Data assimilation and parameter identification for water waves using the nonlinear Schrödinger equation and physics-informed neural networks

TL;DR

This work demonstrates a NLSE-constrained PINN framework for reconstructing deep-water wave envelopes from sparse boundary measurements, enabling between-gap surface reconstruction via PDE residuals. It shows that fixed NLSE coefficients can yield feasible reconstructions for medium-to-narrow-band waves, while treating the coefficients as trainable variables improves accuracy by approximately 8.8% on average. The study highlights the potential and limitations of NLSE-based PINNs for data assimilation in wave tanks and points to directions toward fully nonlinear models and real-data applications. Overall, the approach offers a physics-guided pathway to reduce measurement density and enhance deterministic wave-field predictions.

Abstract

The measurement of deep water gravity wave elevations using in-situ devices, such as wave gauges, typically yields spatially sparse data. This sparsity arises from the deployment of a limited number of gauges due to their installation effort and high operational costs. The reconstruction of the spatio-temporal extent of surface elevation poses an ill-posed data assimilation problem, challenging to solve with conventional numerical techniques. To address this issue, we propose the application of a physics-informed neural network (PINN), aiming to reconstruct physically consistent wave fields between two designated measurement locations several meters apart. Our method ensures this physical consistency by integrating residuals of the hydrodynamic nonlinear Schrödinger equation (NLSE) into the PINN's loss function. Using synthetic wave elevation time series from distinct locations within a wave tank, we initially achieve successful reconstruction quality by employing constant, predetermined NLSE coefficients. However, the reconstruction quality is further improved by introducing NLSE coefficients as additional identifiable variables during PINN training. The results not only showcase a technically relevant application of the PINN method but also represent a pioneering step towards improving the initialization of deterministic wave prediction methods.

Figures (11)

• Figure 1: Measurement setup and corresponding data structure: Wave gauges inside the wave tank acquire the wave elevation at four discrete points in space $x_\mathrm{g}=\{3, 4, 5, 6\} \, \mathrm{m}$ (a), causing a sparse spatio-temporal data structure for each sample (b).
• Figure 2: JONSWAP spectra for different peak enhancement factors $\gamma$ used to initialize the HOSM wave simulations. Higher values of $\gamma$ result in narrower spectra for the generated irregular waves.
• Figure 3: Example of an irregular carrier wave $\eta(x,t)$ measured at one location inside the wave tank. The real and imaginary part of its corresponding complex envelope $A(x,t) = U(x,t) + i V(x,t)$ are visualized, along with its absolute value $|A(x,t)|$.
• Figure 4: Schematic framework of the physics-informed neural network developed to solve the hydrodynamic nonlinear Schrödinger equation. The neural network architecture comprises two input nodes to insert points of the computational domain $(x,t)$ and two output nodes to approximate the real- and imaginary part of the complex-valued NLSE solution $\Tilde{A}(x,t)=\Tilde{U}(x,t)+i\Tilde{V}(x,t)$. Real wave measurement data $\eta_\mathrm{m}$, obtained at the domain boundaries, is transformed into envelope representations $A_\mathrm{m}$ to guide the PINN's solution towards approximating these boundary values. This is achieved by the data loss component $\mathrm{MSE}_\mathrm{data}$ (Eq. \ref{['eq:MSE_Udata']}- \ref{['eq:MSE_Vdata']}). To additionally guide the PINN solution towards ensuring physical consistency inside the entire computational domain, the PDE-loss $\mathrm{MSE}_\mathrm{res}$ incorporates NLSE residuals (Eq. \ref{['eq:MSE_Ures']}-\ref{['eq:res_V']}). The PINNs variables $\boldsymbol{\theta}$ (weights $\mathbf{W}$ and biases $\mathbf{b}$), activation function slopes $\boldsymbol{a}$ and self-adaption weights $\boldsymbol{\lambda}_{\mathrm{d}}, \boldsymbol{\mu}_{\mathrm{d}}, \boldsymbol{\lambda}_{\mathrm{r}}, \boldsymbol{\mu}_{\mathrm{r}}$ are updated iteratively in the training process to minimize the total loss $\mathcal{L}$, which is composed of the data and PDE loss component.
• Figure 5: Exemplary NLSE-PINN training loss curve. The PDE residual error components $\mathrm{MSE}_{U,\mathrm{res}}$ and $\mathrm{MSE}_{V,\mathrm{res}}$ initially exhibit a strong decrease, but slightly increase as the data error components $\mathrm{MSE}_{U,\mathrm{data}}$ and $\mathrm{MSE}_{V,\mathrm{data}}$ are reduced. After around 10,000 epochs, the PDE residual errors reach a plateau, while the data errors continue to gradually improve.