Equation-Coordinate-Invariant Residual Metrics for Physics-Informed Neural Networks

Physics-informed neural networks (PINNs) can learn different solutions from algebraically equivalent formulations of the same partial differential equation (PDE). For coupled systems, invertible scaling and mixing preserve the physical constraints but change standard residual losses and their optimization geometry. We formalize this dependence as equation-coordinate bias and introduce ECI-PINN, which replaces the algebraic residual norm with the minimum correction required to satisfy the locally linearized PDE constraints in a dimensionless jet space. The resulting full residual metric accounts for coupling between equations rather than normalizing each equation independently. When the residual Jacobian with respect to the jet has full row rank, we prove that the unregularized metric preserves loss values and detached parameter gradients under pointwise invertible residual-basis transformations independent of network parameters and jet state. Matched deterministic training therefore follows the same optimization trajectory. We evaluate ECI-PINN using PDE-Rep, a protocol spanning Wave, Reaction-Diffusion, and Kovasznay benchmarks, five equivalent residual representations, and three random seeds. ECI-PINN achieves the lowest identity, average, and worst-case relative L2 errors among the compared methods on all three tasks, reducing the Wave worst-case error by more than 50-fold relative to Vanilla PINN. Comparisons with diagonal metrics and residual normalization support the importance of modeling the full coupled residual geometry. These results establish equation-coordinate invariance as a design principle for reliable physics-informed learning.