Dimensionally consistent loss construction for physics-informed flow and thermomechanical models

Abstract

The loss function of a physics-informed neural network combines quantities that can have different units, scales, sampling  measures, and sensitivities to the network parameters. Consequently, balancing numerical loss values is not equivalent to balancing physical accuracy. This review develops a dimensionally explicit treatment of loss construction for incompressible flow, heat conduction, and elastic-viscoplastic solids. It separates nondimensionalization, residual normalization, statistical weighting, adaptive optimization, and sampling. Two analytical examples show how a change of units can change an uncorrected optimum and how equal loss values can produce unequal parameter gradients. Published flow and plasticity studies are examined alongside gradient-based, relative-progress, and pointwise adaptation methods. The synthesis identifies what each method controls, which comparisons are meaningful, and what evidence is needed to establish an improvement in force, temperature gradient, or stress. A reproducible loss specification should connect every residual and multiplier to a governing equation, observation model, or documented numerical objective.