# A Taxonomic Survey of Physics-Informed Machine Learning
## 1 Introduction
[5] propose the fractional PINN (fPINN), a physics-informed neural network created for the rapidly mutable COVID-19 variants trained on Caputo-Hadamard derivatives in the loss propagation of the training process. [PERSON] et al. [9] provide a summary of several physics-informed machine learning use cases. The wide range of apt applications for physics-informed machine learning further perpetuates the need for qualitative discussion and subclassification. The need for computational methods, especially where problems are modeled by complex and/or multiscale systems of nonlinear equations, is growing expeditiously. An exhaustive amount of scholarly thought has recently been afforded toward methods advancing the data-driven learning of partial differential equations. [PERSON] et al. [10] were able to infer lift and drag from velocity measurements and flow visualizations, with Navier-Stokes being estimated using automatic differentiation to obtain the required derivatives and compute the residuals composing loss step augmentations. A similar process is common for learning process augmentations in physics-informed learning. [PERSON] et al. [11] introduced a hidden physics model for learning nonlinear partial differential equations (PDEs) from noisy and limited experimental data by leveraging the underpinning physical laws. This approach uses the Gaussian process to balance model complexity and data fitting. The hidden physics model was applied to the data-driven discovery of PDEs, such as Navier-Stokes, Schrodinger, and Kuramoto-Sivashinsky equations. Later, in another paper, two neural networks were employed for similar problems [12]. The first neural network models the prior of the unknown solution, and the second neural network models the nonlinear dynamics of the system. In another work, the same group used deep neural networks combined with a multistep time-stepping scheme to identify nonlinear dynamics from noisy data [13]. The effectiveness of this approach was shown for nonlinear and chaotic dynamics, Lorenz system, fluid flow, and Hopf bifurcation. In 2019, [PERSON] et al. [14] also proposed two types of algorithms: continuous-time and discrete-time models. The so-titled physics-informed neural network is tailored to two classes of problems as follows: (a) the data-driven solution of PDEs and (b) the data-driven discovery of PDEs. The approach's effectiveness was demonstrated for several problems, including Navier-Stokes, Burgers'equation, and Schrodinger equations. Consequently, the PINN has been adapted to intuit governing equations or solution spaces for many types of physical systems. For Reynolds-averaged Navier-Stokes problems, [PERSON] et al. [15] propose a physics-informed random forest model for assisting data-driven flow modeling. Other research introduce physics constraints, consequently biasing the learning process driven by prior relevant knowledge. [PERSON] et al. [16] accurately solved highly-dimensional free-boundary partial differential equations and proved the approximation utility of neural networks for quasilinear partial differential equations. [PERSON] et al. [17] proposed a deep learning method for high-dimensional partial differential equations with backward stochastic differential equation reformulations. [PERSON] et al. [18] propose a method for estimating governing equations of numerically derived flow data using automatic model selection. [PERSON] et al. [19] proposed another data-driven approach for governing equation discovery, and [PERSON] et al. [20] introduce the combination of the deep theory of functional connections with neural networks to estimate solutions to partial differential equations. The preceding selection includes examples of the transition from _solving_ partial differential equations with expensive techniques for _learning_ solutions with high-throughput learning machines, and we cover some works on the latter topic next.