Costas Papadimitriou

Omid Sedehi, Antonina M. Kosikova, Costas Papadimitriou et al.

Machine Learning (ML) has widely been used for modeling and predicting physical systems. These techniques offer high expressive power and good generalizability for interpolation within observed data sets. However, the disadvantage of black-box models is that they underperform under blind conditions since no physical knowledge is incorporated. Physics-based ML aims to address this problem by retaining the mathematical flexibility of ML techniques while incorporating physics. In accord, this paper proposes to embed mechanics-based models into the mean function of a Gaussian Process (GP) model and characterize potential discrepancies through kernel machines. A specific class of kernel function is promoted, which has a connection with the gradient of the physics-based model with respect to the input and parameters and shares similarity with the exact Autocovariance function of linear dynamical systems. The spectral properties of the kernel function enable considering dominant periodic processes originating from physics misspecification. Nevertheless, the stationarity of the kernel function is a difficult hurdle in the sequential processing of long data sets, resolved through hierarchical Bayesian techniques. This implementation is also advantageous to mitigate computational costs, alleviating the scalability of GPs when dealing with sequential data. Using numerical and experimental examples, potential applications of the proposed method to structural dynamics inverse problems are demonstrated.

Wang-Ji Yan, Lin-Feng Mei, Jiang Mo et al.

In the era of big data, machine learning (ML) has become a powerful tool in various fields, notably impacting structural dynamics. ML algorithms offer advantages by modeling physical phenomena based on data, even in the absence of underlying mechanisms. However, uncertainties such as measurement noise and modeling errors can compromise the reliability of ML predictions, highlighting the need for effective uncertainty awareness to enhance prediction robustness. This paper presents a comprehensive review on navigating uncertainties in ML, categorizing uncertainty-aware approaches into probabilistic methods (including Bayesian and frequentist perspectives) and non-probabilistic methods (such as interval learning and fuzzy learning). Bayesian neural networks, known for their uncertainty quantification and nonlinear mapping capabilities, are emphasized for their superior performance and potential. The review covers various techniques and methodologies for addressing uncertainties in ML, discussing fundamentals and implementation procedures of each method. While providing a concise overview of fundamental concepts, the paper refrains from in-depth critical explanations. Strengths and limitations of each approach are examined, along with their applications in structural dynamic forward problems like response prediction, sensitivity assessment, and reliability analysis, and inverse problems like system identification, model updating, and damage identification. Additionally, the review identifies research gaps and suggests future directions for investigations, aiming to provide comprehensive insights to the research community. By offering an extensive overview of both probabilistic and non-probabilistic approaches, this review aims to assist researchers and practitioners in making informed decisions when utilizing ML techniques to address uncertainties in structural dynamic problems.

Konstantinos Vogiatzoglou, Costas Papadimitriou, Vasilis Bontozoglou et al.

Forward propagation of input uncertainties in physics-based wildfire models is computationally prohibitive, limiting the use of high-fidelity simulators in risk assessment workflows. This work introduces a geometry-aligned bi-fidelity surrogate framework that addresses the convection-dominated nature of wildfire spread by mapping low- and high-fidelity solution snapshots onto a common reference domain prior to basis selection and reconstruction. Unlike conventional bi-fidelity schemes, which combine spatially shifted snapshots and thus suffer from oscillations and excess basis requirements near sharp fronts, the proposed mapping aligns the dominant front geometry through per-variable shift/stretch transforms in 1D and an activity indicator-based affine alignment in 2D, so that reduced bases compare physically corresponding structures rather than displaced ones. Building on the ADfiRe physics-based simulator, we demonstrate the method on 1D and 2D test cases in which low- and high-fidelity models differ in mesh resolution and physical completeness. Across both settings, the geometry-aligned surrogate reproduces full-field temperature and fuel composition with substantially lower error than its unmapped counterpart, eliminates Gibbs-type oscillations near steep gradients, and recovers high-fidelity probability density functions for key quantities of interest (e.g., maximum temperature, evaporated moisture, and burned area). After offline training, online predictions are roughly three orders of magnitude cheaper than direct high-fidelity evaluation, making the framework a practical building block for many-query uncertainty quantification once the offline cost is amortized over enough queries. We discuss the conditions under which the geometric alignment is most effective, its limitations for non-convex or topologically complex fronts, and the path toward validation against real data.

Lucas Amoudruz, Sergey Litvinov, Costas Papadimitriou et al.

Inverse problems are crucial for many applications in science, engineering and medicine that involve data assimilation, design, and imaging. Their solution infers the parameters or latent states of a complex system from noisy data and partially observable processes. When measurements are an incomplete or indirect view of the system, additional knowledge is required to accurately solve the inverse problem. Adopting a physical model of the system in the form of partial differential equations (PDEs) is a potent method to close this gap. In particular, the method of optimizing a discrete loss (ODIL) has shown great potential in terms of robustness and computational cost. In this work, we introduce B-ODIL, a Bayesian extension of ODIL, that integrates the PDE loss of ODIL as prior knowledge and combines it with a likelihood describing the data. B-ODIL employs a Bayesian formulation of PDE-based inverse problems to infer solutions with quantified uncertainties. We demonstrate the capabilities of B-ODIL in a series of synthetic benchmarks involving PDEs in one, two, and three dimensions. We showcase the application of B-ODIL in estimating tumor concentration and its uncertainty in a patient's brain from MRI scans using a three-dimensional tumor growth model.

Konstantinos Vogiatzoglou, Costas Papadimitriou, Vasilis Bontozoglou et al.

Wildland fires pose a terrifying natural hazard, underscoring the urgent need to develop data-driven and physics-informed digital twins for wildfire prevention, monitoring, intervention, and response. In this direction of research, this work introduces a physics-informed neural network (PiNN) designed to learn the unknown parameters of an interpretable wildfire spreading model. The considered modeling approach integrates fundamental physical laws articulated by key model parameters essential for capturing the complex behavior of wildfires. The proposed machine learning framework leverages the theory of artificial neural networks with the physical constraints governing wildfire dynamics, including the first principles of mass and energy conservation. Training of the PiNN for physics-informed parameter identification is realized using synthetic data on the spatiotemporal evolution of one- and two-dimensional firefronts, derived from a high-fidelity simulator, as well as empirical data (ground surface thermal images) from the Troy Fire that occurred on June 19, 2002, in California. The parameter learning results demonstrate the predictive ability of the proposed PiNN in uncovering the unknown coefficients of the wildfire model in one- and two-dimensional fire spreading scenarios as well as the Troy Fire. Additionally, this methodology exhibits robustness by identifying the same parameters even in the presence of noisy data. By integrating this PiNN approach into a comprehensive framework, the envisioned physics-informed digital twin will enhance intelligent wildfire management and risk assessment, providing a powerful tool for proactive and reactive strategies.