Selecting PINNs and neural operators for repeated engineering analysis

Choosing a learned engineering surrogate requires specifying what is reused across physical cases, what information is available at prediction time, and which errors influence the design decision. Physics-informed neural networks and neural operators address overlapping but nonidentical versions of this task. This review compares coordinate-based field approximations, parameter-conditioned surrogates, function-to-function operators, and models refined with physical equations at deployment. A common mathematical description separates input sampling, learned approximation, output reconstruction, and case-specific adaptation. Analytical examples establish an input-information limit, a cost break-even condition, and a bound connecting surrogate-gradient error to design error under strong convexity. Applications to flow design, thermal loading, and viscoplastic response show why fast field prediction alone does not establish an engineering advantage. A proposed evaluation protocol compares physical accuracy, derivative accuracy, complete computational cost, and performance over genuinely unseen cases.