UMBC Mechanical Engineering Department
Permanent URI for this collectionhttp://hdl.handle.net/11603/52
Mechanical Engineering is one of the oldest and broadest engineering disciplines. In the more traditional sense, Mechanical Engineering employs principles of physics and chemistry and knowledge from the fields of mathematics, mechanics and materials science for the analysis, design, manufacturing and maintenance of mechanical systems. Again, in the traditional sense, a Mechanical Engineer would be the expert in the production and usage of heat and mechanical power which are critical in the design, production and operation of tools, components and machines used in manufacturing durable goods and in using advanced technologies for the benefit of society.
In recent years, the discipline of Mechanical Engineering has taken on an expansive, and critical role in the advancement of transformative new technologies such as the information-, bio-, nano-technologies and MEMS (Micro-Electro-Mechanical Systems). It is clearly a new and exciting world out there. We are delighted to be at the forefront of acquiring new knowledge through cutting edge research in this new and expansive field of Mechanical Engineering. As reflected in their posted profiles and accomplishments, our faculty are recognized nationally and internationally for their research and discoveries. Many of them are Fellows of engineering societies and recipients of prestigious research awards granted for excellence by agencies such as the National Science Foundation and the National Institutes of Health.
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Item Performance enhancement of a magnetoactive elastomer actuator through a coupled magnetoelastic topology optimization scheme(IOP, 2024-11-08) Bergen, Christian; Ounaies, Zoubeida; Lockette, Paris vonWe have found that considering local magnetic fields, large deformations, and magnetoelastic coupling simultaneously significantly affect the resulting shape in magnetoelastic topology optimization in a uniaxial actuator case. In contrast to the work presented here, other works incorporate magnetoelastic formulations that include simplifying assumptions on the local field, and subsequent effects on the magnetization response of the material, or the absence of large deformation mechanics, or both. These assumptions were shown to produce solutions that differ substantively from cases where local fields and large deformations are addressed concurrently. Magnetoelastic topology optimization schemes are needed to optimize magnetoactive elastomer (MAE) devices. MAE devices are magnetic particle-filled polymer matrices designed for specific actuations and controlled remotely by an external magnetic field. They garner considerable research interest as an emerging technology for actuators in soft robots or in applications requiring untethered actuation. The material properties of MAEs are dependent on the volume fraction of particles in the elastomer matrix, where a high-volume fraction increases relative permeability (for soft magnetic particles) but also increases elastic modulus. For optimal actuation, a tradeoff between low stiffness and high magnetic response must be made by adjusting volume fraction and controlling material placement. Using a topology optimization scheme that considers both the magnetic and mechanical properties of the material, the shape and material composition of the device can be tuned to best achieve the desired actuation displacement. In this work, a two-dimensional density-based magnetoelastic multimaterial topology optimization scheme for soft magnetic material is implemented in COMSOL Multiphysics. The optimization scheme uses multiphysics coupling that considers local magnetic fields and large deformations at each iteration through a Maxwell stress tensor formulation. A simulated example is then considered to demonstrate the effectiveness and necessity of a coupled optimization. The effect of considering large deformations during optimization is also investigated. It was found that a coupled topology optimization scheme with large deformations produced shapes with modes of actuation not captured by schemes with simplifying assumptions, leading to better performance at lower material cost. Considering large deformations in the coupled scheme offered significantly better performance, with an increase of 81.3% in a side-by-side performance simulation when compared to uncoupled cases.Item Full-field Modal Analysis of a Tensegrity Column Using a Three-dimensional Scanning Laser Doppler Vibrometer with a Mirror(ASME, 2024-11-05) Yuan, Ke; Yuan, Sichen; Zhu, WeidongTensegrity structures become important components of various engineering structures due to their high stiffness, light weight, and deployable capability. Existing studies on their dynamic analyses mainly focus on responses of their nodal points while overlook deformations of their cable and strut members. This study proposes a non-contact approach for experimental modal analysis of a tensegrity structure to identify its three-dimensional (3D) natural frequencies and full-field mode shapes, which include modes with deformations of its cable and strut members. A 3D scanning laser Doppler vibrometer is used with a mirror for extending its field of view to measure full-field vibration of a novel three-strut metal tensegrity column with free boundaries. Tensions and axial stiffnesses of its cable members are determined using natural frequencies of their transverse and longitudinal modes, respectively, to build its theoretical model for dynamic analysis and model validation purposes. Modal assurance criterion (MAC) values between experimental and theoretical mode shapes are used to identify their paired modes. Modal parameters of the first 15 elastic modes of the tensegrity column identified from the experiment, including those of the overall structure and its cable members, can be classified into five mode groups depending on their types. Modes paired between experimental and theoretical results have MAC values larger than 78%. Differences between natural frequencies of paired modes of the tensegrity column are less than 15%. The proposed non-contact 3D vibration measurement approach allows accurate estimation of 3D full-field modal parameters of the tensegrity column.Item Reinforcement Learning Based Delay Line Design for Crosstalk Minimization(IEEE, 2024-10-31) Jung, Jaeho; Yu, Younggyun; Lee, SoobumReinforcement learning (RL) is one of the artificial intelligence techniques that build an artificial neural network to achieve the most optimum decision. In this study, Deep Q-Network (DQN) RL is applied to the design of delay lines in electrical circuits for signal synchronization. The delay line is usually designed densely in a confined space that results in electrical noise named crosstalk. The challenges of delay line design root from the fact that the line should connect the start and end point using a given length, not to be entangled in a predefined two-dimensional space. The genetic algorithms (GA) or the random exploration method can be used, but their learning efficiency is very low and time-consuming. We propose and implement a novel connected exploration method to significantly expedites the design process. In each state, the direction of the line rendering (left, straight, or right) is considered as an action, and the artificial intelligence agent learns how to design a delay line of the desired length. As a result, we are able to obtain the optimal designs 3,000 times faster than the case of using the GA from our previous study. The proposed method can be applied to various routing design problems such as circuit routing or flow path configuration with seriously reduced design time, and can potentially lead to the discovery of new designs not relied on human intuition.Item Performance enhancement of magnetoactive elastomer soft robots through a coupled magnetoelastic topology optimization scheme(IOP, 2024-11-08) Bergen, Christian Joseph; Ounaies, Zoubeida; von Lockette, ParisWe have found that considering local magnetic fields, large deformations, and magnetoelastic coupling simultaneously significantly affect the resulting shape in magnetoelastic topology optimization in a uniaxial actuator case. In contrast to the work presented here, other works incorporate magnetoelastic formulations that include simplifying assumptions on the local field, and subsequent effects on the magnetization response of the material, or the absence of large deformation mechanics, or both. These assumptions were shown to produce solutions that differ substantively from cases where local fields and large deformations are addressed concurrently. Magnetoelastic topology optimization schemes are needed to optimize magnetoactive elastomer (MAE) devices. MAE devices are magnetic particle-filled polymer matrices designed for specific actuations and controlled remotely by an external magnetic field. They garner considerable research interest as an emerging technology for actuators in soft robots or in applications requiring untethered actuation. The material properties of MAEs are dependent on the volume fraction of particles in the elastomer matrix, where a high-volume fraction increases relative permeability (for soft magnetic particles) but also increases elastic modulus. For optimal actuation, a tradeoff between low stiffness and high magnetic response must be made by adjusting volume fraction and controlling material placement. Using a topology optimization scheme that considers both the magnetic and mechanical properties of the material, the shape and material composition of the device can be tuned to best achieve the desired actuation displacement. In this work, a density-based magnetoelastic multimaterial topology optimization scheme for soft magnetic material is developed in COMSOL Multiphysics. The optimization scheme uses multiphysics coupling that considers local magnetic fields and large deformations at each iteration through a Maxwell stress tensor formulation. A simulated example is then considered to demonstrate the effectiveness and necessity of a coupled optimization. The effect of considering large deformations during optimization is also investigated. It was found that a coupled topology optimization scheme with large deformations produced shapes with modes of actuation not captured by schemes with simplifying assumptions, leading to better performance at lower material cost. Considering large deformations in the coupled scheme offered significantly better performance, with an increase of 81.3% in a side-by-side performance simulation when compared to uncoupled cases.Item RETHINKING DATA SCIENCE PEDAGOGY WITH EMBEDDED ETHICAL CONSIDERATIONS(IATED, 2022) Janeja, Vandana; Sanchez, MariaThe focus of this paper is to present a tool to meet the need of developing ethical critical thinking in data science curricula for undergraduate students. New data science methods impact societies, communities directly or indirectly when dealing with open and other real-world datasets. In particular, for data science there is a need to develop ethical critical thinking while analyzing the data. In the knowledge discovery process there are many opportunities for ethical decision making that a data scientist can evaluate throughout the entire lifecycle of the data to do no harm. To address these concerns within a learning environment focused on skilled workforce development we first introduce a novel ethical data lifecycle framework and then propose a vehicle for implementation through a short term module that can be embedded into a fast paced data science course. The objective of the module is to increase the ethical thinking of students when analyzing data. Pre and post surveys were conducted across each of the two semesters to evaluate students’ attitudes towards ethical thinking. The analysis of the survey results suggests that the objective was achieved based on a positive shift toward agreement with statements related to the importance of ethical thinking.Item MPC-guided, Data-driven Fuzzy Controller Synthesis(2024-10-09) Salazar, Juan Augusto Paredes; Goel, AnkitModel predictive control (MPC) is a powerful control technique for online optimization using system model-based predictions over a finite time horizon. However, the computational cost MPC requires can be prohibitive in resource-constrained computer systems. This paper presents a fuzzy controller synthesis framework guided by MPC. In the proposed framework, training data is obtained from MPC closed-loop simulations and is used to optimize a low computational complexity controller to emulate the response of MPC. In particular, autoregressive moving average (ARMA) controllers are trained using data obtained from MPC closed-loop simulations, such that each ARMA controller emulates the response of the MPC controller under particular desired conditions. Using a Takagi-Sugeno (T-S) fuzzy system, the responses of all the trained ARMA controllers are then weighted depending on the measured system conditions, resulting in the Fuzzy-Autoregressive Moving Average (F-ARMA) controller. The effectiveness of the trained F-ARMA controllers is illustrated via numerical examples.Item Numerical Study of Heat Transfer Enhancement Using Nano-Encapsulated Phase Change (NPC) Slurries in Wavy Microchannels(MDPI, 2024-10-09) Zaw, Myo Min; Zhu, Liang; Ma, RonghuiResearchers have attempted to improve heat transfer in mini/microchannel heat sinks by dispersing nano-encapsulated phase change (NPC) materials in base coolants. While NPC slurries have demonstrated improved heat transfer performance, their applications are limited by decreasing enhancement at increased flow rates. To address this challenge, the present study numerically investigates the effects of wavy channels on the performance of NPC slurries. Simulation results reveal that a wavy channel induces Dean vortices that intensify the mixing of the working fluid and enlarge the melting fractions of the NPC material, thus offering a significantly higher heat transfer efficiency than a straight channel. Moreover, heat transfer enhancement by NPC slurries varies with the imposed heat flux and flow rate. Interestingly, the maximum heat transfer enhancement obtained with the wavy channel not only exceeds the straight one, but also occurs at a higher heat flux and faster flow rate. This finding demonstrates the advantage of wavy channels in management of intensive heat fluxes with NPC slurries. The study also investigates wavy channels with varying amplitude and wavelength. Increasing the wave aspect ratio from 0.2 to 0.588 strengthens Dean vortices and consequently increases the Nusselt number, optimal heat flux, and overall thermal performance factor.Item Sim-to-Real Multirotor Controller Single-shot Learning(2024-10-04) Mirtaba, Mohammad; Oveissi, Parham; Goel, AnkitThis paper demonstrates the sim-to-real capabilities of retrospective cost optimization-based adaptive control for multirotor stabilization and trajectory-tracking problems. First, a continuous-time version of the widely used discrete-time retrospective control adaptive control algorithm is developed. Next, a computationally inexpensive 12-degree-of-freedom model of a multirotor is used to learn the control system in a simulation environment with a single trajectory. Finally, the performance of the learned controller is verified in a complex and realistic multirotor model in simulation and with a physical quadcopter in a waypoint command and a helical trajectory command.Item Model-free, Learning-based Control of LGKS Quantum System(2024-10-03) Delgado, Jhon Manuel Portella; Goel, AnkitThis paper presents a model-free, learning-based adaptive controller for the density tracking problem in a two-level Lindblad-Gorini-Kossakowski-Sudarshan (LGKS) quantum system. The adaptive controller is based on the continuous-time retrospective cost adaptive control. To preserve the geometric properties of the quantum system, an adaptive PID controller driven and optimized by Ulhmann's fidelity is used. The proposed controller is validated in simulation for a low and a high-entropy density-tracking problem.Item High-order Space-time Flux Reconstruction Methods for Moving Domain Simulation(2024-09-21) Yu, MeilinA high-order space-time flux reconstruction (FR) method has been developed to solve conservation laws on moving domains. In the space-time framework, the moving domain simulation is similar to that on a stationary domain, except that the shape of the space-time elements varies with time (and space when a deforming grid is used). The geometric conservation law can be automatically satisfied to the level of the numerical resolution of the space-time schemes when the space-time discretization of the governing partial differential equations can resolve the geometric nonlinearity of curvilinear space-time elements. In this study, a space-time tensor product operation is used to construct the FR formulation, and the Gauss-Legendre quadrature points are used as solution points both in space and time. A dual time stepping method is used to solve the resulting space-time system. As has been proved by Huynh [J Sci Comput 96, 51 (2023)], in the temporal direction, the FR scheme with the Gauss-Legendre solution points is equivalent to the so-called DG-Gauss implicit Runge-Kutta (IRK) scheme when the quadrature rule based on the solution points (i.e. quadrature points used in DG) is sufficiently accurate to integrate the space-time curvilinear elements. Specifically, we show that when linear space-time elements are adopted in moving domain simulations, the temporal FR scheme based on Gauss-Legendre solution points can always guarantee its equivalency to IRK DG-Gauss. The conditions, under which the moving domain simulation with the method of lines are consistent with those using the space-time formulation, are also discussed. The new space-time FR method can achieve arbitrarily high-order spatial and temporal accuracy without numerical constraints on the physical time step in moving domain simulations. The temporal superconvergence property for moving domain simulations have been demonstrated.Item Experimental Investigation of Steel-Borne Acoustic Pulses for Fault Pinpointing in Pipe-Type Cable Systems: A Scaled-Down Model Approach(MDPI, 2024-10-31) Moutassem, Zaki; Li, Gang; Zhu, WeidongPipe-type cable systems, including high-pressure fluid-filled (HPFF) and high-pressure gas-filled cables, are widely used for underground high-voltage transmission. These systems consist of insulated conductor cables within steel pipes, filled with pressurized fluids or gases for insulation and cooling. Despite their reliability, faults can occur due to insulation degradation, thermal expansion, and environmental factors. As many circuits exceed their 40-year design life, efficient fault localization becomes crucial. Fault location involves prelocation and pinpointing. Therefore, a novel pinpointing approach for pipe-type cable systems is proposed, utilizing accelerometers mounted on a steel pipe to capture fault-induced acoustic signals and employing the time difference of arrival method to accurately pinpoint the location of the fault. The experimental investigations utilized a scaled-down HPFF pipe-type cable system setup, featuring a carbon steel pipe, high-frequency accelerometers, and both mechanical and capacitive discharge methods for generating acoustic pulses. The tests evaluated the propagation velocity, attenuation, and pinpointing accuracy with the pipe in various embedment conditions. The experimental results demonstrated accurate fault pinpointing in the centimeter range, even when the pipe was fully embedded, with the acoustic pulse velocities aligning closely with the theoretical values. These experimental investigation findings highlight the potential of this novel acoustic pinpointing technique to improve fault localization in underground systems, enhance grid reliability, and reduce outage duration. Further research is recommended to validate this approach in full-scale systems.Item Singularity-free Backstepping-based Adaptive Control of a Bicopter with Unknown Mass and Inertia(2024-09-19) Delgado, Jhon Manuel Portella; Goel, AnkitThe paper develops a singularity-free backstepping-based adaptive control for stabilizing and tracking the trajectory of a bicopter system. In the bicopter system, the inertial parameters parameterize the input map. Since the classical adaptive backstepping technique requires the inversion of the input map, which contains the estimate of parameter estimates, the stability of the closed-loop system cannot be guaranteed due to the inversion of parameter estimates. This paper proposes a novel technique to circumvent the inversion of parameter estimates in the control law. The resulting controller requires only the sign of the unknown parameters. The proposed controller is validated in simulation for a smooth and nonsmooth trajectory-tracking problem.Item Neural filtering for Neural Network-based Models of Dynamic Systems(2024-09-20) Oveissi, Parham; Rozario, Turibius; Goel, AnkitThe application of neural networks in modeling dynamic systems has become prominent due to their ability to estimate complex nonlinear functions. Despite their effectiveness, neural networks face challenges in long-term predictions, where the prediction error diverges over time, thus degrading their accuracy. This paper presents a neural filter to enhance the accuracy of long-term state predictions of neural network-based models of dynamic systems. Motivated by the extended Kalman filter, the neural filter combines the neural network state predictions with the measurements from the physical system to improve the estimated state's accuracy. The neural filter's improvements in prediction accuracy are demonstrated through applications to four nonlinear dynamical systems. Numerical experiments show that the neural filter significantly improves prediction accuracy and bounds the state estimate covariance, outperforming the neural network predictions.Item Benchmark Study of Melted Track Geometries in Laser Powder Bed Fusion of Inconel 625(Springer Nature, 2021-06-01) Gan, Zhengtao; Jones, Kevontrez K.; Lu, Ye; Liu, Wing KamIn the Air Force Research Laboratory Additive Manufacturing Challenge Series, melted track geometries for a laser powder bed fusion (L-PBF) process of Inconel 625 were used to challenge and validate computational models predicting melting and solidification behavior. The impact of process parameters upon single-track single-layer, multi-track single-layer, and single-track multi-layer L-PBF processes was studied. To accomplish this, a physics-based thermal-fluid model was developed and calibrated using a proper generalized decomposition surrogate model, then compared against the experimental measurements. The thermal-fluid model was enhanced through the usage of an adaptive mesh and residual heat factor (RHF) model, based on the scanning strategy, for improved efficiency and accuracy. It is found that this calibration approach is not only robust and efficient, but it also enables the thermal-fluid model to make predictions which quantitatively agree well with the experimental measurements. The adaptive mesh provides over a 10-times speedup as compared to a uniform mesh. The RHF model improves predictive accuracy by over 60%, particularly near starting and ending points of the melted tracks, which are greatly affected by the thermal behavior of adjacent tracks. Moreover, the thermal-fluid model is shown to potentially predict lack-of-fusion defects and provide insights into the defect generation process in L-PBF.Item Reducing Flow Resistance via Introduction and Enlargement of Microcracks in Convection Enhanced Delivery (CED) in Porous Tumors(MDPI, 2024-09-13) Naseem, Md Jawed; Ma, Ronghui; Zhu, LiangA theoretical simulation is performed to evaluate how microcracks affect the flow resistance in tumors during the convection-enhanced delivery (CED) of nanofluids. Both Darcy’s law and the theory of poroelasticity are used to understand fluid transport with or without microcrack introduction and/or enlargement. The results demonstrate significantly altered pressure and velocity fields in a spherical tumor with a radius of 10 mm due to the presence of a microcrack with a radius of 0.05 mm and length of 3 mm. The non-uniform fluid pressure field enlarges the original cylindrical microcrack to a frustum, with the crack volume more than doubled. Due to the larger permeability and porosity in the microcrack, flow in the tumor is much easier. One finds that the flow resistance with the enlarged microcrack is reduced by 14% from the control without a microcrack. Parametric studies are conducted to show that larger crack radii, longer crack lengths and higher infusing pressures result in further resistance reductions. The largest resistance reduction occurs when the infusing pressure is 4 × 10⁵ Pa and the microcrack is 9 mm long, up to 18% from the control. We conclude that introducing a microcrack is an effective way to facilitate nanofluid delivery in porous tumors using CED.Item A Hammerstein-Weiner Modification of Adaptive Autopilot for Parameter Drift Mitigation with Experimental Results(IEEE, 2024-09-05) Chee, Yin Yong; Oveissi, Parham; Shao, Siyuan; Lee, Joonghyun; Paredes, Juan A.; Bernstein, Dennis S.; Goel, AnkitA crucial challenge in the safe operation of adaptive controllers is the problem of parameter drift, where an underlying optimization problem, if ill-conditioned, may lead to parameter drift. This paper presents a Wiener adaptive autopilot for multicopters to mitigate instabilities caused by adaptive parameter drift and presents simulation and experimental results to validate the modified autopilot. The modified adaptive controller is obtained by including a static nonlinearity in the adaptive loop, updated by the retrospective cost adaptive control algorithm. It is shown in simulation and physical test experiments that the adaptive autopilot with proposed modifications can continually improve the fixed-gain autopilot as well as prevent the drift of the adaptive parameters, thus improving the robustness of the adaptive autopilot.Item Fetal and neonatal echocardiographic analysis of biomechanical alterations for the systemic right ventricle heart(Plos, 2024-09-19) Meyers, Brett A.; Bhattacharya, Sayantan; Brindise, Melissa C.; Loke, Yue-Hin; Payne, R. Mark; Vlachos, Pavlos P.Background - The perinatal transition’s impact on systemic right ventricle (SRV) cardiac hemodynamics is not fully understood. Standard clinical image analysis tools fall short of capturing comprehensive diastolic and systolic measures of these hemodynamics. Objectives - Compare standard and novel hemodynamic echocardiogram (echo) parameters to quantify perinatal changes in SRV and healthy controls. Methods- We performed a retrospective study of 10 SRV patients with echocardiograms at 33-weeks gestation and at day of birth and 12 age-matched controls. We used in-house developed analysis algorithms to quantify ventricular biomechanics from four-chamber B-mode and color Doppler scans. Cardiac morphology, hemodynamics, tissue motion, deformation, and flow parameters were measured. Results - Tissue motion, deformation, and index measurements did not reliably capture biomechanical changes. Stroke volume and cardiac output were nearly twice as large for the SRV compared to the control RV and left ventricle (LV) due to RV enlargement. The enlarged RV exhibited disordered flow with higher energy loss (EL) compared to prenatal control LV and postnatal control RV and LV. Furthermore, the enlarged RV demonstrated elevated vortex strength (VS) and kinetic energy (KE) compared to both the control RV and LV, prenatally and postnatally. The SRV showed reduced relaxation with increased early filling velocity (E) compared prenatally to the LV and postnatally to the control RV and LV. Furthermore, increased recovery pressure (ΔP) was observed between the SRV and control RV and LV, prenatally and postnatally. Conclusions - The novel hydrodynamic parameters more reliably capture the SRV alterations than traditional parameters.Item Heating Induced Nanoparticle Migration and Enhanced Delivery in Tumor Treatment Using Nanotechnology(MDPI, 2024-9-7) Gu, Qimei; Zhu, LiangNanoparticles have been developed as imaging contrast agents, heat absorbers to confine energy into targeted tumors, and drug carriers in advanced cancer treatment. It is crucial to achieve a minimal concentration of drug-carrying nanostructures or to induce an optimized nanoparticle distribution in tumors. This review is focused on understanding how local or whole-body heating alters transport properties in tumors, therefore leading to enhanced nanoparticle delivery or optimized nanoparticle distributions in tumors. First, an overview of cancer treatment and the development of nanotechnology in cancer therapy is introduced. Second, the importance of particle distribution in one of the hyperthermia approaches using nanoparticles in damaging tumors is discussed. How intensive heating during nanoparticle hyperthermia alters interstitial space structure to induce nanoparticle migration in tumors is evaluated. The next section reviews major obstacles in the systemic delivery of therapeutic agents to targeted tumors due to unique features of tumor microenvironments. Experimental observations on how mild local or whole-body heating boosts systemic nanoparticle delivery to tumors are presented, and possible physiological mechanisms are explored. The end of this review provides the current challenges facing clinicians and researchers in designing effective and safe heating strategies to maximize the delivery of therapeutic agents to tumors.Item Learning-based Attitude Estimation with Noisy Measurements and Unknown Gyro Bias(IEEE) Oveissi, Parham; Mirtaba, Mohammad; Goel, AnkitThis paper introduces a learning-based, datadriven attitude estimator, called the retrospective cost attitude estimator (RCAE), for the SO(3) attitude representation. RCAE is motivated by the multiplicative extended Kalman filter (MEKF). However, unlike MEKF, which requires computing a Jacobian to compute the correction signal, RCAC uses retrospective cost optimization that depends only on the measured data. Moreover, due to the structure of the correction signal, RCAE does not require explicit estimation of gyro bias. The performance of RCAE is verified and compared with MEKF through both numerical simulations and physical experiments.Item Matrix-Based Representations and Gradient-Free Algorithms for Neural Network Training(IEEE) Rozario, Turibius; Oveissi, Parham; Goel, AnkitThis paper presents a compact, matrix-based representation of neural networks. Although neural networks are often understood pictorially as interconnected neurons, they are fundamentally mathematical nonlinear functions constructed by composing several vector-valued functions. Using basic results from linear algebra, we represent neural networks as an alternating sequence of linear maps and scalar nonlinear functions, known as activation functions. The training of neural networks involves minimizing a cost function, which typically requires the computation of a gradient. By applying basic multivariable calculus, we show that the cost gradient is also a function composed of a sequence of linear maps and nonlinear functions. In addition to the analytical gradient computation, we explore two gradient-free training methods. We compare these three training methods in terms of convergence rate and prediction accuracy, demonstrating the potential advantages of gradient-free approaches.