Solving the incompressible Navier-Stokes equations is computationally heavy, with the pressure Poisson equation being the most time-consuming step. Iterative linear solvers are typically utilized to solve this equation. Since most solvers are iterative and rely on an initial guess, an opportunity emerges to use machine learning to improve this initial guess, such that fewer iterations are needed, consequently saving time.
A novel graph neural network is designed that employs a custom convolution algorithm, message-passing scheme, and pooling algorithm to maximize its performance. First, a convolution algorithm is proposed that uses interpolation to make a discrete (3x3) CNN kernel continuous. Then, instead of directly computing the kernel weight from the function, an integral over specified bounds is applied to account for the geometrical inhomogeneous distribution of the source nodes. The integral is embedded as the weighted sum of a vector containing learnable parameters, computed through a dot product with the edge attribute vectors. Next to the convolution operation, a message-passing scheme is designed that is compatible with the data format of the finite volume method whilst performing well in terms of the distance information can travel over the mesh. To conclude the design of the model, a custom pooling algorithm is designed that is equivalent to average pooling in CNNs.
A normalization procedure is established that ensures consistency in the model's magnitude. Notably, the ground truth output pressure is normalized using its standard deviation, which is unknown. To estimate this normalization factor, a correction model is established that uses the same convolution algorithm but employs an architecture inspired by classification CNNs.
The model's performance is evaluated based on the reduction in number of iterations required to reach convergence. This is done for both the Preconditioned Conjugate Gradient (PCG) solver and the multigrid Geometric Agglomerated Algebraic Multigrid (GAMG) solver
Across the various tests conducted, the number of iterations needed to reach convergence is reduced by approximately 40%, with the PCG solver performing slightly better than the GAMG solver. However, the PCG solver yields less consistent results, performing very well at samples that closely align with the training data, leading to a reduction of up to 60%. However, its performance drops significantly when tested on data that does not closely resemble the training data, sometimes even increasing the number of iterations. The GAMG solver demonstrates consistent performance, with almost no difference between the training and evaluation data.
In terms of generalization, the model demonstrates promising results, achieving similar performance across datasets with varying levels of complexity. This is interesting as the root mean square error differs significantly across the datasets and individual samples. This suggests that there is no direct relationship between the reduction in the number of iterations and the accuracy of the prediction. Furthermore, the model performs well on unseen meshes, showing that it can handle the unstructured nature of the meshes used throughout this research. This demonstrates the model's ability to be trained on a diverse dataset, after which it can be applied to unseen cases.

This thesis explores the development and application of clustering-based reduced-order modeling (ROM) for chaotic systems, with an emphasis on both predictive modeling and control strategies. Chaotic systems, characterized by their sensitivity to initial conditions and complex spatio-temporal dynamics, present significant challenges in terms of prediction and control. Traditional numerical methods for solving these systems are computationally intensive and often impractical for real-time applications. Reduced-order modeling techniques, which seek to represent high-dimensional systems in lower-dimensional subspaces, offer a promising alternative.

Clustering-based ROM is highlighted as a potentially effective method due to its data-driven nature and computational efficiency. This study focuses on studying the influence of different parameters in the clustering-based ROM approach in modeling and developing a robust control algorithm to mitigate extreme events. This is investigated on three different chaotic systems of increasing complexity: the Lorenz system, the truncated Charney-DeVore (CDV) system and the Moehlis-Faisst-Eckhart (MFE) system. The ability of the clustering-based ROM in reproducing those systems' statistics is confirmed. The influence of the number of clusters and the order of modeling on the ROM accuracy is explored, and a quantitative method to determine the number of clusters when modeling with clustering-based ROM is proposed.

For the control part, a clustering-based control strategy is applied to the CDV system and MFE system. In the CDV system, the control aims to move the system away from the blocked state, and in the MFE system, the objective is to prevent extreme events (which take the form of quasi-relaminarisation events). In both cases, the clustering-based control manages to achieve these objectives, with a reduction of 90% of extreme events in the MFE case. These results highlight the potential of clustering-based control.

This research contributes to the field by providing a viable approach to managing chaotic systems with reduced computational demands, offering potential applications in various engineering and scientific domains.

The motion of liquid metals is described by the equations of magnetohydrodynamics (MHD), that com bine the Maxwell equations and the Navier-Stokes equations. In these type of flows, the magnetic field interacting with the conductive metal induces large pressure losses and unconventional turbulence states such as quasi 2D turbulence, turbulence suppression and flow anisotropy. Currently this turbu lence behaviour can be captured in higher fidelity Computational Fluid Dynamics (CFD) simulations such as Large Eddy Simulation (LES) and Direct Numerical Simulation (DNS), but the high computa tional cost of these simulations make them impractical for industrial applications compared to Reynolds AveragedNavier-Stokes(RANS).However, theeddyviscositymodelswhicharetypicallyusedinRANS are not able to capture anisotropic turbulence states occurring in MHD flows, which results in sig nificant discrepancies in the mean velocity field. Hence, this work presents a data-driven approach to model MHD turbulence. To achieve this time-averaged LES data of annular pipe flow cases at differ ent Hartmann numbers are used to derive corrections for the k − ω SST model. Two correction fields are obtained through a frozen RANS simulation in which the mean LES fields are inserted into the RANS equations. The Reynolds stress anisotropy term is approximated with a modified Tensor Basis Neural Network (TBNN). Moreover, for modelling the turbulence production correction a Scalar Basis Neural Network (SBNN) is proposed and compared to a Sparse Algebraic Regression using the SpaRTA approach. The resulting data driven models are able to reduce the error of the Reynolds stress anisotropy values and the mean flow velocity fields, and can generalise to annular flow cases with different Hartmann numbers from those of the training cases.

This thesis explores the use of physics-informed neural networks (PINNs) to reconstruct the flow fields in a pool fire flame, a canonical configuration in non-premixed combustion. Due to the difficulty in obtaining adequate experimental characterizations of such flows, reacting flows like pool fires stand in need of novel methods capable of reconstructing the principal flow features from limited measurement data.

The present work extends the capabilities of previous PINN reconstruction frameworks through the addition of a new physical prior that embeds information about the flame structure: the mixture fraction. This passive scalar quantifies the local state of mixing in diffusion flames, playing a crucial role in their analysis.

A novel reconstruction framework that incorporates the mixture fraction has been developed. The mixture fraction is found to be a valuable addition to the reconstruction framework: it either reduces the amount of flow variables required to obtain successful velocity reconstructions, leads to higher reconstruction accuracies, alleviates some of the PINN’s well-known failure modes, or a combination of the above. Furthermore, it allows for adequate first order estimates of the heat release rate (HRR) of the flame even in the absence of pressure, density, and temperature data.

This research contributes to addressing climate change challenges through the examination of hydrogen combustion. It investigates the flow dynamics within a simplified model of Ansaldo Energia's GT36 reheat combustor using Large Eddy Simulation (LES) at a high pressure of 20 bar, focusing on the autoignition flashback phenomena observed. More specifically, it explores predicting the apparition of this event through the use of a modularity-based clustering algorithm and its subsequent suppression by using water injection.

In the LES simulation, lean, premixed and high-pressure conditions are used, revealing an unsteady behaviour in the flame dynamics, i.e. an autoignition event in the mixing duct which repeats itself due to high-amplitude pressure waves and their subsequent convergence in the mixing duct. Using this simulation, time-series of multiple variables are acquired over the course of 8 flashback events. These variables are then pruned using a co-kurtosis PCA based dimensionality reduction technique. This method, by measuring the joint occurrence of outliers in the flow variables, identifies successfully which are the most important variables (temperature, density, pressure, velocity in the axial direction, mass fractions of HO2 and OH) that introduce a lasting change in the system and are potentially useful in finding a precursor to the flashback event.

The time-series of these variables are then introduced in the modularity-based clustering algorithm. This algorithm tessellates the phase space of the system and then transforms it into a graph, thus retaining the information about the dynamics of the system. Then, it clusters this graph using a metric called modularity. The method proves effective in finding a precursor for the autoignition event, resulting in an average prediction time for the 8 flashback events of 32.13 microseconds, which is over 50% of the time that the combustor is in the normal operating state. Furthermore, the algorithm performs very well in the number of false positives, and, due to a change made to the algorithm in this study, the number of true positives is increased to 100%. The algorithm is then put through several robustness tests, which include the use of different sampling locations, less features, unseen data and relying only on temperature and pressure information sampled at the combustor walls. Here, the algorithm retained its performance, with only a small decrease in the prediction time, demonstrating its potential towards its use in an online prediction scenario. Lastly, the level of fidelity of the LES simulation was increased by using the digital filter method to simulate turbulent fluctuations at the inlet and fully-developed velocity profile, where once again it was found that the algorithm retains a good prediction time.

For the second part of this research, a flashback event and its afferent prediction time were chosen to investigate the use of water injection and its potential at suppressing the flashback when the water is injected based on the prediction time. Here, following an empirical design approach, where the Sauter Mean Diameter (SMD), the mass flow, the diameter of the nozzle, the angles of the cone were varied, a preliminary design was sought. It was found that the SMD is highly influential towards the spread of the spray, with larger particles being preferred to their ability to retain their momentum and more quickly cover the mixing duct. In addition, large angles for the cone and a high injection velocity are again necessary for a good spread and a quick response time. An attempt was also made to place a set of sprays at the walls of the mixing duct, where it was found that the flashback event now takes place upstream of the spray due to aerodynamic blockage. The design process culminates in a setup where six spray are placed at the inlet of the mixing duct. In this setup, the spray is able to quickly cover the mixing duct and the flashback is suppressed, while also retaining an evaporation efficiency of 96.3%.

While physics-informed neural networks have been shown to accurately solve a wide range of fluid dynamics problems, their effectivity on highly compressible flows is so far limited. In particular, they struggle with transonic and supersonic problems that involve discontinuities such as shocks. While there have been multiple efforts to alleviate the nonphysical phenomena that arise on such problems, current work does not identify and address the underlying failure modes sufficiently. This thesis shows that physics-informed neural networks struggle with highly compressible problems for two independent reasons. Firstly, the differential Euler equations conserve entropy along streamlines, so that physics-informed neural networks try to find an isentropic solution to a non-isentropic problem. Secondly, conventional slip boundary conditions form strong local minima that result in fictive objects that simplify the flow. In response to these failure modes, two new adaptations are introduced, namely a local viscosity method and a streamline output representation. The local viscosity method includes viscosity as an additional network output and adds a viscous loss term to the loss function, resulting in localized viscosity that facilitates the entropy change at shocks. Furthermore, the streamline output representation provides a more natural formulation of the slip boundary conditions, which prevents zero-velocity regions while promoting shock attachment. To the author's best knowledge, this thesis provides the first inviscid steady solutions of curved and detached shocks by physics-informed neural networks.

Most physical systems of interest are chaotic in nature. Quick and reasonably accurate solutions for these systems are essential to various fields such as the effective control mechanism construction and early-stage design. However, their chaotic nature also leads to them being computationally expensive to model using traditional numerical techniques. Reduced Order Modelling (ROM) is an umbrella term that describes any method wherein a system state is projected into a ‘simplified’ state space in order to extract meaningful information about its dominant components. Traditional ROMs whilst effective in capturing statistical information, tend to neglect smaller-energy components responsible for short-term dynamics. With the advances seen in data-driven approaches over the past decade, namely in machine learning (specifically deep learning), these methods seem to offer attractive alternatives to traditional ROM techniques and are the focus of this thesis.
It is found that in existing literature there is no consensus on which ML models to use and no real extensive comparative studies either. Further, there is no unified approach to training the different RNN models, and only one surveyed study utilises an auto-regressive optimization strategy. This is a major pitfall, since the RNNs are only trained for single-step prediction but tested on multi-step prediction - a clear contradiction in the training and testing objectives. Additionally, no studies have been performed on true Runge-Kutta methods-inspired layering architectures, which show initial promise in increasing prediction horizons at little additional compute cost. As such, an extensive comparative study is performed, pitting long-short term memory networks, gated recurrent unit networks and echo state networks against each other. Further, the effect of RK-inspired layering is tested against a standard single-layer network. The effect of incorporating contractive losses in the auto-encoders is also investigated, in addition to multi-regime autoencoders. Every model is tested on chaotic systems of increasing complexity, namely the Lorenz ’63 system, the Charney-DeVore system, the Kuramoto-Sivashinsky system and the Kolmogorov flow system.
It is observed that the contractive loss does not help in modelling a smoother latent space, and instead leads to the unintended minimization of the norm of the latent space variables. Multi-regime auto-encoders are found to work quite well, with well separated latent spaces reflecting the nature of the dynamics (chaotic/periodic). Auto-regressive training is found to be crucial for increasing the prediction performance of back-prop trained RNNs (provided additional care is taken to limit their gradients during training) while it has little to no effect on the ESNs. Further, an initial skip-layer is found to be beneficial whereas, higher order RK layering architectures provide diminishing to no returns.

The computational cost of high-fidelity engineering simulations, for example CFD, is prohibitive if the application requires frequent design iterations or even fully fledged optimization. A popular way to reduce the computational cost and enable fast iteration cycles is to use surrogate models that are trained to predict simulation results from historical simulation data. While most traditional methods are parametric, ANNs are able to process geometries directly and are thus agnostic to the parametrization of the geometric models, which makes them appealing when working on multiple design campaigns. However, ANNs may fail to transfer the learned knowledge when used on new design campaigns that are significantly different from those used to train the model or when the size of the training data set is too small.

The goal of this project is to increase the reliability of ANN-based surrogate models on new design campaigns and on small datasets. One main direction to achieve this goal is to incorporate prior physical knowledge into the learning process. Methods to supplement training data with a simplified solution, meaningful physical scaling, and governing equation-based losses were used. The data set used for this study was based on a real-life-inspired complex 3D parametrized geometry of an automotive HVAC system.

The methods were tested in generalization, transfer learning, and single-campaign inference tasks. Moreover, physics-informed losses were tested in an ill-posed setting as a prediction outlier correction tool. While the initial findings suggest that incorporating prior physical knowledge may improve model performance, especially in low-data regimes, further investigation is needed to draw any definitive conclusions. Additionally, the preliminary results related to the use of physics-informed losses as a correction method are inconclusive and require further experimentation in the future. Therefore, more research is needed to determine the effectiveness of these approaches.

The aerodynamic model of a combat aircraft is essential for its success and competitiveness compared to other combat aircraft. This thesis aims to research the most optimal machine learning model to create an aerodynamic model of a combat aircraft. The very large but still sparse, highly nonlinear dataset forms a challenge for using specific machine learning models. Tree-based models, artificial neural networks (ANNs), and Bayesian neural networks (BNNs) have been identified to be individually capable of modelling the aerodynamics of combat aircraft. For ANNs, additional research was performed into genetic algorithms, as a robust hyperparameter optimisation method. Transfer learning, which reduced the training time by around 14%. Finally, adding gradient information of the training data as an additional input, reduced the mean squared error (MSE) by 7%. Scalable BNNs, which used Bayes-by-backprop, were developed to handle the large dataset. The uncertainties coming from the BNN were overconfident in the results compared to the tolerances of the dataset. The uncertainties showed only a weak correlation with the MSE of a given prediction.
Finally, to leverage each of the model's advantages, a stacked model was created which improves the predictive performance on average by 27% compared to the best base model in terms of the MSE on the test dataset. With the stacked model implemented, each of the aerodynamic coefficients could be predicted with over 97% of the predictions of the test data within the tolerance. This is an average increase of 0.5% for each of the aerodynamic coefficients compared to the best base models. Due to the strict regulations related to this aerodynamic model, the machine learning model that is created cannot replace the aerodynamic model at this time. However, the implementation of this machine learning model allows engineers to design the aerodynamic model faster, and with greater precision.

Chaotic systems are widespread and can be found everywhere, from small scale processes inside the human body to the large scale dynamics of the entire atmosphere. However, modelling these high dimensional chaotic systems is a difficult task due to the intrinsic nonlinear nature of chaos as well as the accompanied computational cost. Therefore, in this research, the predictive potential of hybrid reduced order models consisting of a knowledge-based (proper orthogonal decomposition Galerkin projection) and a deep learning reservoir computing component was studied for high dimensional chaotic systems, for which two hybrid architectures were designed. In the first, the blending of the knowledge-based and deep learning components occurred within the reservoir of the data-driven component (Hybrid-A) and its performance was assessed through the short-term prediction horizon and long-term statistical predictions and compared to the pure knowledge-based and pure deep learning reduced order models. For the second model (Hybrid-FFNN), the blending occurred in a separate feed-forward neural network and the performance was compared to the Hybrid-A model. In order to obtain efficient deep learning predictions for both models, a low dimensional latent space representation of the physical system was obtained using autoencoders. Both models were tested on two-dimensional Kolmogorov flow at a Reynolds number of Re = 20 (periodic regime) and Re = 34 (chaotic regime).

For the periodic case, the short-term performance of the Hybrid-A and deep learning model were comparable, while the knowledge-based model was outperformed due to instabilities as a result of the truncation of the number of modes from proper orthogonal decomposition. Furthermore, an increase in latent space and/or deep learning reservoir size had no consistent influence on the performance of the Hybrid-A model, due to the periodic and thus simple to learn dynamics for the deep learning component. Finally, little influence was found for the accumulation of (de)compression errors occurring in the Hybrid-A model, which was compensated for by the additional information from the knowledge-based model and/or the Hybrid-A model was able to restore the errors due to the periodic nature of the system.

Increasing the Reynolds number to the chaotic regime resulted in the Hybrid-A model to underperform compared to the knowledge-based and deep learning models for smaller latent space due to the (de)compression error accumulation. In addition, the Hybrid-A model was unable to perform better than the knowledge-based model for larger latent spaces and number of retained modes, from which it was concluded that too much information on the physical system was lost in latent space for the Hybrid-A model to benefit from both components, while the knowledge-based model operated on the (full) physical system. For the Hybrid-FFNN it was shown that the feed-forward neural network introduced a too large error to be beneficial over the Hybrid-A model.

For the long-term statistics of both test cases, the importance of the design for the tuning of the hyperparameters of the deep learning model/component was showed. In general, no clear influence was found in terms of the performance of the Hybrid-A model as a function of the latent space and reservoir size. Furthermore, the comparative performance of the knowledge-based, deep learning and Hybrid-A models showed no relation. This behaviour was expected to be the result of a design strategy that biased the short-term performance over the long-term performance of the deep learning model/component. Furthermore, for the periodic regime, such behaviour could also have originated from the periodicity of the flow.

Up to the authors knowledge, this work proposed the first hybrid knowledge-based/deep learning reduced order model using autoencoders for efficient predictions of high dimensional chaotic systems. Even though it was found that the proposed Hybrid models were unable to show increased performance compared to the knowledge-based and deep learning models, the results can be viewed as a start to design other hybrid architectures for potential performance improvement.

This thesis aims to automatically and reliably detect large-scale structures in turbulent flows. To achieve this, a U-net (a type of neural network) is trained using artificially generated data. From the network output, continuous structures are derived and general statistics, including, volume fraction, relative kinetic energy and length scales (using PCA) are computed.

Detections were done on two homogenous isotropic turbulence (HIT) direct numerical simulation (DNS) datasets with Taylor Reynolds numbers of 175 and 1131, respectively. For both cases, the detected structures contained most of the volume and kinetic energy in the domain and were of the integral length scale. However, for the high Reynolds number, there were relatively half as many structures and the structures were roughly 4 times larger compared to those found in the low Reynolds number case.

Physics-informed machine learning is a novel approach to solving flow problems with physics-informed neural networks (PINNs), that combines physical knowledge and machine learning.
This study aims to investigate the potential of the application of PINNs in fluid mechanics problems by solving two practical flow problems.

The first case considers the reconstruction of the full, time accurate flow fields with PINNs from partial data in the wake of an airfoil with periodic vortex shedding.
The results have shown a decent success in flow reconstruction case, but has trouble maintaining a high accuracy at higher time values.
This is likely due to the lack of a mechanism forcing a time marching approach to prevent information flowing in the opposite direction of the positive time axis.

The second case attempts to train PINNs on a steady flow problems with parametric NACA airfoils first to devise a strategy for training PINNs on a parameteric airfoil and show its feasibility.
This is followed by using the found strategy to train PINNs for the more complex flow cases with PARSEC airfoils to use them as fast surrogate flow solvers, while still producing the full flow fields with a comparable accuracy to conventional CFD.

Overall, the flow fields inferred by the PINNs have shown a good qualitative match with the OpenFOAM validation data, even in the most complex case with a PARSEC airfoil, which makes them sufficiently accurate as a surrogate flow solver for a preliminary design optimization.
These results show not only the technical feasibility of PINNs for fluid mechanics, but also a practical value with accelerating the computation time up to a factor of 3.7 when accounting for the training time and producing the OpenFOAM training data.

Recent years have seen an increase in studies focusing on data-driven techniques to enhance modelling approaches like the two-equation turbulence models of Reynolds-averaged Navier-Stokes (RANS). Different techniques have been implemented to improve the results from these simulations. In particular, the main focus has been on overcoming the limitations implied by the Boussinesq assumption. This has been approached by using machine learning techniques as a way of discovering new formulations that could overperform when compared to traditional models.
Despite promising results for steady RANS simulations, little has yet been investigated in URANS applications. In this dissertation, this lack of research will be addressed. The main ideas are then, first, to see how the available information in URANS simulations can be used to improve the anisotropic Reynolds stress tensor prediction, and second if and how this can be done by using a sparse regression technique, whose framework is known as SpaRTA. A procedure involving the triple decomposition of High-Fidelity velocity fields is applied, aiming at finding a model exclusively for the stochastic component of the anisotropy. The test case which is considered is the flow around a cylinder at Re=3900. The High-Fidelity data was collected by running Large Eddy Simulations in OpenFOAM, after which the velocity was split into its different components through Proper Orthogonal Decomposition.
A priori results have shown good performance of the trained models, outperforming the Boussinesq assumption both in the prediction of turbulence componentiality and also on the values of the single anisotropy components.

In many flow experiments it is complex to measure all flow states of interest, leading to the need for a method to retrieve unmeasured flow states from measured ones. This work focuses on Hidden Fluid Mechanics (HFM), which refers to a Physics-Informed Neural Network (PINN) able to incorporate the Navier-Stokes (NS) equations into the loss function of the Neural Network (NN) for the purpose of reconstructing flow fields.

As HFM is a recent framework which has not been applied to many flows yet, it is unknown if HFM is capable of reconstructing flow fields in chemically reacting flows where strong gradients are present. For this reason, the performance of HFM is investigated here using data sets from Direct Numerical Simulations (DNS) of an axisymmetric pool fire in cylindrical coordinates, for which the velocity fields are purposely removed and attempted to be recovered from the density, pressure and temperature fields. The implementation has been performed making use of TensorFlow in Python, where both steady and unsteady pool fires are considered.

Pool fires are simplified representations of real-life accidental and forest fires, illustrating the importance of their research. If the critical pool diameter is not exceeded, a steady flow occurs. However, if surpassed, the flow exhibits an unsteady flickering behaviour known as puffing.

The main finding is that both in the steady and unsteady pool fires, the PINN is capable of reconstructing the most prominent features of the velocity fields when density, pressure and temperature are provided, even though strong gradients are usually underestimated by the PINN, causing wrongly predicted extreme values. This implies that the method can be used mainly for qualitative purposes, as the exact values are not always captured. It is found that the accuracy is lower in the unsteady flow compared to the steady flow. Furthermore, the axial velocity field is generally reconstructed to a higher accuracy than the radial velocity field.

It is found that the accuracy decreases when increasing the spacing between provided data points, from which implications on possible applications to real-life measured data can be drawn, as not all measurement equipment might reach the required spatial resolution. Investigating the effect of adding artificial white noise to mimic measurement noise, it has been found that the method is robust to noise, but less so for the steady case than for the unsteady flow case. Lastly, the effect of removing the density and/or pressure from the provided flow states is investigated, where it is found to have a detrimental effect on the accuracy except if physical assumptions are given to the PINN (i.e. the ideal gas law). Using the right assumptions, it is found that for the steady case providing only the temperature field is deemed enough to reach a relatively similar accuracy compared to not removing any flow states, whereas for the unsteady case at least the temperature and pressure fields need to be provided. The current research shows the capabilities but also the limitations of the HFM framework in case of steady and unsteady flow with chemical reactions and strong gradients, creating opportunities for further research of HFM in other flow cases and for potentially applying the framework to real experimental data of fires or other flows.

Abrupt and rapid high-amplitude changes in a dynamical system’s states known as extreme events appear in many processes occurring in nature, such as drastic climate patterns, rogue waves, or avalanches. These events often entail catastrophic effects, therefore their description and prediction is of great importance. Nevertheless, because of their chaotic nature, their modelling poses a great problem up to this day.

This Master thesis focuses on researching the applicability of a modularity-based clustering technique to identify precursors of rare and extreme events in chaotic systems. A data-driven identification framework based on clustering of system states, probability transition matrices and state space tessellation was developed and tested on six different reduced-order systems. As a final stage, the framework was applied to a two-dimensional model of turbulence (the Kolmogorov flow) which exhibits extreme events in the form of quasi-relaminarisation with extreme dissipation of kinetic energy. The proposed framework provides a way to identify pathways towards extreme events and predict their occurrence from a probabilistic standpoint. The clustering algorithm identifies the precursor states leading to extreme events and allows for a statistical description of the system’s states.