Bridging ocean wave physics and deep learning: Physics-informed neural operators for nonlinear wavefield reconstruction in real-time

Accurate real-time reconstruction of phase-resolved ocean wave fields remains a critical yet largely unsolved problem, primarily due to the absence of practical data assimilation methods for obtaining initial conditions from sparse or indirect wave measurements. While recent advances in supervised deep learning have shown potential for this purpose, they require large labeled datasets of ground truth wave data, which are infeasible to obtain in real-world scenarios. To overcome this limitation, we propose a physics-informed neural operator (PINO) framework for reconstructing spatially and temporally phase-resolved, nonlinear ocean wave fields from sparse measurements, without the need for ground truth data during training. This is achieved by embedding residuals of the free surface boundary conditions of ocean gravity waves into the loss function, constraining the solution space in a soft manner. In the current implementation, the framework is demonstrated for long-crested, unidirectional wave surfaces, where the wave propagation direction is aligned with the radar scanning direction. Within this setting, we validate our approach using highly realistic synthetic wave measurements by demonstrating the accurate reconstruction of nonlinear wave fields from both buoy time series and radar snapshots. Our results indicate that PINOs enable accurate, real-time reconstruction and generalize robustly across a wide range of wave conditions, thereby paving the way for future extensions of this framework toward multidirectional sea states and thus operational wave reconstruction in realistic marine environments.