From Physics-Informed Losses to Equation-Derived Priors and Representations for Neural Operators

When governing equations are known but solution data are scarce, equation knowledge can be used not only as a training constraint but also as an inductive bias for the neural operator. We study three ways of exposing equation information: enforcing the equation as a loss, using it to construct an equation-derived prior, and injecting the structure it reveals into the residual. Our idea is to use the governing equation to reduce the learning problem before training. Across the benchmarks that admit a physics-informed loss, the equation-derived prior compares favorably with imposing the same equation as a training loss, and some problems benefit further from structural injection beyond the prior. These results suggest that equation knowledge is more effective when it is used to shape the representation before training than when used solely as a training constraint.