Learned adaptive weighting for physics-informed Fourier neural operators: Solving discontinuous PDEs with limited data

Physics-informed Fourier neural operators, while promising for limited-data scenarios, struggle to solve partial differential equations with discontinuous solutions. Large, localized residuals at shocks destabilize training, especially when sparse data provides insufficient guidance. Static loss weighting lacks the flexibility to handle dynamically evolving shocks across a solution family. Adaptive weighting methods from physics-informed neural networks offer flexibility but are challenging to apply to operator learning, as they often target single-instance problems and exhibit instability. To address these challenges, we propose the Physics-Informed Fourier Neural Operator with Learned Adaptive Weighting (PIFNO-LAW), a novel dual-operator framework. An auxiliary Fourier neural operator takes the local solution and its derivatives as input to learn a dynamic weight field for the physics loss. This learning is guided by a spatially adaptive dual-penalty regularization, which maintains a relative penalty in smooth regions while attenuating large residuals near shocks. This decoupled design dampens problematic gradients without converging to trivial solutions or suffering from spatial optimization conflicts. On benchmarks including the linear advection, inviscid Burgers’, and Euler equations, PIFNO-LAW captures sharp shock fronts with lower error than data-driven, unweighted, and static heuristic models in most tested limited-data settings.