# A Taxonomic Survey of Physics-Informed Machine Learning
## 3 Discussion of Relevant Applications
### Domain Decomposition  
One important applicability of physics-informed neural networks is in the regime of problems traditionally resolved with expensive meshing methods, such as finite element, for example. In several applications, PINNs reduce the cost with substantial accuracy, while training time and residual calculations can still be concerning. Toward the end of mitigating cost, much work has improved the parallelization of the PINN architecture. [PERSON] et al. [22] introduced XPINN, a generally applicable space-time decomposition for easily parallelized subdomains governed by individually interconnected sub-PINNs. XPINN's wide applicability to forward and inverse modeling problems were displayed. Impressively complex subdomains can be solved reliably with XPINN due to the formulation of relatively simple interfacing conditions. XPINN employs model-driven inductive and learning biases by augmenting network shape and loss functions based upon physical priors. XPINN is a broad example which encompasses each of the branches of the taxonomy. [PERSON] et al. employ XPINN on complex subdomains. Observational bias is introduced through simulated training data produced with a Hopf-Cole-transformation-provided analytical solutions to one-dimensional viscous Burgers equations \(u_{t}+uu_{x}=vu_{x}x\), \(u\in R\), \(t>0\). Similarly to PINN, XPINN includes physical structures in the architecture and loss calculations for physics-model-driven inductive and learning biases. [PERSON] et al. [23] further demonstrated the applicability of domain decomposition in PINNs toward parallel speedup. A general framework of discerning the applicability of the extended PINN paradigm in various modeling problems is presented. Their study provides computational examples on KdV, heat, advection, Poisson, and compressible Euler equations. The Poisson solution employs the [PERSON] regularization method [49]--a regularization technique proposed particularly for PINNs. This study trains on high-fidelity simulations, tweaks model architecture, and incorporates governing equations into optimization, thus exemplifying observational, inductive, and learning biases in one or all of the examples presented. [PERSON] et al. [24] provide a parallel implementation of the previously proposed cPINN [25] and XPINN on two-dimensional steady-state incompressible Navier-Stokes equations, viscous Burgers equation, and a steady-state heat conduction with variable conductivity inverse problem. This work shows the advantages and disadvantages of each method and their use in tandem. Further, optimization of the distributed computing process is given. Each implementation further exemplifies the applicability of domain decomposition methods for physics-informed neural networks toward arbitrarily shaped and complex subdomains. Physics-model-driven inductive and learning biases, as fundamentally similar to XPINN, were discussed by [PERSON] et al. [24] and [PERSON] et al. [25]. Physics-data drivers are available depending upon modeling choices. [PERSON] et al. [26] outlined XPINN's effectiveness toward inverse problems in supersonic flows. Enforced physics information includes governing equations as well as entropy conditions--displaying the utility of additional physics information beyond the model governing system. In this study, the conservative form Euler PDE equations are given as \(\delta_{t}U+\
abla\cdot G(U)=0\). Here, both conservation laws, \(F:=\delta_{t}U+\
abla\cdot G(U)\), and entropy conditions, \(\eta_{t}+\phi_{1x}+\phi+2y\leq 0\), are applied as learning biases. [PERSON] et al. [27] provide an XPINN implementation for steady-state heat transfer in composite materials with interface interaction. The general adaptability of domain decomposition in physics-informed neural networks is further exemplified by the APINNs [28] proposed by [PERSON] et al. ; they provide a variable decomposition technique for fine-tuning subdomain boundaries. APINN utilizes a gating network to mimic XPINN and provide soft domain decomposition.