Breaking the Translation Equivariance Barrier: Coordinate Channels and Bounded Error Propagation in Multi-Physics Neural Operator Cascades

Neural operators such as the Fourier Neural Operator (FNO) achieve remarkable accuracy in approximating solutions to partial differential equations (PDEs), but two fundamental challenges hinder their deployment in safety-critical engineering: (1) a structural approximation barrier arising from FNO's translation equivariance that prevents learning Dirichlet boundary conditions, and (2) the lack of uncertainty bounds when multiple operators are chained in multi-physics cascades. We address both. First, we prove that FNO's translation equivariance creates a fundamental approximation barrier for PDEs with non-trivial boundary conditions (Theorem 1), and that coordinate channels resolve this (Theorem 2), validated by up to 63x error reduction across five scenarios. Second, we formalize error propagation in operator cascades: we prove that when K neural operators are composed sequentially, each with conformal prediction coverage (1-alpha_k), the cascade coverage degrades due to exchangeability violations between stages (Theorem 3), and show that coordinate channels mitigate this degradation by reducing systematic boundary errors that compound across stages. We validate on a 3D electromagnetic-thermal-structural cascade, where coordinate channels reduce coupling error from 18.5% to 5.0%. For per-stage UQ, we propose Physics- Informed Conformal Prediction (PI-CP), which embeds PDE residuals into the nonconformity score—complementary to recent physics-residual CP (Gopakumar et al., 2025)—and achieves 89-91% coverage across six physics scenarios with spatially adaptive intervals. We further identify a fundamental boundary of diffusion-based UQ: on well-conditioned forward PDEs, the trained FNO's residual is too small for diffusion models to capture meaningful uncertainty (38-53% coverage). An industrial application—fiber-optic gyroscope bias instability prediction—validates the framework with 3.7-8.0x improvement over analytical baselines.