A significant advancement in Neural Network (NN) research is the integration of domain-specific knowledge through custom loss functions. This approach addresses a crucial challenge: how can models utilize physics or mathematical principles to enhance predictions when dealing with sparse, noisy, or incomplete data? Physics-Informed Neural Networks (PINNs) put this idea into practice by incorporating physical equations, such as Partial Differential Equations (PDEs), as soft constraints. This guidance helps the networks find solutions that align with established laws. Recently, researchers have expanded this framework to include Bayesian NNs (BNNs), which allow for uncertainty quantification while still adhering to physical principles. But what happens when the governing equations of a system are not known? In this work, we introduce methods to automatically extract PDEs from historical data. We then integrate these learned equations into three different modeling approaches: PINNs, Bayesian-PINNs (B-PINNs), and Bayesian Linear Regression (BLR). To assess these frameworks, we evaluate them on a real-world Multivariate Time Series (MTS) dataset. We compare their effectiveness in forecasting future states under different scenarios: with and without PDE constraints and accuracy considerations. This research aims to bridge the gap between data-driven discovery and physics-guided learning, providing valuable insights for practical applications.

Received 17 February 2025, accepted 5 March 2025, date of publication 13 March 2025, date of current version 21 March 2025. Continual Learning with Quasi-Newton Methods STEVEN VANDER EECKT and HUGO VAN HAMME (Senior, IEEE) Department Electrical Engineering ESAT-PSI, KU Leuven, B-3001 Leuven, Belgium Corresponding author: Steven Vander Eeeckt (e-mail: steven.vandereeckt@esat.kuleuven.be).ABSTRACT Catastrophic forgetting remains a major challenge when neural networks learn tasks sequentially. Elastic Weight Consolidation (EWC) attempts to address this problem by introducing a Bayesian-inspired regularization loss to preserve knowledge of previously learned tasks. However, EWC relies on a Laplace approximation where the Hessian is simplified to the diagonal of the Fisher information matrix, assuming uncorrelated model parameters. This overly simplistic assumption often leads to poor Hessian estimates, limiting its effectiveness. To overcome this limitation, we introduce Continual Learning with Sampled Quasi-Newton (CSQN), which leverages Quasi-Newton methods to compute more accurate Hessian approximations. Experimental results across four benchmarks demonstrate that CSQN consistently outperforms EWC and other state-of-the-art baselines, including rehearsal-based methods. CSQN reduces EWC's forgetting by 50% and improves its performance by 8% on average. Notably, CSQN achieves superior results on three out of four benchmarks, including the most challenging scenarios, highlighting its potential as a robust solution for continual learning.INDEX TERMS artificial neural networks, catastrophic forgetting, continual learning, quasi-Newton methods I. INTRODUCTION Since the 2010s, Artificial Neural Networks (ANNs) have been able to match or even surpass human performance on a wide variety of tasks. However, when presented with a set of tasks to be learned sequentially--a setting referred to as Continual Learning (CL)--ANNs suffer from catastrophic forgetting [1]. Unlike humans, ANNs struggle to retain previously learned knowledge when extending their knowledge. Naively adapting an ANN to a new task generally leads to a deterioration in the network's performance on previous tasks. Many CL methods have been proposed to alleviate catastrophic forgetting. One of the most well-known is Elastic Weight Consolidation (EWC) [2], which approaches CL from a Bayesian perspective. After training on a task, EWC uses Laplace approximation [3] to estimate a posterior distribution over the model parameters for that task. When training on the next task, this posterior is used via a regularization loss to prevent the model from catastrophically forgetting the previous task. To estimate the Hessian, which is needed in the Laplace approximation to measure the (un)certainty of the model parameters, EWC uses the Fisher Information Matrix (FIM). Furthermore, to simplify the computation, EWC assumes that the FIM is approximately diagonal.

Causal Bayesian Optimization (CBO) is a methodology designed to optimize an outcome variable by leveraging known causal relationships through targeted interventions. Traditional CBO methods require a fully and accurately specified causal graph, which is a limitation in many real-world scenarios where such graphs are unknown. To address this, we propose a new method for the CBO framework that operates without prior knowledge of the causal graph. Consistent with causal bandit theory, we demonstrate through theoretical analysis and that focusing on the direct causal parents of the target variable is sufficient for optimization, and provide empirical validation in the context of CBO. Furthermore we introduce a new method that learns a Bayesian posterior over the direct parents of the target variable. This allows us to optimize the outcome variable while simultaneously learning the causal structure. Our contributions include a derivation of the closed-form posterior distribution for the linear case. In the nonlinear case where the posterior is not tractable, we present a Gaussian Process (GP) approximation that still enables CBO by inferring the parents of the outcome variable. The proposed method performs competitively with existing benchmarks and scales well to larger graphs, making it a practical tool for real-world applications where causal information is incomplete.

Depression disorder is a serious health condition that has affected the lives of millions of people around the world. Diagnosis of depression is a challenging practice that relies heavily on subjective studies and, in most cases, suffers from late findings. Electroencephalography (EEG) biomarkers have been suggested and investigated in recent years as a potential transformative objective practice. In this article, for the first time, a detailed systematic review of EEG-based depression diagnosis approaches is conducted using advanced machine learning techniques and statistical analyses. For this, 938 potentially relevant articles (since 1985) were initially detected and filtered into 139 relevant articles based on the review scheme 'preferred reporting items for systematic reviews and meta-analyses (PRISMA).' This article compares and discusses the selected articles and categorizes them according to the type of machine learning techniques and statistical analyses. Algorithms, preprocessing techniques, extracted features, and data acquisition systems are discussed and summarized. This review paper explains the existing challenges of the current algorithms and sheds light on the future direction of the field. This systematic review outlines the issues and challenges in machine intelligence for the diagnosis of EEG depression that can be addressed in future studies and possibly in future wearable technologies.

Deep neural networks (DNNs) have become powerful tools for modeling complex data structures through sequentially integrating simple functions in each hidden layer. In survival analysis, recent advances of DNNs primarily focus on enhancing model capabilities, especially in exploring nonlinear covariate effects under right censoring. However, deep learning methods for interval-censored data, where the unobservable failure time is only known to lie in an interval, remain underexplored and limited to specific data type or model. This work proposes a general regression framework for interval-censored data with a broad class of partially linear transformation models, where key covariate effects are modeled parametrically while nonlinear effects of nuisance multi-modal covariates are approximated via DNNs, balancing interpretability and flexibility. We employ sieve maximum likelihood estimation by leveraging monotone splines to approximate the cumulative baseline hazard function. To ensure reliable and tractable estimation, we develop an EM algorithm incorporating stochastic gradient descent. We establish the asymptotic properties of parameter estimators and show that the DNN estimator achieves minimax-optimal convergence. Extensive simulations demonstrate superior estimation and prediction accuracy over state-of-the-art methods. Applying our method to the Alzheimer's Disease Neuroimaging Initiative dataset yields novel insights and improved predictive performance compared to traditional approaches.

This work addresses the problem of graph learning from data following a Gaussian Graphical Model (GGM) with a time-varying mean. Graphical Lasso (GL), the standard method for estimating sparse precision matrices, assumes that the observed data follows a zero-mean Gaussian distribution. However, this assumption is often violated in real-world scenarios where the mean evolves over time due to external influences, trends, or regime shifts. When the mean is not properly accounted for, applying GL directly can lead to estimating a biased precision matrix, hence hindering the graph learning task. To overcome this limitation, we propose Graphical Lasso with Adaptive Targeted Adaptive Importance Sampling (GL-ATAIS), an iterative method that jointly estimates the time-varying mean and the precision matrix. Our approach integrates Bayesian inference with frequentist estimation, leveraging importance sampling to obtain an estimate of the mean while using a regularized maximum likelihood estimator to infer the precision matrix. By iteratively refining both estimates, GL-ATAIS mitigates the bias introduced by time-varying means, leading to more accurate graph recovery. Our numerical evaluation demonstrates the impact of properly accounting for time-dependent means and highlights the advantages of GL-ATAIS over standard GL in recovering the true graph structure.

Learning human preferences is essential for human-robot interaction, as it enables robots to adapt their behaviors to align with human expectations and goals. However, the inherent uncertainties in both human behavior and robotic systems make preference learning a challenging task. While probabilistic robotics algorithms offer uncertainty quantification, the integration of human preference uncertainty remains underexplored. To bridge this gap, we introduce uncertainty unification and propose a novel framework, uncertainty-unified preference learning (UUPL), which enhances Gaussian Process (GP)-based preference learning by unifying human and robot uncertainties. Specifically, UUPL includes a human preference uncertainty model that improves GP posterior mean estimation, and an uncertainty-weighted Gaussian Mixture Model (GMM) that enhances GP predictive variance accuracy. Additionally, we design a user-specific calibration process to align uncertainty representations across users, ensuring consistency and reliability in the model performance. Comprehensive experiments and user studies demonstrate that UUPL achieves state-of-the-art performance in both prediction accuracy and user rating. An ablation study further validates the effectiveness of human uncertainty model and uncertainty-weighted GMM of UUPL.

Recovering the underlying Directed Acyclic Graph (DAG) structures from observational data presents a formidable challenge, partly due to the combinatorial nature of the DAG-constrained optimization problem. Recently, researchers have identified gradient vanishing as one of the primary obstacles in differentiable DAG learning and have proposed several DAG constraints to mitigate this issue. By developing the necessary theory to establish a connection between analytic functions and DAG constraints, we demonstrate that analytic functions from the set $\{f(x) = c_0 + \sum_{i=1}^{\infty}c_ix^i | \forall i > 0, c_i > 0; r = \lim_{i\rightarrow \infty}c_{i}/c_{i+1} > 0\}$ can be employed to formulate effective DAG constraints. Furthermore, we establish that this set of functions is closed under several functional operators, including differentiation, summation, and multiplication. Consequently, these operators can be leveraged to create novel DAG constraints based on existing ones. Using these properties, we design a series of DAG constraints and develop an efficient algorithm to evaluate them. Experiments in various settings demonstrate that our DAG constraints outperform previous state-of-the-art comparators. Our implementation is available at https://github.com/zzhang1987/AnalyticDAGLearning.

The appropriateness of the Poisson model is frequently challenged when examining spatial count data marked by unbalanced distributions, over-dispersion, or under-dispersion. Moreover, traditional parametric models may inadequately capture the relationships among variables when covariates display ambiguous functional forms or when spatial patterns are intricate and indeterminate. To tackle these issues, we propose an innovative Bayesian hierarchical modeling system. This method combines non-parametric techniques with an adapted dispersed count model based on renewal theory, facilitating the effective management of unequal dispersion, non-linear correlations, and complex geographic dependencies in count data. We illustrate the efficacy of our strategy by applying it to lung and bronchus cancer mortality data from Iowa, emphasizing environmental and demographic factors like ozone concentrations, PM2.5, green space, and asthma prevalence. Our analysis demonstrates considerable regional heterogeneity and non-linear relationships, providing important insights into the impact of environmental and health-related factors on cancer death rates. This application highlights the significance of our methodology in public health research, where precise modeling and forecasting are essential for guiding policy and intervention efforts. Additionally, we performed a simulation study to assess the resilience and accuracy of the suggested method, validating its superiority in managing dispersion and capturing intricate spatial patterns relative to conventional methods. The suggested framework presents a flexible and robust instrument for geographical count analysis, offering innovative insights for academics and practitioners in disciplines such as epidemiology, environmental science, and spatial statistics.

The study proposes an advanced machine learning approach to predict spine surgery outcomes by incorporating oversampling techniques and grid search optimization. A variety of models including GaussianNB, ComplementNB, KNN, Decision Tree, and optimized versions with RandomOverSampler and SMOTE were tested on a dataset of 244 patients, which included pre-surgical, psychometric, socioeconomic, and analytical variables. The enhanced KNN models achieved up to 76% accuracy and a 67% F1-score, while grid-search optimization further improved performance. The findings underscore the potential of these advanced techniques to aid healthcare professionals in decision-making, with future research needed to refine these models on larger and more diverse datasets.