information, as more stable and parsimonious representations of the data. Our approach employs REDNet for data fitting, wherein multiple reservoirs are organized along intrinsic correlation directions to establish intra- and inter-dimensional dependencies, thereby capturing multi-directional dynamics in high-dimensional data. Representing each data instance with an independently fitted REDNet model maps these instances into a class-separable REDNet model space, where FD could be performed on the models rather than the original data. Concentrating on the data-intrinsic dynamics, our method achieves rapid training speeds, and maintains robust performance even with minimal training data. Experiments on several datasets demonstrate its effectiveness.

Anomaly detection (AD) is gaining prominence, especially in situations with limited labeled data or unknown anomalies, demanding an efficient approach with minimal reliance on labeled data or prior knowledge. Building upon the framework of Learning in the Model Space (LMS), this paper proposes conducting AD through Learning in the Multi-Level Model Spaces (MLMS). LMS transforms the data from the data space to the model space by representing each data instance with a fitted model. In MLMS, to fully capture the dynamic characteristics within the data, multi-level details of the original data instance are decomposed. These details are individually fitted, resulting in a set of fitted models that capture the multi-level dynamic characteristics of the original instance. Representing each data instance with a set of fitted models, rather than a single one, transforms it from the data space into the multi-level model spaces. The pairwise difference measurement between model sets is introduced, fully considering the distance between fitted models and the intra-class aggregation of similar models at each level of detail. Subsequently, effective AD can be implemented in the multi-level model spaces, with or without sufficient multi-class labeled data. Experiments on multiple AD datasets demonstrate the effectiveness of the proposed method.

Ground Penetrating Radar (GPR) has been widely used in pipeline detection and underground diagnosis. In practical applications, the characteristics of the GPR data of the detected area and the likely underground anomalous structures could be rarely acknowledged before fully analyzing the obtained GPR data, causing challenges to identify the underground structures or anomalies automatically. In this article, a GPR B-scan image diagnosis method based on learning in the model space is proposed. The idea of learning in the model space is to use models fitted on parts of data as more stable and parsimonious representations of the data. For the GPR image, 2-Direction Echo State Network (2D-ESN) is proposed to fit the image segments through the next item prediction. By building the connections between the points on the image in both the horizontal and vertical directions, the 2D-ESN regards the GPR image segment as a whole and could effectively capture the dynamic characteristics of the GPR image. And then, semi-supervised and supervised learning methods could be further implemented on the 2D-ESN models for underground diagnosis. Experiments on real-world datasets are conducted, and the results demonstrate the effectiveness of the proposed model.

Maintaining and upgrading underground pipelines are major undertakings in urban operations, where accurately locating buried pipelines has long been an issue. In this article, we propose a pipeline mapping method based on integrating multipositional pipeline data, which includes the multisensor data acquisition (MDA) platform and the scalable probability-based pipeline mapping (SP-PM) model. To effectively collect pipeline data at multiple positions, several pipeline detecting and positioning sensors are equipped in the MDA platform. Different types of sensor data are synchronously collected and processed to obtain manageable pipeline data at multiple positions within the detected area, such as the radius, depth, and positioned points of underground pipelines. The SP-PM model is then proposed, where the obtained pipeline data are probabilistically described and classified into classes. Each class contains the pipeline data possibly generated by the same pipeline. The classified data are then iteratively integrated to estimate the pipeline map with the maximum probability. The SP-PM model probabilistically estimates the degree of correlation between the multipositional pipeline data and each potential pipeline, and it has no strict requirement on existing statutory records or limits on the number of detections per pipeline. Unrecorded pipelines could be identified and involved into the generated pipeline map, along with continuous adjusting of the pipelines’ number. We conducted experiments on real-world environments. The experimental results verify the accuracy and efficiency of the proposed method for buried pipeline mapping.

Ground Penetrating Radar (GPR) provides an effective means for underground anomaly detection, but is also accompanied by some practical issues such as the lack of prior knowledge, insufficient labeled data, and timeliness constraints. In this paper, we propose detecting underground anomalies by adequately capturing multi-directional changing information within GPR B-scan data, which extends the framework of Model-Space Learning (MSL). MSL aims to transform the data from the data space to the model space by representing the original data with fitted models that capture the changes within the data. In GPR data, due to the continuity of the underground medium and electromagnetic wave, there is not only effective changing information in each column of data along the vertical direction, but also in the horizontal detecting direction according to the subsurface environments and existing underground structures. To fully capture the changing information within GPR data, we fit the GPR data along multiple directions respectively, and synthesize the fitted models into a Comprehensive-Change-Captured model (C3 model) wherein multi-directional changing information within the original data is captured. Representing the original data with the C3 model transforms the data from the data space to the C3 model space. The distance metric between C3 models is then introduced, and learning methods could be efficiently implemented on the models. Experiments are conducted on GPR data collected along urban roads, with or without prior knowledge about the existing subsurface anomalies. The obtained results demonstrate the effectiveness of the proposed method.