A Normal Equation-Based Extreme Learning Machine for Solving Linear Partial Differential Equations

Dwivedi, Vikas; Srinivasan, Balaji

doi:10.1115/1.4051530

Cited by 13 publications

(3 citation statements)

References 22 publications

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“…The physics loss ensures that the system's governing equations are enforced. Various case studies and engineering problems where this approach has been applied can be found in [10,12,[63][64][65][66][67][68]. Additionally, Figure 11 illustrates a continuous PINN architecture demonstrating the physics-guided loss formulation.…”

Section: Physics-informed Neural Networkmentioning

confidence: 99%

Time-Series Machine Learning Techniques for Modeling and Identification of Mechatronic Systems with Friction: A Review and Real Application

Ayankoso,

Olejnik

2023

Electronics

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Developing accurate dynamic models for various systems is crucial for optimization, control, fault diagnosis, and prognosis. Recent advancements in information technologies and computing platforms enable the acquisition of input–output data from dynamical systems, resulting in a shift from physics-based methods to data-driven techniques in science and engineering. This review examines different data-driven modeling approaches applied to the identification of mechanical and electronic systems. The approaches encompass various neural networks (NNs), like the feedforward neural network (FNN), convolutional neural network (CNN), long short-term memory (LSTM), transformer, and emerging machine learning (ML) techniques, such as the physics-informed neural network (PINN) and sparse identification of nonlinear dynamics (SINDy). The main focus is placed on applying these techniques to real-world problems. A real application is presented to demonstrate the effectiveness of different machine learning techniques, namely, FNN, CNN, LSTM, transformer, SINDy, and PINN, in data-driven modeling and the identification of a geared DC motor. The results show that the considered ML techniques (traditional and state-of-the-art methods) perform well in predicting the behavior of such a classic dynamical system. Furthermore, SINDy and PINN models stand out for their interpretability compared to the other data-driven models examined. Our findings explicitly show the satisfactory predictive performance of six different ML models while also highlighting their pros and cons, such as interpretability and computational complexity, using a real-world case study. The developed models have various applications and potential research areas are discussed.

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Section: Physics-informed Neural Networkmentioning

confidence: 99%

Time-Series Machine Learning Techniques for Modeling and Identification of Mechatronic Systems with Friction: A Review and Real Application

Ayankoso,

Olejnik

2023

Electronics

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“…For example, the nonlinear function F (u) = u ∂u ∂x (as in the Burgers' equation) leads to F (u)φ = ∂u ∂x φ + u ∂φ ∂x . Therefore, to solve the problem (15) by HLConcELM, the input training data (denoted by X) to the neural network is a Q×d matrix, consisting of the coordinates of all the collocation points, X = x p Q×d (for all x p ∈ X). The output data (denoted by U) of the neural network is a Q×m L matrix, representing the field solution u(x) on the collocation points, U = u(x p ) Q×m L…”

Section: Solving Linear/nonlinear Pdes With Hidden-layer Concatenated...mentioning

confidence: 99%

“…Interestingly, the authors set the number of hidden nodes to be equal to the total number of conditions in the problem. A solution strategy based on the normal equation associated with the linear system is studied in [15].…”

Section: Introductionmentioning

confidence: 99%

Numerical Computation of Partial Differential Equations by Hidden-Layer Concatenated Extreme Learning Machine

Ni¹,

Dong²

2022

Preprint

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The extreme learning machine (ELM) method can yield highly accurate solutions to linear/nonlinear partial differential equations (PDEs), but requires the last hidden layer of the neural network to be wide to achieve a high accuracy. If the last hidden layer is narrow, the accuracy of the existing ELM method will be poor, irrespective of the rest of the network configuration. In this paper we present a modified ELM method, termed HLConcELM (hidden-layer concatenated ELM), to overcome the above drawback of the conventional ELM method. The HLConcELM method can produce highly accurate solutions to linear/nonlinear PDEs when the last hidden layer of the network is narrow and when it is wide. The new method is based on a type of modified feedforward neural networks (FNN), termed HLConcFNN (hidden-layer concatenated FNN), which incorporates a logical concatenation of the hidden layers in the network and exposes all the hidden nodes to the output-layer nodes. We show that HLConcFNNs have the remarkable property that, given a network architecture, when additional hidden layers are appended to the network or when extra nodes are added to the existing hidden layers, the approximation capacity of the HLConcFNN associated with the new architecture is guaranteed to be not smaller than that of the original network architecture. We present ample benchmark tests with linear/nonlinear PDEs to demonstrate the computational accuracy and performance of the HLConcELM method and the superiority of this method to the conventional ELM from previous works.

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A machine-learning-based peridynamic surrogate model for characterizing deformation and failure of materials and structures

Wang,

Wu,

Huang

et al. 2024

Engineering with Computers

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A Normal Equation-Based Extreme Learning Machine for Solving Linear Partial Differential Equations

Cited by 13 publications

References 22 publications

Time-Series Machine Learning Techniques for Modeling and Identification of Mechatronic Systems with Friction: A Review and Real Application

Time-Series Machine Learning Techniques for Modeling and Identification of Mechatronic Systems with Friction: A Review and Real Application

Numerical Computation of Partial Differential Equations by Hidden-Layer Concatenated Extreme Learning Machine

A machine-learning-based peridynamic surrogate model for characterizing deformation and failure of materials and structures

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