Wrong Operator or Blind Design? A Reference-Free Diagnostic for Physics-Informed Coefficient Learning

Neural operators evaluate parametric partial differential equations cheaply but degrade sharply outside their training distribution. Physics-informed neural networks avoid dependence on labeled data, yet their optimization can be basin-fragile: when the governing residual admits multiple solutions, a PINN trained from scratch may converge to a physically incorrect state despite achieving a small residual. We show that these failure modes can be addressed jointly: an imperfect NO provides the structural prior needed to place a PINN in the correct solution basin, while the PDE residual refines the solution beyond the operator's accuracy. We introduce a three-stage framework that freezes the spatial basis of a physics-informed NO, extrapolates its solution branch to an out-of-distribution parameter using a polynomial continuation prior, and distills the resulting field into a fresh PINN. The NO need not be accurate at the target; it transfers solution-branch information, while PDE residual minimization in the PINN governs convergence. We evaluate the framework on three nonlinear PDEs: 1D viscous Burgers, 2D steady Allen-Cahn near a pitchfork bifurcation, and 2D steady lid-driven cavity flow. For Allen-Cahn, where the trivial solution satisfies the PDE residual exactly, a standard PINN collapses to the trivial zero branch, whereas distillation from the crude extrapolated operator recovers the non-trivial branch that matches the finite-difference reference. For the lid-driven cavity, extrapolating to a Reynolds number of Re = 3200 accelerates convergence to the correct physical state, achieving competitive accuracy using fewer parameters and optimization steps than recent literature baselines. These results establish a simple principle: an NO need not accurately predict the solution to be useful; it only needs to identify the correct basin from which PINN optimization can recover it.

Physics-informed neural networks (PINNs) embed governing equations in their loss function, enabling mesh-free solutions to partial differential equations. Low training loss is treated as evidence that the learned solution is physically correct. This paper shows that assumption breaks down when encoded physics are incorrect. By perturbing PDE parameters before training, a setting we describe as physics parameter poisoning or parameter misspecification, we produce models that train to low loss but give incorrect answers; we treat the perturbation schedule as sensitivity analysis rather than only as a security threat, and none of our claims requires an adversary. Achieving low residual loss does not discriminate accurate from inaccurate solutions: poisoned models reach losses at or below the clean baseline yet differ by large margins, so driving the residual down is not evidence of physical accuracy. Across three PDE systems (Burgers equation, Navier-Stokes cavity, and convection-diffusion), poisoned models match or beat the clean-model training loss while their solutions differ by up to 71% in the fixed sweep and up to 128% under adversarial search; at Cavity Re=400 the poisoned loss falls below the clean baseline. We define a detection difficulty ratio R (solution error divided by training loss) to summarize how invisible the corruption is, though cross-PDE comparison is complicated by differences in loss scale. We test six candidate defenses, none of which reliably detects corruption across all regimes. We propose a post-hoc defense: sweeping the PDE residual loss across parameter values without retraining. The loss minimum recovers the true training parameter without external data, and generalizes across all three PDE systems. The effect holds across five network architectures (8.7K to 133K parameters), is bidirectional, and is confirmed across multiple random seeds.

Physics-informed neural operators (PINOs) aim to learn solution operators for partial differential equations by using the governing physics as supervision, rather than relying solely on paired input-output simulation data. By incorporating physical constraints into the training objective, PINOs combine the cross-instance generalization of neural operators with the data efficiency of physics-informed learning. Despite this promise, how to train PINOs efficiently and robustly remains less well-understood than the training of either data-driven neural operators or physics-informed neural networks (PINNs). To bridge this gap, we examine key components of the PINO training pipeline, including architecture design, optimizer choice, loss balancing, and collocation-point sampling strategy. We study three representative operator backbones, Deep Operator Network (DeepONet), Fourier Neural Operator (FNO), and Continuous Vision Transformer (CViT), across five diverse parametric PDE systems. Our results show that CViT provides consistently strong and stable performance across the considered benchmarks. Beyond architecture, we find that several optimization pathologies previously identified in PINN training naturally arise in PINOs, including gradient conflicts and causal violation. We also find that mitigation algorithms developed for PINNs remain effective in the PINO setting. We further compare physics-informed and data-driven training under different data regimes, revealing that a carefully designed physics-informed training pipeline can match, and in some cases, outperform purely data-driven neural operators. Taken together, these findings provide a systematic empirical understanding of the optimization challenges in PINO training and inform a practical pipeline for efficient and robust physics-informed operator learning. Code and data are available at https://github.com/NanxiiChen/PI-CViT.