Physics-informed neural networks for phase-resolved data assimilation and prediction of nonlinear ocean waves

TL;DR

This work introduces a physics-informed neural network framework that couples two neural networks to solve fully nonlinear potential flow equations for phase-resolved ocean waves. By hard-constraining sparse surface measurements and using adaptive loss balancing, the method reconstructs the surface elevation η and infers the full velocity potential Φ across depth, even without direct potential measurements. Validation against linear theory and laboratory wave-flume data shows accurate reconstruction (low SSP) and physically consistent velocity fields, with successful short-term prediction within a defined assimilation region. The approach promises real-time data assimilation and deeper physical insight, including potential extensions to 2D+ t, bathymetry inversion, and broader coastal engineering applications.

Abstract

The assimilation and prediction of phase-resolved surface gravity waves are critical challenges in ocean science and engineering. Potential flow theory (PFT) has been widely employed to develop wave models and numerical techniques for wave prediction. However, traditional wave prediction methods are often limited. For example, most simplified wave models have a limited ability to capture strong wave nonlinearity, while fully nonlinear PFT solvers often fail to meet the speed requirements of engineering applications. This computational inefficiency also hinders the development of effective data assimilation techniques, which are required to reconstruct spatial wave information from sparse measurements to initialize the wave prediction. To address these challenges, we propose a novel solver method that leverages physics-informed neural networks (PINNs) that parameterize PFT solutions as neural networks. This provides a computationally inexpensive way to assimilate and predict wave data. The proposed PINN framework is validated through comparisons with analytical linear PFT solutions and experimental data collected in a laboratory wave flume. The results demonstrate that our approach accurately captures and predicts irregular, nonlinear, and dispersive wave surface dynamics. Moreover, the PINN can infer the fully nonlinear velocity potential throughout the entire fluid volume solely from surface elevation measurements, enabling the calculation of fluid velocities that are difficult to measure experimentally.

Figures (12)

• Figure 1: In the laboratory wave flume (a), lateral cameras capture the wave elevation with high resolution (b), resulting in a spatio-temporal data structure after preprocessing (c) that provides a ground truth ($\eta_\mathrm{true}$) for comparing the PINN reconstruction ($\tilde{\eta}$) between sparse buoy time series data ($\eta_\mathrm{m}$).
• Figure 2: Physics-informed neural network (PINN) framework to solve the potential flow theory (PFT) of ocean gravity waves. Two neural networks run in parallel to approximate the solutions of the velocity potential $\tilde{\Phi}(x,t,z)$ and surface elevation $\tilde{\eta}(x,t)$. Collocation points at the instantaneous free surface $(\mathcal{P}_\mathrm{s})$, seabed $(\mathcal{P}_\mathrm{b})$ and in the fluid domain $(\mathcal{P}_\mathrm{L})$ serve as inputs $\mathbf{v}$ for the NNs, are normalized to $\hat{\mathbf{v}}$ and lifted to a higher-dimensional feature space by Fourier embedding $\mu(\hat{\mathbf{v}})$. Surface measurements $\eta_\mathrm{m}$ at sparse points $(\mathcal{P}_\mathrm{d})$ hard-constrain the direct output of the elevation network $\mathcal{N}_{\tilde{\eta}}$ with a smooth distance function $R(x,t)$ and extension $M(x,t)$ of $\eta_\mathrm{m}$, both approximated from pre-trained low-capacity NNs. Parametrizing the PFT-solutions as neural networks $\tilde{\eta}(x,t)$ and $\tilde{\Phi}(x,t,z)$ enables continuous differentiation, allowing for computation of loss components using automatic differentiation (AD). These loss components include residuals of the Laplace equation $(\mathcal{L}_\mathrm{Lap})$, and boundary conditions $(\mathcal{L}_\mathrm{BC,kin}, \, \mathcal{L}_\mathrm{BC,dyn}, \, \mathcal{L}_\mathrm{BC,bot})$. The total loss $\mathcal{L}$ is minimized during training to approximate $\tilde{\eta}(x,t)$ and $\tilde{\Phi}(x,t,z)$.
• Figure 3: Training loss curve of the PFT-PINN provided with sparse measurements from a superposition of three wave components for an assimilation task. The Adam optimizer for 5,000 epochs is followed by L-BFGS optimization. ReLoBRaLo loss balancing ensures that each loss component $\mathcal{L}_i$ progresses with an approximately consistent rate relative to its initial MSE.
• Figure 4: Ground truth surface elevation $\eta_\mathrm{true}(x,t)$ (upper left) from a superposition of three wave components, used to extract synthetic measurements $\eta_\mathrm{m}(x=x_{\mathrm{wb},j},t)$. The PFT-PINN's resulting solution for elevation $\tilde{\eta}(x,t)$ (lower left) closely matches the ground truth, as further confirmed by the alignment of $\eta_\mathrm{true}$ and $\tilde{\eta}$ in the cross-sections at five spatial points (right).
• Figure 5: Ground truth potential $\Phi_\mathrm{true}(x,t, z=\eta_\mathrm{true})$ (upper left) from a superposition of three wave components. No potential measurements were provided for PFT-PINN training, its solution $\tilde{\Phi}(x,t, z=\tilde{\eta})$ (lower left) is entirely inferred from the coupling of $\tilde{\eta}$ and $\tilde{\Phi}$ in the surface BCs. Despite this, $\tilde{\Phi}$ closely matches $\Phi_\mathrm{true}$, as confirmed by the cross-section comparison (right).