Background: Patients suffering from critical congenital heart disease (cCHD) require cardiac intervention within the first year of life. During the postoperative period, patients are at risk of haemodynamic instability resulting in insufficient organ perfusion and subsequent organ failure. To prevent this, patients are placed in the paediatric intensive care unit (PICU) where various vital parameters can be monitored. However, interpreting these continuous data streams can be challenging. Machine learning offers the potential to support clinical decision-making in this setting, but challenges remain, particularly in labelling haemodynamic instability and accounting for varying physiology of this patient demographic. The goal of this study was to improve the retrospective labelling of haemodynamic instability and evaluate the affect of age-stratified subpopulation on model performance.

Methods: This study used a retrospective dataset of continuously measured parameters (heart rate, respiratory rate, mean arterial pressure, central venous pressure, oxygen saturation, and perfusion index) collected from post-operative cCHD patients admitted to the PICU of Erasmus MC Sophia Children's Hospital, the Netherlands, between January 2016 and April 2025. A new scoring system was developed to quantify the haemodynamic support received by patients and to identify intervention times at which support was increased. These interventions were used to label haemodynamic instability in the a period dT prior to intervention. The resulting labelling was applied to train a random forest algorithm to predict haemodynamic instability, and the model was subsequently retrained on age-based subpopulations.

Results: A total of 425 patients were included for this study. The new labelling method resulted in 5.7% of the data being labelled as haemodynamically unstable. The random forest using the new labelling achieved an average (SD) area under precision-recall curve (AUCPR) of 0.233 (0.041) on the test set during cross-validation and final test AUCPR of 0.203. The largest age subpopulations were 0--30 days and 90--180 days. The class prior of instability was 9.2% in the 0--30 days subpopulation and the prediction model achieved an AUCPR of 0.244 (0.064). In the 90--180 day subpopulation the class prior was 4.9% and an AUCPR of 0.221 (0.105) was achieved.

Conclusion: This study proposed a new method of retrospectively labelling haemodynamic instability with the goal of training a predictive model to predict these instabilities. A random forest model trained using the new labelling showed limited improvement, with an AUCPR of 0.204 and an AUCROC of 0.762. Age-based subpopulation analysis indicated potential for reduced data variation, though larger cohorts are needed for better generalisation. Further refinements in the retrospective labelling of haemodynamic instability are required for an effective prediction model to be developed.

Introduction: In critically ill paediatric patients, sleep is essential for recovery and development, yet sleep disturbances are common in the paediatric intensive care unit (PICU), which highlight the need to integrate sleep monitoring into clinical practice. While automated sleep stage classification using machine learning (ML) on single-channel electroencephalography (EEG) data has shown promise in mostly healthy adult populations, its application to critically ill children is challenged by age-specific sleep architecture, medication effects, pathological conditions, and artefacts. This study evaluates whether deep learning (DL) feature extraction and dynamic models can improve sleep staging performance in this population and explores the use of an unsupervised ML model to gain deeper insight into the complex sleep structures in this population.

Methods: This study utilised EEG recordings from three datasets—healthy adults, non-critically ill children, and critically ill children—to train and evaluate supervised and unsupervised sleep stage classification models. As supervised models, a convolutional neural network (CNN) was used for feature extraction, followed by dynamic models including a long short-term memory (LSTM) network and a hidden Markov model (HMM) to account for temporal dependencies in sleep. Additionally, an unsupervised HMM was applied to explore underlying structures in the sleep EEG data without predefined labels.

Results: Supervised models achieved good performance in healthy adults and non-critically ill children, with maximum accuracies of 90.2\% and 77.4\%, respectively, for three-state classification. The added value of dynamic models over the CNN alone varied per dataset and model type and was not consistent. In critically ill children, classification performance was low, with a maximum accuracy of 61.4\%, and notably low macro-F1 and Cohen’s kappa scores (45.9\% and 26.5\%, respectively). The unsupervised HMM revealed that identifying distinct and stable clusters over time was challenging in all datasets. For critically ill children, the model often failed to identify multiple distinct clusters within individual patients, and substantial variability in cluster assignments was observed across patients.

Discussion: This study demonstrates that DL-based feature extraction and dynamic modelling using single-channel EEG can achieve strong sleep staging performance in healthy adults and non-critically ill children, highlighting the potential for (semi-)automated scoring tools in more stable populations. In contrast, performance in critically ill children was notably lower, likely due to factors such as high variability in sleep architecture, signal artefacts, limited data quality, and the uncertain reliability of manually assigned labels. These results suggest that conventional sleep stages do not generalise well to this population, and a purely data-driven, unsupervised approach does not offer a viable alternative. Overall, the findings emphasise the need for a larger dataset of critically ill children, further evaluation of relevant patient and data characteristics, the inclusion of alternative signals such as electrocardiography, and greater focus on model interpretability.

As Deep Neural Networks (DNNs) continue to be deployed in safety-critical domains, two specific concerns — adversarial examples and Out-of-Distribution (OoD) data — pose significant threats to their reliability. This thesis proposes novel methods to enhance the robustness of deep learning by detecting such inputs and mitigating their impact.

A central insight of this work is that algorithmic stability plays a crucial role in generalizing to in-distribution data. Motivated by this, the thesis formulates a dual perspective on stability with respect to the hypotheses and explores whether this perspective facilitates the separation of problematic inputs under two main lenses: epistemic uncertainty estimation and the choice of an appropriate inductive bias. By grounding our approach in generative modeling with a latent variable based on an information bottleneck and, specifically, employing Variational Autoencoders (VAEs), we first leverage Bayesian inference over model parameters to estimate the model’s uncertainty with respect to a particular input. Second, we investigate the required properties of both VAE maps and latent representations from a topological perspective. This reveals how OoD inputs predominantly map onto empty regions — or “holes” — in the latent manifold. Finally, we discover that adversarial examples likewise exhibit similar behavior. This finding is then used to craft new scoring functions that reliably distinguish between inliers, outliers, and adversarial attacks.

The integration of wearable technology into healthcare is revolutionizing health monitoring by enabling continuous tracking of vital metrics like heart rate and blood sugar. Devices such as smartwatches and glucose monitors empower proactive interventions, reducing hospital visits and personalizing care. For instance, wearables can detect irregular heart rhythms for early cardiovascular disease detection or assist individuals with diabetes in managing glucose levels. These advancements are enabled by technologies like photoplethysmography (PPG), a non-invasive method for real-time monitoring of physiological signals. Continuous monitoring generates time series data that captures dynamic health fluctuations over time. This data allows for identifying irregularities and deviations that isolated measurements might miss. Detecting anomalies, such as abrupt changes in heart rate or prolonged abnormal patterns, is essential for timely interventions in managing chronic conditions like hypertension and cardiovascular diseases.

However, the analysis of time series data introduces challenges. For instance, label scarcity arises because labeling health anomalies requires expert input, which is often infeasible for large datasets. Inter-subject variability becomes a concern as physiological patterns differ significantly across individuals, complicating model generalization. Furthermore, temporal dependencies in time series data add complexity, as observations are sequentially related and anomalies may not manifest as isolated points but as patterns or sequences deviating from normal behavior. Detecting subtle anomalies, minor deviations that accumulate over time but may signal early-stage conditions, becomes particularly challenging due to their resemblance to normal temporal variations and noise in the data. For example, a gradual change in heart rate variability might indicate the onset of an irregular rhythm but could easily blend into inherent variability if not carefully analyzed. Moreover, evaluation metrics for time series data are insufficient, failing to capture the temporal complexities of real-world applications. Consequently, conventional metrics can misrepresent model performance, leading to unreliable or misleading assessments.

This thesis addresses these challenges by advancing time series anomaly detection through innovative methodologies. A key focus is on addressing the limitations of existing evaluation metrics by introducing new evaluation metrics to better capture temporal complexities, ensuring reliable and meaningful performance assessments. Beyond evaluation, this thesis is guided by several core principles to address the challenges inherent in time series data analysis. Central to this is the use of unsupervised representation learning to tackle label scarcity and variability, enabling robust feature extraction from unlabeled data while maintaining generalizability. Finally, the thesis develops strategies for increasing sensitivity to subtle anomalies, providing effective solutions for identifying small yet significant deviations in complex datasets. Together, these contributions present a comprehensive framework for improving anomaly detection systems across diverse applications, bridging theoretical advancements with practical, real-world needs.

Introduction:
Pulmonary exacerbations are critical events in paediatric patients with asthma or cystic fibrosis (CF). These exacerbation events are often associated with sudden health deterioration and increased healthcare burden. The early prediction of exacerbations events could allow for timely interventions, and thus improved patient outcomes. This thesis attempted to develop a machine learning (ML) model to predict pulmonary exacerbations before they occur in a paediatric population using remote patient monitoring (RPM) data.

Methods:
A retrospective study was conducted using continuous data from wearable devices, daily spirometry, environmental data, and patient-reported outcomes. Predictions were focused on the occurrence of an exacerbation within three prediction windows (1-day, 3-day, and 7-day). Two ML approaches were considered: anomaly detection (using Gaussian mixture model, Isolation forest, One-class-SVM, and Local outlier factor), and classification models (Logistic regression, Random forest), using 5-fold nested cross-validation. Time-related transformations were performed to capture the temporal dependency of time-series data, including the feature engineering of clinical features related to heart rate and physical activity.

Results:
A total of 2401 home monitoring days of 90 paediatric patients, with 10 observed exacerbation events were included in the analysis. All models struggled to achieve high predictive value, with PR-AUC values below 0.20 and ROC-AUC values ranging from 0.43 to 0.72 across different time windows. No single model consistently outperformed the others. Despite the low performance, the models demonstrated better than random prediction for secondary outcomes, such as weekends and holidays, suggesting the ability to capture patterns in the data.

Conclusion:
This thesis shows the potential and limitations of using ML techniques for predicting pulmonary exacerbations using RPM data. The current anomaly detection and classification model performances are insufficient for clinical application. The low incidence of exacerbation events and the limitations in data quality contribute to these results. These findings point to the need for further refinement and more robust datasets to fully realise the potential of ML in the context of predicting pulmonary exacerbations.

Weather forecasting has always been a critical issue across various fields, and traditionally, weather predictions have relied on solving complex atmospheric equations using supercomputers. However, the rise of machine learning has introduced a more efficient method that utilizes historical data to forecast weather patterns. Nonetheless, there is a problem with auto-correlation, where values at a specific time point correlate with those at a previous time. Machine learning models assume independence between data points, which can lead to overfitting if the data is not split properly. This happens because training data might not be independent of test data and models can learn the noise rather than the underlying pattern in data. Also, auto-correlation can apply when having multiple locations with data (named spatial data), because nearby locations share similar characteristics.

This thesis investigates temporal and spatial data splitting strategies to address this challenge to determine which methods provide the most reliable performance estimates. Temporal splitting strategies involve dividing the data using time intervals, while spatial splitting strategies involve dividing based on the geographical location of weather stations. The goal is to identify strategies that minimize bias and variance in the error estimates of weather forecasting models. This will ensure less overfitting and robust and generalisable predictions across different climatic conditions and data regimes.

By systematically evaluating various data splitting strategies, this thesis aims to provide insights into the best approaches for preparing meteorological data for Machine Learning-based weather prediction models, thereby contributing to the advancement of more accurate and efficient weather forecasting techniques. Four different strategies for temporal data splitting were evaluated: random splitting, testing at the end of the dataset, testing in the middle of the dataset, and splitting the test year into multiple parts. Three different strategies for splitting spatial data were evaluated: random splitting, the use of one cluster of cities as the test data with the other clusters as training data, and lastly, choosing only the non-neighbouring clusters as training data. Some of these strategies, such as the random strategy, do not take into account auto-correlation when splitting the data.

Results indicate that the random strategy in both scenarios yields a relatively modest error rate. However, it is advisable to employ a strategy that organizes the test data into a cohesive block and positions it in the centre of the dataset time. Hence, this thesis contributes to the development of more refined weather prediction by shedding light on how data splitting strategies influence outcomes.

Physics-Informed Neural Networks (PINNs) offer a promising approach to solving partial differential
equations (PDEs). In PINNs, physical laws are incorporated into the loss function, guiding the network to learn a model that adheres to these laws as defined by the PDEs. Training PINNs involves using three types of points: collocation points within the domain of the PDEs, boundary points on the edges of the domain, and initial points at the starting time (t = 0). A common challenge in training PINNs is the imbalance between the number of boundaries and initial points compared to collocation points, which can negatively impact the training process. Typically, the number of each type of point is manually set, and the number of collocation points is usually ten times that of boundary and initial points, leading to an uneven loss distribution over time. Our work introduces a method called Relative Residual Resampling (R3) to address this issue. This method dynamically adjusts the number of each type of training point to ensure a more balanced distribution. Consequently, the loss is more evenly spread across the time domain, enhancing the performance of the PINN. We tested our method on the 1D Heat Equation and 1D Diffusion Equation. The results demonstrate that our approach reduces the overall loss by at least 6%, and balances the loss distribution over time by reducing the loss range for at least 65% compared with the state-of-the-art residual-based resampling strategy. This improvement makes PINNs more accurate and efficient for solving time-dependent PDEs, providing a practical solution for applications where understanding temporal dynamics is crucial.

When addressing combinatorial optimization problems, the focus is predominantly on their computational complexity, and it is often forgotten to look at the bigger picture. As a result, it is common to miss critical details which could play a major role in the overall process. One such detail is the presence of uncertainty in the real world. A naive approach might directly predict values for the uncertain parameters, without taking into account that the ultimate goal is to obtain sound decisions. Consequently, in many cases, the resulting solutions are suboptimal. This challenge is precisely the premise behind Decision-Focused Learning (DFL) framework, which is a core of this work. This study pioneers the application of the DFL framework to scheduling problems with uncertain processing times, utilizing contextual features to predict these uncertainties. By employing the promising Score Function Gradient Estimation method, the research tackles the issue of non-differentiable regret loss functions in DFL. Key contributions include the development of techniques to enhance the performance of the Score Function, an in-depth analysis of DFL's applicability to complex scheduling scenarios, and a detailed evaluation of its strengths and weaknesses. This work not only demonstrates the potential of DFL in this new context but also lays the groundwork for future research and improvements in handling uncertainty in combinatorial optimization.

Federated Learning (FL), is a distributed learning approach where multiple clients collaboratively train a model whilst maintaining data security and privacy. One significant challenge in FL that must be addressed is statistical heterogeneity within the data. This occurs because data across different clients may not come from the same distribution, potentially leading to sub-optimal performance. To address this, we examine how insights gained from a generative model’s latent space can mitigate these problems by adjusting the aggregation weight (influence) assigned to each client during the training process. We leverage information derived from a Variational Autoencoder (VAE) trained in a federated manner and propose a method to modify the aggregation weight of each client in FL. This method considers local discrepancies, resulting from differences between the local latent space distributions and global latent space distributions, together with the dataset sizes of each client. Experiments were conducted on the MNIST and Fashion-MNIST datasets. Our results indicate that our method enhance the model’s performance by up to 6.76% in the best case, in terms of reducing the average test VAE loss and accelerating the convergence of the β-VAE in scenarios characterised by severe data imbalances among clients. It worsens performance when all clients have an equal level of imbalance. The source code for our research is available at https://github.com/FederatedRP2024Delft/
Federated-Learning-PyTorch-Weight-Modification

Federated learning (FL) enables privacy-preserving collaboration among numerous clients for training machine learning models. In FL, a server coordinates model aggregation while preserving data privacy. However, non-identically and independently distributed (non-IID) local data label distributions degrade the performance of the global model. This paper investigates the impact of synthetic data on mitigating non-IID data distributions in federated learning. We explore data-based augmentation techniques, including uniform and minority imputation, utilizing conditional variational autoencoders (CVAEs) to generate synthetic data.
Additionally, we examine a framework-based approach where a pre-trained model, centrally trained on synthetic data, is distributed to clients for finetuning on their original datasets. Our results, which use the binarized MNIST dataset, demonstrate a quality gap between synthetic and original datasets, leading to diminished classification performance when trained on only synthetic data. Integrating both original and synthetic data improves performance on heavily imbalanced label distributions. At the same time, uniform imputation experiments reveal that optimal imputation must strike a balance, with performance degradation being noticeable when datasets consist of more than 45\% synthetic images. Synthetic imputation did not suffer from degradation in the explored range of imputation amounts and achieved an average F1 score improvement of 0.015 over uniform imputation.

Federated learning stands as an approach to train machine learning models on data residing at multiple clients, but where data must remain private to the client it belongs to. Despite its promise, federated learning faces significant challenges, particularly when dealing with non-IID and non-stationary data. A model trained on non-stationary data can be subject to concept shift, where the data used for training faces a sudden change of concept, leading to a large performance degradation when classifying data under the new concept. This research focuses on comparing the performance of federated and centralized models under such conditions. Our objective is to evaluate the extent to which federated models are more affected by concept shift than their centralized counterparts. Through a series of experiments involving image (CIFAR-10) and tabular data (2-dimensional, linearly separable, binary-classification), we demonstrate that while federated models can achieve performances close to centralized models, they exhibit greater sensitivity to data complexity and distribution shifts. Our findings suggest that, despite centralized models being better than federated ones, the gain in performance from gathering data in one place might not outweigh the privacy concerns. Furthermore, we also find that the accuracy under concept shift is dependent on the performance on original data.

Federated learning (FL) is an innovative approach in machine learning that enables model training across multiple decentralized devices or servers without sharing local data, thus preserving privacy and utilizing decentralized data. However, a significant challenge in FL is handling non-IID (Non-Identical and Independently Distributed) data, which can adversely affect performance. This paper investigates the impact of federated learning on the performance of various generative models, including Generative Adversarial Networks (GANs) and Variational Autoencoders (VAEs), specifically in the context of image and tabular data generation tasks. Our study aims to determine how these generative models perform when trained in a federated manner compared to centralized training. We evaluate the models using several metrics, including classifier accuracy on generated images, Earth Mover’s Distance (EMD) for distribution comparison, resemblance, discriminability, downstream utility, and privacy metrics for tabular data. Experiments conducted on the MNIST and CIFAR-10 datasets for image generation, and the Adult and Abalone datasets for tabular data generation, reveal that VAEs exhibit robust and consistent performance across federated and centralized setups. In contrast, GANs show significant performance degradation under federated non-IID conditions. The results indicate that VAEs can effectively address the non-IID data challenge in FL by generating high-quality synthetic data, thereby enhancing model generalizability and stability. The framework used for executing the experiments in this study can be found at https://github.com/alexojica/research-project-experiments.

Federated Learning (FL) is a machine learning approach that has gained considerable interest over the years. FL allows global models to train without compromising the data privacy of the clients' training datasets by sending the global model to each client to learn the weights and propagating only the learned weights back to a central location. However, it is not without limitations as several challenges hinder the model's performance. One of those challenges is the presence of non-IID (Independent and Identically Distributed) properties in the training data. Most real-world data is non-IID, and this imbalance in data distribution has been shown to significantly affect the model's performance. To address this issue, we propose a generative federated learning by pre-training the global model on synthetic data created by a generative model that follows the collective distribution of all clients' training datasets. Our research shows that this approach bridges the performance gap between IID and non-IID in FL, except for certain extreme non-IID cases.

Introduction
Patient-ventilator asynchrony (PVA) poses a significant challenge in the management of mechanically ventilated patients, contributing to adverse clinical outcomes. Current methods of detecting PVA rely on visual assessment by clinicians, leading to subjectivity and inconsistency. Therefore, there is a need for automated techniques to identify PVA accurately and efficiently. In this study, we explore the application of supervised and unsupervised machine learning algorithms to develop an automatic detection system for PVA.

Methods
This study was conducted at the ICU of the LUMC in Leiden, the Netherlands. Patients eligible for inclusion were mechanically ventilated with an esophageal balloon inserted. Data collected included flow, Paw, and Pes curves, which were labelled using an open-source data labeling platform and processed in Python. Supervised CNN models were trained for different ventilation modes, while unsupervised techniques, utilizing Mahalanobis distance, were explored for data pre-labeling. The discriminative capability of the models was assessed using AUROC values.
Results
25 patients were included in this study and labelled by clinicians. Using an unsupervised machine learning technique based on the Mahalanobis distance for data pre-labeling, a threshold of 3.5 was selected, resulting in a 95% accuracy in correctly identifying normal breaths. Creating different CNN models for automating the detection of PVA the results demonstrate the discriminative capability of the various models across all ventilation modes, PSV and PCV ventilation. They can differentiate between normal and abnormal breaths, as indicated by the AUROC values of 0.85(±0.08), 0.83 (±0.12), and 0.80 (±0.28) respectively.

Discussion
This study investigated the application of machine learning techniques to analyse ventilation data in critical care settings. Through a combination of unsupervised and supervised learning methods, we have explored the automation of the labelling process and the development of predictive models for identifying patient-ventilator asynchronies (PVAs).

Introduction
Approximately 9 in 1000 children are born with congenital heart disease (CHD), of whom a quarter are classified as critical CHD (CCHD) and require an intervention within their first year. Monitoring these patients in the Paediatric Intensive Care Unit (PICU) is crucial, yet with increasing amounts of data, detecting subtle changes that are important for the disease progression interpretation of all vital signs becomes difficult, even for skilled physicians. Machine learning (ML) offers potential solutions, however, challenges such as inter-patient variability and the absence of clear definitions for haemodynamic instability persist. This study aims to develop a ML algorithm for early prediction of haemodynamic instability in CCHD patients with high frequency vital-signs, addressing these challenges through objective labelling methods and stratification approaches.

Methods
Two approaches, on population and patient level, were developed with nested cross-validation (CV). Due to a high inter-patient variability, the patient specific approach was added. A first iteration of objectively labelling haemodynamic instability was proposed, based on medical interventions such as medication administration and fluid therapy. Since it is difficult to retrospectively determine for how long patients were unstable, multiple values for instability duration (dT ) were added to the analysis.
To capture the temporal dependency of time-series data, lag-analysis was performed, adding the relation between the vital signs and their previous values to the model development. Lag-analysis included a sliding window that moved over the data. The width of sliding window (W ) was optimised during the model development. Additionally, a horizon (r ) was implemented, so the data within the sliding window were predicting future timestamps.

Results
This retrospective study included a total of 224 admissions in the analysis. Two random forest classifiers were trained using a nested CV structure to detect haemodynamic instability in CCHD patients. For both approaches the same temporal settings (W : 50 minutes, r : 45 minutes, dT: 120 minutes) were used. This study has shown that the between-patient approach had notable differences between the mean train (85%, AUCPR) and test performance (40%, AUCPR). The in-patient approach, while using 20% and 10% of the test data for training, still yielded a test performance of 96% (AUCPR) and 90% (AUCPR), respectively.

Discussion and conclusion
Generally speaking, the experiments suggest that the first iterations of the models were not robust and generalised poorly. It is most likely caused by a large inter-patient variability and a simple labelling system that is still depending on subjectivity.
This study has shown that the proposed prediction model, which combines high frequency vital signs, labels, and temporal settings (W, r, dT ), requires additional refinement before it can be considered clinically feasible to implement this model as a reliable bedside tool for predicting haemodynamic instability.

Data-driven approaches are a promising new addition to the list of available strategies for solving Partial Differential Equations (PDEs). One such approach, the Principal Component Analysis-based Neural Network PDE solver, can be used to learn a mapping between two function spaces, corresponding to a PDE. However, the practical limitations of this approach are unclear. This paper seeks to investigate for which types of inputs and outputs this type of solver gives useful results. Using a dataset with inputs sampled from Gaussian Random Fields with different parameters, and outputs for Poisson's equation and the Heat equation, obtained by using a Finite Element solver, neural networks are trained, and their performance is evaluated. The method performs adequately for the chosen inputs, and patterns are found in the resulting error, which differ for each set of input parameters. Thus, for these equations, it seems that this method performs differently for different input distributions, but further research is necessary to investigate if these patterns will hold for other equations.

Solving Partial Differential Equations (PDEs) in engineering such as Navier-Stokes is incredibly computationally expensive and complex. Without analytical solutions, numerical solutions can take ages to simulate at great expense. In order to reduce this cost, neural networks may be used to compute approximations of the solution for use during engineering processes. PCA-net is a neural network approach that reduces the dimensionality of the input and output data for PDEs in order to allow mapping from a high-dimensional input and output function with a fully connected neural network through the use of Principal Component Analysis (PCA). In this paper, PCA-net is applied to Navier-Stokes with varying viscosities to test the generalization of PCA-net on viscosity parameters. Training is done on four discrete viscosities, while testing is done on continuous viscosities, extrapolating and interpolating around the training set. Results shows good performance on low viscosities, both with interpolation and extrapolation. Mid-to-high viscosity interpolation shows lesser performance, with high viscosity extrapolation diverging to great error. Omitting high viscosities, performance over varying viscosities is close to that shown by previous research.

This paper presents a comprehensive exploration of a novel method combining Principal Component Analysis (PCA) and Neural Networks (NN) to efficiently solve Partial Differential Equations (PDEs), a fundamental challenge in modeling a wide range of real-world phenomena. Our research extends the work of Bhattacharya et al. by focusing on PCA for effective dimensionality reduction and utilizing NN for mapping in the reduced dimension. This approach addresses the significant computational challenges and inaccuracies often encountered with classical numerical techniques in solving PDEs.

We specifically investigate the still-water equation, employing our PCA-NN method to learn a reduced order mapping of PDE solutions and evaluate its robustness in diverse noisy environments. Our findings reveal a notable relationship between noise intensity and error, indicating a linear trend for Gaussian and Salt and Pepper noise, and an exponential trend for Uniform noise. Furthermore, this study uncovers a critical weakness of the model in predicting points with a high rate of change.

Overall, our research significantly contributes to understanding the practical applicability and limitations of PCA-NN methods in real-world, noisy settings, offering valuable insights for future applications in this domain.