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Recent breakthroughs in computing power have made it feasible to use machinelearning and deep learning to advance scientific computing in many fields,including fluid mechanics, solid mechanics, materials science, etc. Neuralnetworks, in particular, play a central role in this hybridization. Due totheir intrinsic architecture, conventional neural networks cannot besuccessfully trained and scoped when data is sparse, which is the case in manyscientific and engineering domains. Nonetheless, neural networks provide asolid foundation to respect physics-driven or knowledge-based constraintsduring training. Generally speaking, there are three distinct neural networkframeworks to enforce the underlying physics: (i) physics-guided neuralnetworks (PgNNs), (ii) physics-informed neural networks (PiNNs), and (iii)physics-encoded neural networks (PeNNs). These methods provide distinctadvantages for accelerating the numerical modeling of complex multiscalemulti-physics phenomena. In addition, the recent developments in neuraloperators (NOs) add another dimension to these new simulation paradigms,especially when the real-time prediction of complex multi-physics systems isrequired. All these models also come with their own unique drawbacks andlimitations that call for further fundamental research. This study aims topresent a review of the four neural network frameworks (i.e., PgNNs, PiNNs,PeNNs, and NOs) used in scientific computing research. The state-of-the-artarchitectures and their applications are reviewed, limitations are discussed,and future research opportunities in terms of improving algorithms, consideringcausalities, expanding applications, and coupling scientific and deep learningsolvers are presented. This critical review provides researchers and engineerswith a solid starting point to comprehend how to integrate different layers ofphysics into neural networks.

Physics-Guided, Physics-Informed, and Physics-Encoded Neural Networks in Scientific Computing