Daniël N. Wilke

Jesse Stevens, Daniel N. Wilke, Itumeleng Setshedi

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. The data is then projected onto the latent directions to obtain their projected representations (or scores). For example, PCA solvers usually rank the principal directions by explaining the most to least variance, while ICA solvers usually return independent directions unordered and often with single sources spread across multiple directions as multiple sub-sources, which is of severe detriment to their usability and interpretability. This paper proposes a general framework to enhance latent space representations for improving the interpretability of linear latent spaces. Although the concepts in this paper are language agnostic, the framework is written in Python. This framework automates the clustering and ranking of latent vectors to enhance the latent information per latent vector, as well as, the interpretation of latent vectors. Several innovative enhancements are incorporated including latent ranking (LR), latent scaling (LS), latent clustering (LC), and latent condensing (LCON). For a specified linear latent variable model, LR ranks latent directions according to a specified metric, LS scales latent directions according to a specified metric, LC automatically clusters latent directions into a specified number of clusters, while, LCON automatically determines an appropriate number of clusters into which to condense the latent directions for a given metric. Additional functionality of the framework includes single-channel and multi-channel data sources, data preprocessing 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, and LCON are showcased on two crafted foundational problems with two applied latent variable models, namely, PCA and ICA.

Daniel N. Wilke

We present a solver-agnostic framework in which coordinated large language model (LLM) agents autonomously execute the complete computational mechanics workflow, from perceptual data of an engineering component through geometry extraction, material inference, discretisation, solver execution, uncertainty quantification, and code-compliant assessment, to an engineering report with actionable recommendations. Agents are formalised as conditioned operators on a shared context space with quality gates that introduce conditional iteration between pipeline layers. We introduce a mathematical framework for extracting engineering information from perceptual data under uncertainty using interval bounds, probability densities, and fuzzy membership functions, and introduce task-dependent conservatism to resolve the ambiguity of what `conservative' means when different limit states are governed by opposing parameter trends. The framework is demonstrated through a finite element analysis pipeline applied to a photograph of a steel L-bracket, producing a 171,504-node tetrahedral mesh, seven analyses across three boundary condition hypotheses, and a code-compliant assessment revealing structural failure with a quantified redesign. All results are presented as generated in the first autonomous iteration without manual correction, reinforcing that a professional engineer must review and sign off on any such analysis.

Ryan Balshaw, P. Stephan Heyns, Daniel N. Wilke et al.

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.

Stephan Schmidt, Daniel N. Wilke, Konstantinos C. Gryllias

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 squared envelope spectrum, quantified by a squared 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 squared 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.

Marc Fransen, Andreas Fürst, Deepak Tunuguntla et al.

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. Attended by researchers from both the granular materials (GM) and machine learning (ML) communities, a recent Lorentz Center Workshop on "Machine Learning for Discrete Granular Media" brought the ML community up to date with GM challenges. This position paper emerged from the workshop discussions. 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 digital twins for 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. We then explore graph neural networks and recent advances in neural operator learning. Lastly, we discuss model-order reduction and probabilistic learning techniques for high-dimensional parameterised systems, which are crucial for quantifying uncertainties arising from physics-based and data-driven models. We present a 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.

Daniel N Wilke

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.

Jay Raut, Daniel N. Wilke, Stephan Schmidt

Data normalisation, a common and often necessary preprocessing step in engineering and scientific applications, can severely distort the discovery of governing equations by magnitudebased sparse regression methods. This issue is particularly acute for the Sparse Identification of Nonlinear Dynamics (SINDy) framework, where the core assumption of sparsity is undermined by the interaction between data scaling and measurement noise. The resulting discovered models can be dense, uninterpretable, and physically incorrect. To address this critical vulnerability, we introduce the Sequential Thresholding of Coefficient of Variation (STCV), a novel, computationally efficient sparse regression algorithm that is 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 terms in the model library. This shift from magnitude to statistical significance makes the discovery process invariant to arbitrary data scaling. Through comprehensive benchmarking on canonical dynamical systems and practical engineering problems, including a physical mass-spring-damper experiment, we demonstrate that STCV consistently and significantly outperforms standard Sequential Thresholding Least Squares (STLSQ) and Ensemble-SINDy (E-SINDy) on normalised, noisy datasets. The results show that STCV-based methods can successfully identify the 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, thereby enhancing model interpretability and trustworthiness.

Younghwan Chae, Daniel N. Wilke, Dominic Kafka

Mini-batch sub-sampling (MBSS) is favored in deep neural network training to reduce the computational cost. Still, it introduces an inherent sampling error, making the selection of appropriate learning rates challenging. The sampling errors can manifest either as a bias or variances in a line search. Dynamic MBSS re-samples a mini-batch at every function evaluation. Hence, dynamic MBSS results in point-wise discontinuous loss functions with smaller bias but larger variance than static sampled loss functions. However, dynamic MBSS has the advantage of having larger data throughput during training but requires the complexity regarding discontinuities to be resolved. This study extends the gradient-only surrogate (GOS), a line search method using quadratic approximation models built with only directional derivative information, for dynamic MBSS loss functions. We propose a gradient-only approximation line search (GOALS) with strong convergence characteristics with defined optimality criterion. We investigate GOALS's performance by applying it on various optimizers that include SGD, RMSprop and Adam on ResNet-18 and EfficientNetB0. We also compare GOALS's against the other existing learning rate methods. We quantify both the best performing and most robust algorithms. For the latter, we introduce a relative robust criterion that allows us to quantify the difference between an algorithm and the best performing algorithm for a given problem. The results show that training a model with the recommended learning rate for a class of search directions helps to reduce the model errors in multimodal cases.

Dominic Kafka, Daniel Nicolas Wilke

Gradient-only and probabilistic line searches have recently reintroduced the ability to adaptively determine learning rates in dynamic mini-batch sub-sampled neural network training. However, stochastic line searches are still in their infancy and thus call for an ongoing investigation. We study the application of the Gradient-Only Line Search that is Inexact (GOLS-I) to automatically determine the learning rate schedule for a selection of popular neural network training algorithms, including NAG, Adagrad, Adadelta, Adam and LBFGS, with numerous shallow, deep and convolutional neural network architectures trained on different datasets with various loss functions. We find that GOLS-I's learning rate schedules are competitive with manually tuned learning rates, over seven optimization algorithms, three types of neural network architecture, 23 datasets and two loss functions. We demonstrate that algorithms, which include dominant momentum characteristics, are not well suited to be used with GOLS-I. However, we find GOLS-I to be effective in automatically determining learning rate schedules over 15 orders of magnitude, for most popular neural network training algorithms, effectively removing the need to tune the sensitive hyperparameters of learning rate schedules in neural network training.

D. Kafka, Daniel. N. Wilke

Gradient-only line searches (GOLS) adaptively determine step sizes along search directions for discontinuous loss functions resulting from dynamic mini-batch sub-sampling in neural network training. Step sizes in GOLS are determined by localizing Stochastic Non-Negative Associated Gradient Projection Points (SNN-GPPs) along descent directions. These are identified by a sign change in the directional derivative from negative to positive along a descent direction. Activation functions are a significant component of neural network architectures as they introduce non-linearities essential for complex function approximations. The smoothness and continuity characteristics of the activation functions directly affect the gradient characteristics of the loss function to be optimized. Therefore, it is of interest to investigate the relationship between activation functions and different neural network architectures in the context of GOLS. We find that GOLS are robust for a range of activation functions, but sensitive to the Rectified Linear Unit (ReLU) activation function in standard feedforward architectures. The zero-derivative in ReLU's negative input domain can lead to the gradient-vector becoming sparse, which severely affects training. We show that implementing architectural features such as batch normalization and skip connections can alleviate these difficulties and benefit training with GOLS for all activation functions considered.

Dominic Kafka, Daniel N. Wilke

Learning rates in stochastic neural network training are currently determined a priori to training, using expensive manual or automated iterative tuning. This study proposes gradient-only line searches to resolve the learning rate for neural network training algorithms. Stochastic sub-sampling during training decreases computational cost and allows the optimization algorithms to progress over local minima. However, it also results in discontinuous cost functions. Minimization line searches are not effective in this context, as they use a vanishing derivative (first order optimality condition), which often do not exist in a discontinuous cost function and therefore converge to discontinuities as opposed to minima from the data trends. Instead, we base candidate solutions along a search direction purely on gradient information, in particular by a directional derivative sign change from negative to positive (a Non-negative Associative Gradient Projection Point (NN- GPP)). Only considering a sign change from negative to positive always indicates a minimum, thus NN-GPPs contain second order information. Conversely, a vanishing gradient is purely a first order condition, which may indicate a minimum, maximum or saddle point. This insight allows the learning rate of an algorithm to be reliably resolved as the step size along a search direction, increasing convergence performance and eliminating an otherwise expensive hyperparameter.

Younghwan Chae, Daniel N. Wilke

Choosing appropriate step sizes is critical for reducing the computational cost of training large-scale neural network models. Mini-batch sub-sampling (MBSS) is often employed for computational tractability. However, MBSS introduces a sampling error, that can manifest as a bias or variance in a line search. This is because MBSS can be performed statically, where the mini-batch is updated only when the search direction changes, or dynamically, where the mini-batch is updated every-time the function is evaluated. Static MBSS results in a smooth loss function along a search direction, reflecting low variance but large bias in the estimated "true" (or full batch) minimum. Conversely, dynamic MBSS results in a point-wise discontinuous function, with computable gradients using backpropagation, along a search direction, reflecting high variance but lower bias in the estimated "true" (or full batch) minimum. In this study, quadratic line search approximations are considered to study the quality of function and derivative information to construct approximations for dynamic MBSS loss functions. An empirical study is conducted where function and derivative information are enforced in various ways for the quadratic approximations. The results for various neural network problems show that being selective on what information is enforced helps to reduce the variance of predicted step sizes.

Dominic Kafka, Daniel Wilke

Step sizes in neural network training are largely determined using predetermined rules such as fixed learning rates and learning rate schedules. These require user input or expensive global optimization strategies to determine their functional form and associated hyperparameters. Line searches are capable of adaptively resolving learning rate schedules. However, due to discontinuities induced by mini-batch sub-sampling, they have largely fallen out of favour. Notwithstanding, probabilistic line searches, which use statistical surrogates over a limited spatial domain, have recently demonstrated viability in resolving learning rates for stochastic loss functions. This paper introduces an alternative paradigm, Gradient-Only Line Searches that are Inexact (GOLS-I), as an alternative strategy to automatically determine learning rates in stochastic loss functions over a range of 15 orders of magnitude without the use of surrogates. We show that GOLS-I is a competitive strategy to reliably determine step sizes, adding high value in terms of performance, while being easy to implement.

Dominic Kafka, Daniel Wilke

Mini-batch sub-sampling in neural network training is unavoidable, due to growing data demands, memory-limited computational resources such as graphical processing units (GPUs), and the dynamics of on-line learning. In this study we specifically distinguish between static mini-batch sub-sampled loss functions, where mini-batches are intermittently fixed during training, resulting in smooth but biased loss functions; and the dynamic sub-sampling equivalent, where new mini-batches are sampled at every loss evaluation, trading bias for variance in sampling induced discontinuities. These render automated optimization strategies such as minimization line searches ineffective, since critical points may not exist and function minimizers find spurious, discontinuity induced minima. This paper suggests recasting the optimization problem to find stochastic non-negative associated gradient projection points (SNN-GPPs). We demonstrate that the SNN-GPP optimality criterion is less susceptible to sub-sampling induced discontinuities than critical points or minimizers. We conduct a visual investigation, comparing local minimum and SNN-GPP optimality criteria in the loss functions of a simple neural network training problem for a variety of popular activation functions. Since SNN-GPPs better approximate the location of true optima, particularly when using smooth activation functions with high curvature characteristics, we postulate that line searches locating SNN-GPPs can contribute significantly to automating neural network training