# A Taxonomic Survey of Physics-Informed Machine Learning
## 1 Introduction  
Building reliable multiphysics models is an essential operation in nearly any given area of scientific research. However, solving and interpreting said models can be an essential limitation for many. As complexity and dimensionality in physical models, so to does the expense associated with traditional methods towards differentiation--such as finite element mesh construction. In addition to numerical hurdles, challenging experimental observation or otherwise unavailable data can also limit the applicability of certain modeling approaches. Furthermore, traditional learning machines fail to learn relationships in many complex physical systems due to the typical imbalance of data for one and the lack of physically relevant knowledge for another. Problems lacking trustworthy observational data are typically modeled by precise systems of complex equations with initial conditions and tuned coefficients. This work details recent advancements in physics-informed machine learning. Physics-informed machine learning is a tool by which researchers can extract physically relevant solutions to multiscale modeling problems. Crucially, physics-informed learning machines have been shown to accurately learn general solutions to complex physical processes having sparse multifidelity and/or otherwise incomplete data by leveraging the knowledge of the underlying physical features. What differentiates physics-informed learning from traditional statistical model is the somehow tangible inclusion of physically relevant prior knowledge. The constitutional need for qualitatively defined, physically relevant learning interventions further increases the need for a qualitative taxonomy. Most notable among the recent advancements, we focus on the increasing parallelism of the physics-informed neural network algorithm and the introduction of neural operators for learning systems of differential equations. In 2021, [PERSON] et al. [1] provided a comprehensive review of the methods leveraged in physics-informed learning and formed an outline of biases catalyzed by prior physical knowledge. [PERSON] et al. assert as a key point, \"Physics-informed machine learning integrates seamlessly data and mathematical physics models, even in partially understood, uncertain and high-dimensional contexts\" [1]. This comprehensive review primarily details physics-informed neural learning machines for applicability to a diverse set of difficult, ill-posed, and inverse problems. [PERSON] et al. continue to discuss domain decomposition for scalability and operator learning as future areas of research. Toward a recent expansion in a wide range of application, the use of physics-informed learning machines has seen the method's application in diverse fields including fluids [2], heat transfer [3], COVID-19 spread [4; 5], and cardiac modeling [6; 7; 8]. [PERSON] et al. [2] offer a review of physics-informed machine learning implementations for three-dimensional wake flows, supersonic flows, and biomedical flows. High-dimensional and noisy data from fluid flows are prohibitively difficult to train with traditional learning algorithms; this review highlights the applicability of physics-informed neural networks tackling this problem in fluid flow modeling. For heat transfer problems, [PERSON] et al. [3] discuss a variety of physics-informed machine learning approaches in convection heat transfer problems with unknown boundary conditions, including several forced convection and mixed convection problems. Again, [PERSON] et al. showcase diverse applications of physics-informed neural networks to apply neural learning machines in traditionally impractical settings where injecting physically relevant prior information makes neural network modeling viable. In 2022, [PERSON] et al. [4] provided an SEIRP-informed neural with architecture and training routine changes defined by governing compartmental infection model equations. Additionally, [PERSON] et al.