This paper proposes a novel framework for simulating the dynamics of beams on elastic foundations. Specifically, partial differential equations modeling Euler–Bernoulli and Timoshenko beams on the Winkler foundation are simulated using a causal physics-informed neural network (PINN) coupled with transfer learning. Conventional PINNs encounter challenges in handling large space–time domains, even for problems with closed-form analytical solutions. A causality-respecting PINN loss function is employed to overcome this limitation, effectively capturing the underlying physics. However, it is observed that the causality-respecting PINN lacks generalizability. We propose using solutions to similar problems instead of training from scratch by employing transfer learning while adhering to causality to accelerate convergence and ensure accurate results across diverse scenarios. The primary contribution of this paper lies in introducing a causality-respecting PINN loss function in the context of structural engineering and coupling it with transfer learning to enhance the generalizability of PINNs in simulating the dynamics of beams on elastic foundations. Numerical experiments on the Euler–Bernoulli beam highlight the efficacy of the proposed approach for various initial conditions, including those with noise in the initial data. Furthermore, the potential of the proposed method is demonstrated for the Timoshenko beam in an extended spatial and temporal domain. Several comparisons suggest that the proposed method accurately captures the inherent dynamics, outperforming the state-of-the-art physics-informed methods under standard L²-norm metric and accelerating convergence.

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Beams are the fundamental structural engineering element, supporting and stabilizing various structures ranging from suspension bridges to buildings and railways. Modeling and analyzing these structures necessitates a comprehensive understanding of the underlying beam dynamics constituting the structures. Simulating and predicting the beam dynamics is pivotal in ensuring structural integrity, optimizing structure design, and selecting appropriate materials. For instance, in railways, tracks and catenary contact wires are conceptualized as beams, allowing for the application of renowned beam theories like Euler-Bernoulli and Timoshenko. These theories provide a foundation for formulating partial differential equations (PDEs) that govern the dynamic behaviors of these beam systems.

These PDEs could be leveraged to simulate the underlying scenarios. The dissertation introduces physics-informed machine learning (PIML) based approaches tailored to simulate the dynamics of beam structures. The aim is to incorporate the physical laws in the neural networks training for more accurate and realistic simulations, handle noisy data effectively, and improve prediction accuracy while mitigating challenges such as multiscale problems and generalization. Chapter 1 outlines the primary challenges tackled in the dissertation. Chapters 2 through 5 detail the methodologies developed to address each challenge.

Chapter 2 presents a physics-informed neural network (PINN) based methodology to simulate complex beam systems with real-world mate- rial properties. In addition, inverse problems are solved in the presence of noisy data to predict unknown parameters, including force acting on the beam systems. It is essential to consider the real-world material parameters to simulate the dynamics of the modeled system and ensure the digital model represents the ground truth. However, incorporating material characteristics leads to multiscale PDE coefficients in the physical model, posing difficulty in training for PINNs. Subsequently, a frame- work is proposed to incorporate nondimensional PDEs into the PINN loss function. This approach facilitates efficient forward and inverse simulations while robust to noise and uncertainty in measurement data. The efficacy of this approach is demonstrated through simulations of Euler- Bernoulli and Timoshenko beam systems, contributing to the challenge of simulating large-scale systems with multiple interconnected components.

Chapter 3 investigates beam dynamic simulations on Winkler foundations for large spatiotemporal domains using PIML. Predictions on expansive spatiotemporal domains are vital for structural integrity, design optimization, and control mechanisms. A causality-respecting PINN frame- work is introduced, enhancing prediction accuracy. Furthermore, integrating transfer learning addresses the need to re-train the network for different initial conditions and computational domains. Numerical experiments based on Euler-Bernoulli and Timoshenko theories validate the methodology for respecting the causality and generalizing the beam dynamics across similar problems. The approach efficiently predicts beam dynamics under diverse engineering scenarios, reducing computational costs and improving convergence.

Chapter 4 explores the generalization abilities of PIML, essential for practical applications requiring accurate predictions in unexplored regions. The proposed framework exploits the inherent causality in the PDE solutions by merging PIML models with recurrent neural architectures, namely neural oscillators. The neural ordinary differential equations in the form of neural oscillators effectively handle long-time dependencies and address gradient-related issues, fostering improved generalization in PIML tasks. Benchmark equations like viscous Burgers, Allen-Cahn, Schrödinger, and biharmonic Euler-Bernoulli beam equations are used to demonstrate the effectiveness of the proposed approach. Through ex- tensive experimentation with time-dependent nonlinear PDEs, the study showcases superior performance compared to existing state-of-the-art methods. The proposed method provides accurate solutions for extrapolation and prediction beyond the training data by enhancing the generalization capabilities of PIML, promising advancements in complex system simulations.

Chapter 5 follows up on generalization of beam dynamics beyond PIML- based approaches. Computer-aided simulations are crucial for advancing engineering industries, but existing simulators often struggle to generalize beyond their training domain. The chapter proposes a two-stage methodology to tackle this challenge. Firstly, it utilizes specialized simulators tailored to the application, such as causal PINNs and black-box finite element simulations. Secondly, it integrates predictions from the first stage into a recurrent neural architecture, incorporating ordinary differential equations to capture intrinsic dynamics and enhance generalization. The approach efficiently captures causality and generalizes dynamics across various data sources. Numerical experiments cover fundamental structural engineering scenarios, including real-world catenary contact wire uplift predictions, and demonstrate superior performance compared to conventional methods, and promise for diverse industrial applications. This dissertation concludes with Chapter 6.

In particular, this dissertation introduces PIML methodologies for simulating complex beam structures, addressing key challenges such as incorporating real material properties, handling noisy data, and improving prediction accuracy. Chapter 2 introduces a PINN-based methodology that efficiently simulates beam systems and predicts unknown parameters, mitigating the difficulties posed by multiscale PDE coefficients. Chapter 3 tackles the challenge of large-domain beam dynamics predictions on the Winkler foundations by using causality-respecting PINNs and integrating transfer learning to reduce computational costs. Chapter 4 addresses the challenge of out-of-domain predictions in PIML by introducing neural oscillators. Chapter 5 proposes a two-stage methodology to generalize beam dynamics simulations, integrating beam dynamics solvers and recurrent neural-based architectures, showcasing its efficacy in real-world applications such as catenary contact wire uplift predictions.
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Computer-aided simulations are routinely used to predict a prototype's performance. High-fidelity physics-based simulators might be computationally expensive for design and optimization, spurring the development of cheap deep-learning surrogates. The resulting surrogates often struggle to generalize and predict novel scenarios beyond their training domain. We propose a two-stage methodology addressing the challenge of generalization. It employs physics-based simulators, supplemented with ordinary differential equations integrated into the recurrent architecture, to learn the intrinsic dynamics. The proposed approach captures the inherent causality and generalizes the dynamics irrespective of a data source. The presented numerical experiments encompass five fundamental structural engineering scenarios, including beams on Winkler foundations based on Euler-Bernoulli and Timoshenko theories, beams under moving loads, and catenary-pantograph interactions in railways. The proposed methodology outperforms conventional recurrent methods and remains invariant to data sources, showcasing its efficacy. Numerical experiments highlight its prospects for design optimization, predictive maintenance, and enhancing safety measures.

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Nonlinear hysteresis modeling is essential for estimating, controlling, and characterizing the behavior of piezoelectric material-based devices. However, current deep-learning approaches face challenges in generalizing effectively to previously unseen voltage profiles. This Letter tackles the limitation of generalization by introducing the notion of neural operators for modeling the nonlinear constitutive laws governing inverse piezoelectric hysteresis, specifically focusing on the relationship between voltage inputs and displacement responses. The study utilizes two neural operators—Fourier neural operator and the deep operator network—to predict material responses to unseen voltage profiles that are not part of the training data. Numerical experiments, including butterfly-shaped hysteresis curves, show that in accuracy and generalization to unseen voltage profiles, neural operators outperform traditional recurrent neural network-based models, including conventional gated networks. The findings highlight the potential of neural operators for modeling hysteresis in piezoelectric materials, offering advantages over existing methods in varying voltage scenarios.@en

This paper presents a new approach to simulate forward and inverse problems of moving loads using physics-informed machine learning (PIML). Physics-informed neural networks (PINNs) utilize the underlying physics of moving load problems and aim to predict the deflection of beams and the magnitude of the loads. The mathematical representation of the moving load considered involves a Dirac delta function, to capture the effect of the load moving across the structure. Approximating the Dirac delta function with PINNs is challenging because of its instantaneous change of output at a single point, causing difficulty in the convergence of the loss function. We propose to approximate the Dirac delta function with a Gaussian function. The incorporated Gaussian function physical equations are used in the physics-informed neural architecture to simulate beam deflections and to predict the magnitude of the load. Numerical results show that PIML is an effective method for simulating the forward and inverse problems for the considered model of a moving load.

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A primary challenge of physics-informed machine learning (PIML) is its generalization beyond the training domain, especially when dealing with complex physical problems represented by partial differential equations (PDEs). This paper aims to enhance the generalization capabilities of PIML, facilitating practical, real-world applications where accurate predictions in unexplored regions are crucial. We leverage the inherent causality and temporal sequential characteristics of PDE solutions to fuse PIML models with recurrent neural architectures based on systems of ordinary differential equations, referred to as neural oscillators. Through effectively capturing long-time dependencies and mitigating the exploding and vanishing gradient problem, neural oscillators foster improved generalization in PIML tasks. Extensive experimentation involving time-dependent nonlinear PDEs and biharmonic beam equations demonstrates the efficacy of the proposed approach. Incorporating neural oscillators outperforms existing state-of-the-art methods on benchmark problems across various metrics. Consequently, the proposed method improves the generalization capabilities of PIML, providing accurate solutions for extrapolation and prediction beyond the training data.

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This article proposes a new framework using physics-informed neural networks (PINNs) to simulate complex structural systems that consist of single and double beams based on Euler–Bernoulli and Timoshenko theories, where the double beams are connected with a Winkler foundation. In particular, forward and inverse problems for the Euler–Bernoulli and Timoshenko partial differential equations (PDEs) are solved using nondimensional equations with the physics-informed loss function. Higher order complex beam PDEs are efficiently solved for forward problems to compute the transverse displacements and cross-sectional rotations with less than 1e−3 % error. Furthermore, inverse problems are robustly solved to determine the unknown dimensionless model parameters and applied force in the entire space–time domain, even in the case of noisy data. The results suggest that PINNs are a promising strategy for solving problems in engineering structures and machines involving beam systems.@en

This paper addresses the problem of determining the distribution of the return current in electric railway traction systems. The dynamics of traction return current are simulated in all three space dimensions by informing the neural networks with the Partial Differential Equations (PDEs) known as telegraph equations. In addition, this work proposes a method of choosing optimal activation functions for training the physics-informed neural network to solve higher-dimensional PDEs. We propose a Monte Carlo based framework to choose the activation function in lower dimensions, mitigating the need for ensemble training in higher dimensions. To further strengthen the applicability of the Monte Carlo based framework, experiments are presented under two loss functions governed by L2 and L∞ norms. The presented method efficiently simulates the traction return current for electric railway systems, even for three-dimensional problems.

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