WINO: A Weak-Form Physics Informed Neural Operator for Hyperelasticity on Variable Domains

We propose a new weak-form Physics-Informed Neural Network approach (named INI-VPINN). INI-VPINN naturally incorporates Neumann boundary and interface conditions into the variational formulation. It removes the need for additional loss terms or multiple subdomain networks. This framework employs compact support weighting functions and integration by parts to implicitly impose flux and continuity constraints. In this way, it implicitly ensures physical consistency across material boundaries. The proposed method is tested on Poisson and Laplace problems with sharp interfaces and complex geometries. Results show that, compared with several other Physics Informed Neural Networks-based formulations, the INI-VPINN consistently achieves higher accuracy, smoother and faster convergence. The proposed framework provides a general approach for solving multimaterial problems with complex geometries and mixed Neumann-Dirichlet boundary conditions using neural networks. The implementation is publicly available in a GitHub repository.

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.

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.