# RESEARCH PAPERS

International journals | Conference proceedings | Preprints

## 16. UNCERTAINTY QUANTIFICATION IN SCIENTIFIC MACHINE LEARNING: METHODS, METRICS, AND COMPARISONS

2023 - Journal of Computational Physics - arXiv version

Psaros A.F., Meng X., Zou Z., Guo L., Karniadakis G.Em.

Neural networks (NNs) are currently changing the computational paradigm on how to combine data with mathematical laws in physics and engineering in a profound way, tackling challenging inverse and ill-posed problems not solvable with traditional methods. However, quantifying errors and uncertainties in NN-based inference is more complicated than in traditional methods. This is because in addition to aleatoric uncertainty associated with noisy data, there is also uncertainty due to limited data, but also due to NN hyperparameters, overparametrization, optimization and sampling errors as well as model misspecification. Although there are some recent works on uncertainty quantification (UQ) in NNs, there is no systematic investigation of suitable methods towards quantifying the total uncertainty effectively and efficiently even for function approximation, and there is even less work on solving partial differential equations and learning operator mappings between infinite-dimensional function spaces using NNs. In this work, we present a comprehensive framework that includes uncertainty modeling, new and existing solution methods, as well as evaluation metrics and post-hoc improvement approaches. To demonstrate the applicability and reliability of our framework, we present an extensive comparative study in which various methods are tested on prototype problems, including problems with mixed input-output data, and stochastic problems in high dimensions. In the Appendix, we include a comprehensive description of all the UQ methods employed. Further, to help facilitate the deployment of UQ in Scientific Machine Learning research and practice, we present and develop in [1] an open-source Python library (github.com/Crunch-UQ4MI/neuraluq), termed NeuralUQ, that is accompanied by an educational tutorial and additional computational experiments.

## 15. A WIENER PATH INTEGRAL FORMALISM FOR TREATING NONLINEAR SYSTEMS WITH NON-MARKOVIAN RESPONSE PROCESSES

2023 - Journal of Engineering Mechanics

Mavromatis I. G., Psaros A. F., Kougioumtzoglou I. A.

A novel formalism of the Wiener path integral (WPI) technique for determining the stochastic response of diverse dynamical systems is developed. It can be construed as a generalization of earlier efforts to account, in a direct manner, also for systems with non-Markovian response processes. Specifically, first, the probability of a path and the associated transition probability density function (PDF) corresponding to the Wiener excitation process are considered. Next, a functional change of variables is employed, in conjunction with the governing stochastic differential equation, for deriving the system response joint transition PDF as a functional integral over the space of possible paths connecting the initial and final states of the response vector. In comparison to alternative derivations in the literature, which resort to the Chapman-Kolmogorov equation as the starting point, the herein-developed novel formalism circumvents the Markovian assumption for the system response process. Overall, the veracity and mathematical legitimacy of the WPI technique to treat also non-Markovian system response processes are demonstrated. In this regard, nonlinear systems with a history-dependent state, such as hysteretic structures or oscillators endowed with fractional derivative elements, can be accounted for in a direct manner—that is, without resorting to any ad hoc modifications of the WPI technique pertaining, typically, to employing additional auxiliary filter equations and state variables. A Biot hysteretic oscillator with cubic nonlinearities and an oscillator with asymmetric nonlinearities and fractional derivative elements are considered as illustrative numerical examples for demonstrating the reliability of the developed technique. Comparisons with relevant Monte Carlo simulation (MCS) data are included as well.

## 14. NEURALUQ: A COMPREHENSIVE LIBRARY FOR UNCERTAINTY QUANTIFICATION IN NEURAL DIFFERENTIAL EQUATIONS AND OPERATORS

2022 - Preprint

Zou Z., Meng X., Psaros A.F., Karniadakis G.Em.

Uncertainty quantification (UQ) in machine learning is currently drawing increasing research interest, driven by the rapid deployment of deep neural networks across different fields, such as computer vision, natural language processing, and the need for reliable tools in risk-sensitive applications. Recently, various machine learning models have also been developed to tackle problems in the field of scientific computing with applications to computational science and engineering (CSE). Physics-informed neural networks and deep operator networks are two such models for solving partial differential equations and learning operator mappings, respectively. In this regard, a comprehensive study of UQ methods tailored specifically for scientific machine learning (SciML) models has been provided in [45]. Nevertheless, and despite their theoretical merit, implementations of these methods are not straightforward, especially in large-scale CSE applications, hindering their broad adoption in both research and industry settings. In this paper, we present an open-source Python library (this https URL), termed NeuralUQ and accompanied by an educational tutorial, for employing UQ methods for SciML in a convenient and structured manner. The library, designed for both educational and research purposes, supports multiple modern UQ methods and SciML models. It is based on a succinct workflow and facilitates flexible employment and easy extensions by the users. We first present a tutorial of NeuralUQ and subsequently demonstrate its applicability and efficiency in four diverse examples, involving dynamical systems and high-dimensional parametric and time-dependent PDEs.

## 13. META-LEARNING PINN LOSS FUNCTIONS

2022 - Journal of Computational Physics - arXiv version

Psaros A.F., Kawaguchi K., Karniadakis G.Em.

We propose a meta-learning technique for offline discovery of physics-informed neural network (PINN) loss functions. We extend earlier works on meta-learning, and develop a gradient-based meta-learning algorithm for addressing diverse task distributions based on parametrized partial differential equations (PDEs) that are solved with PINNs. Furthermore, based on new theory we identify two desirable properties of meta-learned losses in PINN problems, which we enforce by proposing a new regularization method or using a specific parametrization of the loss function. In the computational examples, the meta-learned losses are employed at test time for addressing regression and PDE task distributions. Our results indicate that significant performance improvement can be achieved by using a shared-among-tasks offline-learned loss function even for out-of-distribution meta-testing. In this case, we solve for test tasks that do not belong to the task distribution used in meta-training, and we also employ PINN architectures that are different from the PINN architecture used in meta-training. To better understand the capabilities and limitations of the proposed method, we consider various parametrizations of the loss function and describe different algorithm design options and how they may affect meta-learning performance.

## 12. A QUADRATIC WIENER PATH INTEGRAL APPROXIMATION FOR STOCHASTIC RESPONSE DETERMINATION OF MULTI-DEGREE-OF-FREEDOM NONLINEAR SYSTEMS

2022 - Probabilistic Engineering Mechanics

Zhao Y., Psaros A.F., Petromichelakis I., Kougioumtzoglou I.A.

A Wiener path integral (WPI) technique is developed for determining the stochastic response of multi-degree-of-freedom (MDOF) nonlinear systems. Specifically, the nonlinear system response joint transition probability density function (PDF) is expressed as a WPI over the space of paths satisfying the initial and final conditions in time. Next, a functional series expansion is considered for the WPI and a quadratic approximation is employed. Further, relying on a variational principle yields a functional optimization problem to be solved for the most probable path, which is used for determining approximately the joint response transition PDF. It is shown that compared to the standard (semiclassical) WPI solution approach, which accounts only for the most probable path, the quadratic approximation developed herein exhibits enhanced accuracy. This is due to the fact that fluctuations around the most probable path are also accounted for by considering a localized state-dependent factor in the calculation of the WPI. Furthermore, the PDF normalization step of the most probable path approach is bypassed, and thus, probabilities of rare events (e.g., failures) can be determined in a direct manner without the need for obtaining the complete joint response PDF first. The herein developed technique can be construed as an extension of earlier efforts in the literature to account for MDOF systems. Several numerical examples are considered for demonstrating the accuracy of the technique. These pertain to various dynamical systems exhibiting diverse nonlinear behaviors. Comparisons with pertinent Monte Carlo simulation data are included as well.

## 11. UNCERTAINTY QUANTIFICATION OF NONLINEAR SYSTEM STOCHASTIC RESPONSE ESTIMATES BASED ON THE WIENER PATH INTEGRAL TECHNIQUE: A BAYESIAN COMPRESSIVE SAMPLING TREATMENT

2022 - Probabilistic Engineering Mechanics - Full text

Katsidoniotaki M., Psaros A.F., Kougioumtzoglou I.A.

The Wiener path integral (WPI) technique for determining the stochastic response of diverse nonlinear systems is enhanced herein based on a Bayesian compressive sampling (CS) treatment. Specifically, first, sparse expansions for the system response joint probability density function (PDF) are employed. Next, exploiting the localization capabilities of the WPI technique for direct evaluation of specific PDF points leads to an underdetermined linear system of equations for the expansion coefficients. Further, relying on a Bayesian CS solution formulation yields a posterior distribution for the expansion coefficient vector. In this regard, a significant advantage of the herein developed methodology relates to the fact that the uncertainty of the response PDF estimates obtained by the WPI technique is quantified. Furthermore, an adaptive scheme is proposed based on the quantified uncertainty of the estimates for optimal selection of PDF sample points. This yields considerably fewer boundary value problems to be solved as part of the WPI technique, and thus, the associated computational cost is significantly reduced. Two indicative numerical examples pertaining to a Duffing nonlinear oscillator and to an oscillator with asymmetric nonlinearities are considered for demonstrating the capabilities of the developed technique. Comparisons with pertinent Monte Carlo simulation data are included as well.

## 10. STOCHASTIC RESPONSE ANALYSIS AND RELIABILITY-BASED DESIGN OPTIMIZATION OF NONLINEAR ELECTROMECHANICAL ENERGY HARVESTERS WITH FRACTIONAL DERIVATIVE ELEMENTS

2021 - ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part B: Mechanical Engineering - Full text

Petromichelakis I., Psaros A.F., Kougioumtzoglou I.A.

A methodology based on the Wiener path integral technique (WPI) is developed for stochastic response determination and reliability-based design optimization of a class of nonlinear electromechanical energy harvesters endowed with fractional derivative elements. In this regard, first, the WPI technique is appropriately adapted and enhanced to account both for the singular diffusion matrix and for the fractional derivative modeling of the capacitance in the coupled electromechanical governing equations. Next, a reliability-based design optimization problem is formulated and solved, in conjunction with the WPI technique, for determining the optimal parameters of the harvester. It is noted that the herein proposed definition of the failure probability constraint is particularly suitable for harvester configurations subject to space limitations. Several numerical examples are included, while comparisons with pertinent Monte Carlo simulation data demonstrate the satisfactory performance of the methodology.

## 9. SPARSE REPRESENTATIONS AND COMPRESSIVE SAMPLING APPROACHES IN ENGINEERING MECHANICS: A REVIEW OF THEORETICAL CONCEPTS AND DIVERSE APPLICATIONS

2020 - Probabilistic Engineering Mechanics - Full text

Kougioumtzoglou I.A., Petromichelakis I., Psaros A.F.

A review of theoretical concepts and diverse applications of sparse representations and compressive sampling (CS) approaches in engineering mechanics problems is provided from a broad perspective. First, following a presentation of well-established CS concepts and optimization algorithms, attention is directed to currently emerging tools and techniques for enhancing solution sparsity and for exploiting additional information in the data. These include alternative to l1-norm minimization formulations and iterative re-weighting solution schemes, Bayesian approaches, as well as structured sparsity and dictionary learning strategies. Next, CS-based research work of relevance to engineering mechanics problems is categorized and discussed under three distinct application areas: a) inverse problems in structural health monitoring, b) uncertainty modeling and simulation, and c) computationally efficient uncertainty propagation.

## 8. FUNCTIONAL SERIES EXPANSIONS AND QUADRATIC APPROXIMATIONS FOR ENHANCING THE ACCURACY OF THE WIENER PATH INTEGRAL TECHNIQUE

2020 - Journal of Engineering Mechanics - Full text

Psaros A.F., Kougioumtzoglou I.A.

A novel Wiener path integral (WPI) technique is developed for determining the response of stochastically excited nonlinear oscillators. This is done by resorting to functional series expansions in conjunction with quadratic approximations. The technique can be construed as an extension and enhancement in terms of accuracy of the standard (semi-classical) WPI solution approach where only the most probable path connecting initial and final states is considered for determining the joint response probability density function (PDF). In contrast, the herein developed technique accounts also for fluctuations around the most probable path; thus, yielding an increased accuracy degree. An additional significant advantage of the proposed enhancement as compared to the most probable path approach relates to the fact that low probability events (e.g., failure probabilities) can be estimated directly in a computationally efficient manner by determining a few only points of the joint response PDF.

## 7. AN EXACT CLOSED-FORM SOLUTION FOR LINEAR MULTI-DEGREE-OF-FREEDOM SYSTEMS UNDER GAUSSIAN WHITE NOISE VIA THE WIENER PATH INTEGRAL TECHNIQUE

2020 - Probabilistic Engineering Mechanics - Full text

Psaros A.F., Zhao Y., Kougioumtzoglou I.A.

The exact joint response transition probability density function (PDF) of linear multi-degree-of-freedom oscillators under Gaussian white noise is derived in closed-form based on the Wiener path integral (WPI) technique. Specifically, in the majority of practical implementations of the WPI technique, only the first couple of terms are retained in the functional expansion of the path integral related stochastic action. The remaining terms are typically omitted since their evaluation exhibits considerable analytical and computational challenges. Obviously, this approximation affects, unavoidably, the accuracy degree of the technique. However, it is shown herein that, for the special case of linear systems, higher than second order variations in the path integral functional expansion vanish, and thus, retaining only the first term (most probable path approximation) yields the exact joint response transition Gaussian PDF. Both single- and multi degree-of-freedom linear systems are considered as illustrative examples for demonstrating the exact nature of the derived solutions.

## 6. STOCHASTIC RESPONSE DETERMINATION OF NONLINEAR SYSTEMS WITH SINGULAR DIFFUSION MATRICES: A WIENER PATH INTEGRAL VARIATIONAL FORMULATION WITH CONSTRAINTS

2020 - Probabilistic Engineering Mechanics - Full text

Petromichelakis I., Psaros A.F., Kougioumtzoglou I.A.

The Wiener path integral (WPI) approximate semi-analytical technique for determining the joint response probability density function (PDF) of stochastically excited nonlinear oscillators is generalized herein to account for systems with singular diffusion matrices. Indicative examples include (but are not limited to) systems with only some of their degrees-of-freedom excited, hysteresis modeling via additional auxiliary state equations, and energy harvesters with coupled electro-mechanical equations. Several numerical examples pertaining to both linear and nonlinear constraint equations are considered, including various multi-degree-of-freedom systems, a linear oscillator under earthquake excitation and a nonlinear oscillator exhibiting hysteresis following the Bouc-Wen formalism.

## 5. WIENER PATH INTEGRALS AND MULTI-DIMENSIONAL GLOBAL BASES FOR NON-STATIONARY STOCHASTIC RESPONSE DETERMINATION OF STRUCTURAL SYSTEMS

2019 - Mechanical Systems and Signal Processing - Full text

Psaros A.F., Petromichelakis I., Kougioumtzoglou I.A.

A novel approximate technique based on Wiener path integrals (WPIs) is developed for determining, in a computationally efficient manner, the non-stationary joint response probability density function (PDF) of nonlinear multi-degree-of-freedom dynamical systems. Specifically, appropriate multi-dimensional (time-dependent) global bases are constructed for approximating the non-stationary joint response PDF. Several numerical examples pertaining to diverse structural systems are considered, including both single-and multi-degree-of-freedom nonlinear dynamical systems subject to non-stationary excitations, as well as a bending beam with a non-Gaussian and non-homogeneous Young's modulus.

## 4. WIENER PATH INTEGRAL BASED RESPONSE DETERMINATION OF NONLINEAR SYSTEMS SUBJECT TO NON-WHITE, NON-GAUSSIAN, AND NON-STATIONARY STOCHASTIC EXCITATION

2018 - Journal of Sound and Vibration - Full text

Psaros A.F., Brudastova O., Malara G., Kougioumtzoglou I.A.

The recently developed Wiener Path Integral (WPI) technique for determining the joint response probability density function of nonlinear systems subject to Gaussian white noise excitation is generalized herein to account for non-white, non-Gaussian, and non-stationary excitation processes. Specifically, modeling the excitation process as the output of a filter equation with Gaussian white noise as its input, it is possible to define an augmented response vector process to be considered in the WPI solution technique. Several numerical examples pertaining to both single-and multi-degree-of-freedom systems are considered, including a marine structural system exposed to flow-induced non-white excitation, as well as a bending beam with a non-Gaussian and non-homogeneous Young's modulus.

## 3. SPARSE REPRESENTATIONS AND COMPRESSIVE SAMPLING FOR ENHANCING THE COMPUTATIONAL EFFICIENCY OF THE WIENER PATH INTEGRAL TECHNIQUE

2018 - Mechanical Systems and Signal Processing - Full text

Psaros A.F., Kougioumtzoglou I.A., Petromichelakis I.

The computational efficiency of the Wiener path integral (WPI) technique for determining the stochastic response of diverse dynamical systems is enhanced by exploiting recent developments in the area of sparse representations. Specifically, an appropriate basis for expanding the system joint response probability density function (PDF) is utilized. Next, only very few PDF points are determined based on the localization capabilities of the WPI technique. Further, compressive sampling procedures in conjunction with group sparsity concepts and appropriate optimization algorithms are employed for efficiently determining the coefficients of the system response PDF expansion. It is shown that the herein developed enhancement renders the technique capable of treating readily relatively high-dimensional stochastic systems. Two illustrative numerical examples are considered.

## 2. STOCHASTIC RESPONSE DETERMINATION AND OPTIMIZATION OF A CLASS OF NONLINEAR ELECTROMECHANICAL ENERGY HARVESTERS: A WIENER PATH INTEGRAL APPROACH

2018 - Probabilistic Engineering Mechanics - Full text

Petromichelakis I., Psaros A.F., Kougioumtzoglou I.A.

A methodology based on the Wiener path integral technique (WPI) is developed for stochastic response determination and optimization of a class of nonlinear electromechanical energy harvesters. To this aim, first, the WPI technique is extended to address the particular form of the coupled electromechanical governing equations, which possess a singular diffusion matrix. Specifically, a constrained variational problem is formulated and solved for determining the joint response probability density function (PDF) of the nonlinear energy harvesters. This is done either by resorting to a Lagrange multipliers approach, or by utilizing the nullspace of the constraint equation. Next, the herein extended WPI technique is coupled with an appropriate optimization algorithm for determining optimal energy harvester parameters. It is shown that due to the relatively high accuracy exhibited in determining the joint response PDF, the WPI technique is particularly well-suited for constrained optimization problems , where the constraint refers to low probability events (e.g. probabilities of failure).

## 1. STRENGTH, STIFFNESS AND CYCLIC DEFORMATION CAPACITY OF RC FRAMES CONVERTED INTO WALLS BY INFILLING WITH RC

2016 - Bulletin of Earthquake Engineering

Biskinis D., Fardis M.N., Psaros A.F.

In seismic retrofitting of concrete buildings, frame bays are converted into reinforced concrete (RC) walls by infilling the space between the frame members with RC of a thickness of not more than their width. The cyclic behavior of the resulting wall depends on the connection between the RC infill and the surrounding RC members. The paper uses the results from 56 cyclic tests on such composite walls to express their properties in terms of the geometry, the reinforcement and the connection.