Data normalisation, a common, often necessary preprocessing step in engineering and scientific applications, can severely distort the discovery of governing equations by magnitude-based sparse regression methods. This is particularly acute for the Sparse Identification of Nonlinear Dynamics (SINDy) framework, where the core sparsity assumption is undermined by the interaction between data scaling and measurement noise, yielding dense, uninterpretable, physically incorrect models. To address this vulnerability, we introduce Sequential Thresholding of Coefficient of Variation (STCV), a novel, computationally efficient sparse regression algorithm inherently robust to data scaling. STCV replaces conventional magnitude-based thresholding with a dimensionless statistical metric, the Coefficient Presence (CP), which assesses the statistical validity and consistency of candidate library terms. This shift from magnitude to statistical significance makes term selection largely insensitive to arbitrary per-variable scaling. We say "largely insensitive" rather than "invariant" because STCV's inner loop still relies on a Sequentially Thresholded Least Squares (STLSQ) step with a small fixed near-zero threshold; this step is magnitude-dependent and can in principle retain residual scale sensitivity, though in practice the dimensionless CP metric dominates selection, as shown empirically in Section 4. Through comprehensive benchmarking on canonical dynamical systems and practical engineering problems, including a physical mass-spring-damper experiment, we demonstrate STCV consistently and significantly outperforms standard STLSQ and Ensemble-SINDy (E-SINDy) on normalised, noisy datasets, identifying correct, sparse physical laws even when other methods fail. By mitigating the distorting effects of normalisation, STCV makes sparse system identification a more reliable and automated tool for real-world applications, enhancing model interpretability and trustworthiness.

Latent variable models (LVMs) are commonly used to capture the underlying dependencies, patterns, and hidden structure in observed data. Source duplication is a by-product of the data hankelisation pre-processing step common to single channel LVM applications, which hinders practical LVM utilisation. In this article, a Python package titled spectrally-regularised-LVMs is presented. The proposed package addresses the source duplication issue via the addition of a novel spectral regularisation term. This package provides a framework for spectral regularisation in single channel LVM applications, thereby making it easier to investigate and utilise LVMs with spectral regularisation. This is achieved via the use of symbolic or explicit representations of potential LVM objective functions which are incorporated into a framework that uses spectral regularisation during the LVM parameter estimation process. The objective of this package is to provide a consistent linear LVM optimisation framework which incorporates spectral regularisation and caters to single channel time-series applications.

Micro-scale mechanisms, such as inter-particle and particle-fluid interactions, govern the behaviour of granular systems. While particle-scale simulations provide detailed insights into these interactions, their computational cost is often prohibitive. At a recent Lorentz Center Workshop on “ Machine Learning for Discrete Granular Media ”, researchers explored how machine learning approaches can aid the development of constitutive laws and efficient data-driven surrogates for granular materials while also addressing uncertainty quantification. Attended by researchers from both the granular materials (GM) and machine learning (ML) communities, the workshop brought the ML community up to date with GM challenges. This position paper emerged from the workshop discussions. In this position paper, we define granular materials and identify seven key challenges that characterise their distinctive behaviour across various scales and regimes–ranging from gas-like to fluid-like and solid-like. Addressing these challenges is essential for developing robust and efficient models for the digital twinning of granular systems in various industrial applications. To showcase the potential of ML to the GM community, we present classical and emerging machine/deep learning techniques that have been, or could be, applied to granular materials. We reviewed sequence-based learning models for path-dependent constitutive behaviour, followed by encoder-decoder type models for representing high-dimensional data in reduced spaces. We then explore graph neural networks and recent advances in neural operator learning. The latter captures the emerging field evolution of interacting particles via efficient latent space representation. Lastly, we discuss model-order reduction and probabilistic learning techniques for high-dimensional parameterised systems, both of which are crucial for quantifying and incorporating uncertainties arising from physics-based and data-driven models. We present a typical workflow aimed at unifying data structures and modelling pipelines and guiding readers through the selection, training, and deployment of ML surrogates for granular material simulations. Finally, we illustrate the workflow’s practical use with two representative examples, focusing on granular materials in solid-like and fluid-like regimes.

Optimal filters are designed in vibration-based condition monitoring to enhance weak fault signatures for improved diagnosis. While optimisation-based filter design approaches have matured, their validation has typically focused on final objective values and constraint satisfaction. However, ensuring robust and reliable results requires verifying the correctness of design sensitivities, i.e., the gradients of both objective and constraint functions with respect to design variables, as well as hyperparameter sensitivities. This paper emphasises the importance of confirming and quantifying a filter's response to varying hyperparameters to ensure it meets design specifications. By rigorously verifying sensitivities, engineers can more confidently deploy optimal filter designs that enhance fault-related features, resulting in more effective fault detection and diagnosis in complex engineering systems.

In vibration-based condition monitoring, optimal filter design improves fault detection by enhancing weak fault signatures within vibration signals. This process involves optimising a derived objective function from a defined objective. The objectives are often based on proxy health indicators to determine the filter's parameters. However, these indicators can be compromised by irrelevant extraneous signal components and fluctuating operational conditions, affecting the filter's efficacy. Fault detection primarily uses the fault component's prominence in the envelope spectrum, quantified by an envelope spectrum-based signal-to-noise ratio. New optimal filter objective functions are derived from the proposed generalised envelope spectrum- based signal-to-noise objective for machines operating under variable speed conditions. Instead of optimising proxy health indicators, the optimal filter coefficients of the formulation directly maximise the envelope spectrum-based signal-to-noise ratio over targeted frequency bands using standard gradient-based optimisers. Four derived objective functions from the proposed objective effectively outperform five prominent methods in tests on three experimental datasets.

Guided wave ultrasound field measurements capture reflections from aluminothermic welds in the rail track at unknown distances from a transducer. The measured guided wave signals are complex and challenging to interpret due to multiple dispersive modes propagating in the rail, changing environmental conditions, and noise. Data-driven machine learning techniques have been applied to complex signal-processing problems and have shown significant potential in learning and resolving complex problems. This study aims to understand the implications of using various simulated data sets for application within an appropriate learning framework to capture the underlying features of the field measurements and maximise the performance of these techniques. The use of simulated spectrograms, including variations of signal attributes (attenuation, positions of welds, noise, and mode reflection coefficients) generally observed in experimental measurements, allows the reflections of individual modes to be isolated or combined for training. This allows us to present the training data in three distinct forms. The first dataset has the highest mode reflection information density per sample and consists of simulated spectrogram data with multiple reflections of modes from multiple welds, like experimentally obtained spectrograms. The second training dataset consists of spectrograms with multiple mode reflections; however, only for a single weld reflection per spectrogram. The third training set contains a reflection of a single mode from a single weld in each spectrogram. The data-driven models applied are principal component autoregression and variational auto-encoders. The reconstruction error and latent space interpretability were considered as metrics for the algorithms’ ability to learn using test sets, i.e., unseen data. The results show that datasets with sufficient feature variation and higher mode reflection information density better construct the test set of simulated and experimental spectrograms. However, training using the third dataset shows more interpretable latent variables for an artificial growing defect attached to the rail. Furthermore, data-driven machine learning methods trained using simulated spectrogram data are useful for reconstructing and learning features from experimental measurements, provided that the training data have representative mode feature variation and noise.

Linear latent variable models such as principal component analysis (PCA), independent component analysis (ICA), canonical correlation analysis (CCA), and factor analysis (FA) identify latent directions (or loadings) either ordered or unordered. These data are then projected onto the latent directions to obtain their projected representations (or scores). For example, PCA solvers usually rank principal directions by explaining the most variance to the least variance. In contrast, ICA solvers usually return independent directions unordered and often with single sources spread across multiple directions as multiple sub-sources, severely diminishing their usability and interpretability. This paper proposes a general framework to enhance latent space representations to improve the interpretability of linear latent spaces. Although the concepts in this paper are programming language agnostic, the framework is written in Python. This framework simplifies the process of clustering and ranking of latent vectors to enhance latent information per latent vector and the interpretation of latent vectors. Several innovative enhancements are incorporated, including latent ranking (LR), latent scaling (LS), latent clustering (LC), and latent condensing (LCON). LR ranks latent directions according to a specified scalar metric. LS scales latent directions according to a specified metric. LC automatically clusters latent directions into a specified number of clusters. Lastly, LCON automatically determines the appropriate number of clusters to condense the latent directions for a given metric to enable optimal latent discovery. Additional functionality of the framework includes single-channel and multi-channel data sources and data pre-processing strategies such as Hankelisation to seamlessly expand the applicability of linear latent variable models (LLVMs) to a wider variety of data. The effectiveness of LR, LS, LC, and LCON is shown in two foundational problems crafted with two applied latent variable models, namely, PCA and ICA.

Discontinuous loss functions are prevalent in engineering, sciences, data science, and machine learning applications. In engineering and the sciences, loss or constraint functions that require differential or partial differential equations to be solved often manifest discontinuities because of adaptive time stepping or remeshing. This results in abrupt changes in the discretization error, resulting in discontinuities. In the data sciences, loss functions computed from batches with different samples are discontinuous. These discontinuities are the result of abrupt changes in the sampling error. Alternatively, the loss and activation functions can be the source of discontinuities. This chapter gives illustrative examples of some of the origins of discontinuous loss functions and some basic strategies for exploiting gradients to optimize loss functions induced with discretization and sampling errors, i.e., gradient-only optimization.

Developing reliable ultrasonic-guided wave monitoring systems requires a significant amount of inspection data for each application scenario. Experimental investigations are fundamental but require a long period and are costly, especially for real-life testing. This is exacerbated by a lack of experimental data that includes damage. In some guided wave applications, such as pipelines, it is possible to introduce artificial damage and perform lab experiments on the test structure. However, in rail track applications, laboratory experiments are either not possible or meaningful. The generation of synthetic data using modelling capabilities thus becomes increasingly important. This paper presents a variational autoencoder (VAE)-based deep learning approach for generating synthetic ultrasonic inspection data for welded railway tracks. The primary aim is to use a VAE model to generate synthetic data containing damage signatures at specified positions along the length of a rail track. The VAE is trained to encode an input damage-free baseline signal and decode to reconstruct an inspection signal with damage by adding a damage signature on either side of the transducer by specifying the distance to the damage signature as an additional variable in the latent space. The training data was produced from a physics-based model that computes virtual experimental response signals using the semi-analytical finite element and the traditional finite element procedures. The VAE reconstructed response signals containing damage signatures were almost identical to the original target signals simulated using the physics-based model. The VAE was able to capture the complex features in the signals resulting from the interaction of multiple propagating modes in a multi-discontinuous waveguide. The VAE model successfully generated synthetic inspection data by fusing reflections from welds with the reflection from a crack model at specified distances from the transducer on either the right or left side. In some cases, the VAE did not exactly reconstruct the peak amplitude of the reflections. This study demonstrated the potential and highlighted the benefit of using a VAE to generate synthetic data with damage signatures as opposed to using superposition to fuse the damage-free responses containing reflections from welds with a damage signature. The results show that it is possible to generate realistic inspection data for unavailable damage scenarios.

Inverse problems in engineering form routinely part of larger engineering simulations.Therefore, the quality of the solution to an inverse problem directly influences the quality of the larger simulation and, ultimately, the ability to solve an engineering problem.Inverse problems can be challenging and time-consuming to solve, as most inverse strategies require iteration due to the non-linear nature of the problem.As a result, they often remain poorly solved before proceeding to the larger analysis.The quality of the solution to an inverse problem is influenced by the inverse strategy, scaling of the problem, scaling of the data, and initial guesses employed for iterative strategies.Research has focussed considerably on inverse strategies and scaling.However, research into strategies that improve initial guesses of an inverse problem has been largely neglected.This study proposes an elegant strategy to improve the initial guesses for conventional optimizationbased inverse strategies, namely direct inverse maps (DIMs) or inverse regression.DIMs form part of modern multivariate statistics.DIM approximates the solution to an inverse problem using regression; popular choices are linear regression, e.g., partial least squares regression (PLSR).These strategies are not iterative but require several independent apriori simulations to have been conducted.As they are not iterative, one way to improve the solution is to increase the number of independent a-priori simulations to be conducted.Our proposed strategy is to use DIM to generate initial guesses for optimization-based inverse strategies.We conduct a parameter investigation on a truss structure's virtual vibrationbased damage identification problem.

Multifidelity surrogate modelling combines data of varying accuracy and cost from different sources. It strategically uses low-fidelity models for rapid evaluations, saving computational resources, and high-fidelity models for detailed refinement. It improves decision-making by addressing uncertainties and surpassing the limits of single-fidelity models, which either oversimplify or are computationally intensive. Blending high-fidelity data for detailed responses with frequent low-fidelity data for quick approximations facilitates design optimisation in various domains. Despite progress in interpolation, regression, enhanced sampling, error estimation, variable fidelity, and data fusion techniques, challenges persist in selecting fidelity levels and developing efficient data fusion methods. This study proposes a new fusion approach to construct multi-fidelity surrogate models by constructing gradient-only surrogates that use only gradients to construct regression surfaces. Results are demonstrated on foundational example problems that isolate and illustrate the fusion approach's efficacy, avoiding the need for complex examples that obfuscate the main concept.

The Latent Space Perspicacity and Interpretation Enhancement (LS-PIE) framework enhances dimensionality reduction methods for linear latent variable models (LVMs). This paper extends LS-PIE by introducing an optimal latent discovery strategy to automate identifying optimal latent dimensions and projections based on user-defined metrics. The latent condensing (LCON) method clusters and condenses an extensive latent space into a compact form. A new approach, latent expansion (LEXP), incrementally increases latent dimensions using a linear LVM to find an optimal compact space. This study compares these methods across multiple datasets, including a simple toy problem, mixed signals, ECG data, and simulated vibrational data. LEXP can accelerate the discovery of optimal latent spaces and may yield different compact spaces from LCON, depending on the LVM. This paper highlights the LS-PIE algorithm’s applications and compares LCON and LEXP in organising, ranking, and scoring latent components akin to principal component analysis or singular value decomposition. This paper shows clear improvements in the interpretability of the resulting latent representations allowing for clearer and more focused analysis.

Recent developments in surrogate construction predominantly focused on two strategies to improve surrogate accuracy. Firstly, component-wise domain scaling informed by cross-validation. Secondly, regression to construct response surfaces using additional information in the form of additional function-values sampled from multi-fidelity models and gradients. Component-wise domain scaling reliably improves the surrogate quality at low dimensions but has been shown to suffer from high computational costs for higher dimensional problems. The second strategy, adding gradients to train surrogates, typically results in regression surrogates. Counter-intuitively, these gradient-enhanced regression-based surrogates do not exhibit improved accuracy compared to surrogates only interpolating function values. This study empirically establishes three main findings. Firstly, constructing the surrogate in poorly scaled domains is the predominant cause of deteriorating response surfaces when regressing with additional gradient information. Secondly, surrogate accuracy improves if the surrogates are constructed in a fully transformed domain, by scaling and rotating the original domain, not just simply scaling the domain. The domain transformation scheme should be based on the local curvature of the approximation surface and not its global curvature. Thirdly, the main benefit of gradient information is to efficiently determine the (near) optimal domain in which to construct the surrogate. This study proposes a foundational transformation algorithm that performs near-optimal transformations for lower dimensional problems. The algorithm consistently outperforms cross-validation-based component-wise domain scaling for higher dimensional problems. A carefully selected test problem set that varies between 2 and 16-dimensional problems is used to clearly demonstrate the three main findings of this study.

Latent variable models can effectively determine the condition of essential rotating machinery without needing labelled data. These models analyse vibration data via an unsupervised learning strategy. Temporal preservation is necessary to obtain an informative latent manifold for the fault diagnosis task. In a temporal-preserving context, two approaches exist to develop a condition-monitoring methodology: offline and online. For latent variable models, the available training modes are no different. While many traditional methods use offline training, online training can dynamically adjust the latent manifold, possibly leading to better fault signature extraction from the vibration data. This study explores online training using temporal-preserving latent variable models. Within online training, there are two main methods: one focuses on reconstructing data and the other on interpreting the data components. Both are considered to evaluate how they diagnose faults over time. Using two experimental datasets, the study confirms that models from both training modes can detect changes in machinery health and identify faults even under varying conditions. Importantly, the complementarity of offline and online models is emphasised, reassuring their versatility in fault diagnostics. Understanding the implications of the training approach and the available model formulations is crucial for further research in latent variable model-based fault diagnostics. Conflict of Interest Statement The authors declare no conflicts of interest.

This work describes a post-processing scheme for multiphase flow systems to characterize primary atomization. The scheme relies on the 2D fast Fourier transform (FFT) to separate the inherently multi-scale features present in the flow results. Emphasis is put on the robust quantitative analysis enabled by this scheme, with this work specifically focusing on comparing atomizer nozzle designs. The generalized finite difference (GFD) method is used to simulate a high pressure gas injected into a viscous liquid stream. The proposed scheme is applied to time-averaged results exclusively. The scheme is used to evaluate both the surface and volume features of the fluid system. Due to the better recovery of small-scale features using the proposed scheme, the benefits of post-processing multiphase surface information rather than fluid volume information was shown. While the volume information lacks the fine-scale details of the surface information, the duality between interfaces and fluid volumes leads to similar trends related to the large-scale spatial structure recovered from both surface- and volume-based data sets.

Operational risk has been widely studied, and international guidelines provide procedures for the correct management of operational risk; however, this has not been studied from a corporate sustainability point of view. Therefore, this work seeks to find a way to model and optimize the impact of operational risks on corporate sustainability. The methodology used is based on the assignment of two distribution functions for the creation of a probabilistic model that allows quantifying the probability of occurrence (frequency) and the expected monetary impact (severity) on the sustainability variables (environmental, social, and economic). The result is a statistical convolution through Monte Carlo simulation, which makes it possible to quantify aggregate losses to finally make an optimization process of the variables and estimate the financial impact. Therefore, this study extends the literature on risk quantification, proposing a stochastic model that quantifies and optimizes the operational risks that are related to corporate sustainability. The proposed model offers a practical way to quantify operational risks related to corporate sustainability while also being flexible, as it does not require historical information and can be used with data collected from the company based on the proposed probability distributions. Finally, the proposed model has three limitations: the distribution functions, use of Solver (Excel), and exclusion of some risk management strategies, which future research can consider.

This paper presents a GPU-based, incompressible, multiphase generalized finite difference solver for simulating multiphase flow. The method includes a dampening scheme that allows for large density ratio cases to be simulated. Two verification studies are performed by simulating the relaxation of a square droplet surrounded by a light fluid and a bubble rising in a denser fluid. The scheme is also used to simulate the collision of binary droplets at moderate Reynolds numbers (250–550). The effects of the surface tension and density ratio are explored in this work by considering cases with Weber numbers of 8 and 180 and density ratios of 2:1 and 1000:1. The robustness of the multiphase scheme is highlighted when resolving thin fluid structures arising in both high and low density ratio cases at We = 180.

Guided wave ultrasound (GWU) systems have been widely used for monitoring structures such as rails, pipelines, and plates. In railway tracks, the monitoring process involves the complicated propagation of waves over several hundred meters. The propagating waves are multi-modal and interact with discontinuities differently, increasing complexity and leading to different response signals. When the researcher wants to gain insight into the behavior of guided waves, predicting response signals for different combinations of modes becomes necessary. However, the task can become computationally costly when physics-based models are used. Digital twins can enable a practitioner to deal systematically with the complexities of guided wave monitoring in practical or user-specified settings. This paper investigates the use of a hybrid digital model of an operational rail track to predict response signals for varying user-specified settings, specifically, the prediction of response signals for various combinations of modes of propagation in the rail. The digital twin hybrid model employs a physics-based model and a data-driven model. The physics-based model simulates the wave propagation response using techniques developed from the traditional 3D finite element method and the 2D semi-analytical finite element method (FEM). The physics-based model is used to generate virtual experimental signals containing different combinations of modes of propagation. These response signals are used to train the data-driven model based on a variational auto-encoder (VAE). Given an input baseline signal containing only the most dominant mode excited by a transducer, the VAE is trained to predict an inspection signal with increased complexity according to the specified combination of modes. The results show that, once the VAE has been trained, it can be used to predict inspection signals for different combinations of propagating modes, thus replacing the physics-based model, which is computationally costly. In the future, the VAE architecture will be adapted to predict response signals for varying environmental and operational conditions.

This work extends a multi-phase mixing model framework designed for a Smoothed Particle Hydrodynamics context. Specifically, we propose a higher-order variation using the first-order accurate Generalised Finite Difference differential operators to construct an incompressible scheme for simulating fluid-solid coupled systems resolved via a continuum mixture model. The proposed scheme incorporates inter-phase shear between phases and the viscosity dependency of the solid phase concentration. The scheme is verified by simulating a modified lid-driven cavity case at Re = 1000. In this simulation, our method was capable of treating initially discontinuous concentration fields with a maximum solid volume concentration of 0.5 and a solid-to-fluid density ratio of 4.