The SINDy (Sparse Identification of Non-linear Dynamics) algorithm is a method of turning a set of data representing non-linear dynamics into a much smaller set of equations comprised of non-linear functions summed together. This provides a human readable system model the represents the dynamic system analyzed. The SINDy algorithm is important for a variety of applications, including high precision industrial and robotic applications. A Hardware Accelerator was designed to decrease the time spent doing calculations. This thesis proposes an efficient hardware accelerator approach for a broad range of applications that use SINDy and similar system identification algorithms. The accelerator is leverages both systolic arrays for integrated neural network models with other numerical solvers. The novel and efficient reuse of similar processing elements allows this approach to only use a minimal footprint, so that it could be added to microcontroller devices or implemented on lower cost FPGA devices. Our proposed approach also allows the designer to offload calculations onto edge devices from controller nodes and requires less communication from those edge devices to the controller due to the reduced equation space.

Scientific machine learning (SciML) is a new branch of AI research at the edge of scientific computing (Sci) and machine learning (ML). It deals with efficient amalgamation of data-driven algorithms along with scientific computing to discover the dynamics of the time-evolving process. The output of such algorithms is represented in the form of a governing equation(s) (e.g., ordinary differential equation(s), ODE(s)), which one can solve then for any time point and, thus, obtain a rigorous prediction.  In this thesis, we present a methodology on how to incorporate the SciML approach in the context of clinical trials to predict IPF disease progression in the form of governing equation. Our proposed methodology also quantifies the uncertainties associated with the model by fitting 95\% high density interval (HDI) for the ODE parameters and 95\% posterior prediction interval for posterior predicted samples. We have also investigated the possibility of predicting later outcomes by using the observations collected at early phase of the study. We were successful in combining ML techniques, statistical methodologies and scientific computing tools such as bootstrap sampling, cubic spline interpolation, Bayesian inference and sparse identification of nonlinear dynamics (SINDy) to discover the dynamics behind the efficacy outcome as well as in quantifying the uncertainty of the parameters of the governing equation in the form of 95 \% HDI intervals. We compared the resulting model with the existed disease progression model described by the Weibull function. Based on the mean squared error (MSE) criterion between our ODE approximated values and population means of respective datasets, we achieved the least possible MSE of 0.133,0.089,0.213 and 0.057. After comparing these MSE values with the MSE values obtained after using Weibull function, for the third dataset and pooled dataset, our ODE model performed better in reducing error than the Weibull baseline model by 7.5\% and 8.1\%, respectively. Whereas for the first and second datasets, the Weibull model performed better in reducing errors by 1.5\% and 1.2\%, respectively. Comparing the overall performance in terms of MSE, our proposed model approximates the population means better in all the cases except for the first and second datasets, assuming the latter case's error margin is very small. Also, in terms of interpretation, our dynamical system model contains the mechanistic elements that can explain the decay/acceleration rate of the efficacy endpoint, which is missing in the Weibull model. However, our approach had a limitation in predicting final outcomes using a model derived from  24, 36, 48 weeks observations with good accuracy where as on the contrast, the Weibull model do not possess the predicting capability. However, the extrapolated trend based on 60 weeks of data was found to be close to population mean and the ODE model built on 72 weeks of data. Finally we highlight potential questions for the future work.

Given the importance of diagnosing internal combustion engines and their expandingmarket, it is crucial to investigate the complex dynamics of these engines and developa model to detect and localize component and sensor faults, independent of operatingconditions. This master’s thesis explores capturing dynamic relationships in engine signalsto detect faults. Internal combustion engines have highly nonlinear dynamics that arenot easy to capture with basic system identification methods. Consequently, some newlydeveloped methods, such as Sparse Identification of Nonlinear Dynamics (SINDy) andAuto Regressive (AR) models, have been implemented to address the governing equations.Following previous research in this field, some analytical relations between the sensorswere known. Additionally, these equations provide the sensitivity of residuals based onthe input sensor fault. By implementing the mentioned methods and using informationfrom the analytical equations, some relationships between the sensor values and the inputfault have been identified.After finding these dynamic relationships between the sensor values, a classificationalgorithm was selected to classify the sensor faults. Additionally, to estimate the faultseverities (sensor inaccuracies), some regression models have been implemented. Further-more, it was desired to evaluate the isolability of the faults, and in this regard, some newconstraints have been considered to reconstruct the relations with the desired isolability.Finally, through several evaluations, it was shown that the proposed method is not af-fected by the driving cycle. However, this research established these methods based on aspecific case study, which is a specific turbocharged internal combustion engine.