A Neural Operator-Based Emulator for Regional Shallow Water Dynamics

TL;DR

The paper tackles real-time regional coastal forecasting by learning the time-evolution operator of the shallow-water equations under variable IC, BC, and a domain parameter. It introduces MITONet, a neural operator framework that fuses latent-space autoencoders, multi-input branches, and temporal bundling to produce autoregressive forecasts with long horizon accuracy. Empirical results on the Shinnecock Inlet ADCIRC dataset show MITONet achieves 300x–600x speed-ups over traditional solvers while maintaining high accuracy (high ACC, low RMSE) over a 55-day rollout and across unseen initial conditions and parameter values. The work demonstrates substantial potential for fast, flexible coastal-hydrodynamics emulation and points to future work in extreme events and uncertainty quantification.

Abstract

Coastal regions are particularly vulnerable to the impacts of rising sea levels and extreme weather events. Accurate real-time forecasting of hydrodynamic processes in these areas is essential for infrastructure planning and climate adaptation. In this study, we present the Multiple-Input Temporal Operator Network (MITONet), a novel autoregressive neural emulator that employs dimensionality reduction to efficiently approximate high-dimensional numerical solvers for complex, nonlinear problems that are governed by time-dependent, parameterized partial differential equations. Although MITONet is applicable to a wide range of problems, we showcase its capabilities by forecasting regional tide-driven dynamics described by the two-dimensional shallow-water equations, while incorporating initial conditions, boundary conditions, and a varying domain parameter. We demonstrate MITONet's performance in a real-world application, highlighting its ability to make accurate predictions by extrapolating both in time and parametric space.

Figures (9)

• Figure 1: Schematic representation of the MITONet framework. First, (a) an autoencoder maps the high-dimensional solution snapshots to a low-dimensional latent space. Then, (b) the MITONet is provided with relevant input functions, such as domain parameters, initial conditions or their latent representation at a given time $t$, and boundary conditions for time $t+\beta \Delta t$ to predict the latent representation of the solution snapshot at time $t + \beta \Delta t$, where $\beta = 1, \ldots, \tau$ and $\tau$ is a chosen look-forward window. This procedure is repeated autoregressively to generate a time series of snapshots in the latent space, which are then passed through the decoder network to recover the predictions in the high-dimensional space. To train the MITONet model, a temporal bundling approach is adopted to split the time series of training snapshots into sub-trajectories consisting of $\tau$ outputs for each input snapshot. For $\tau$=$5$, panel (c) illustrates the different sub-trajectories using different colors.
• Figure 2: Mesh and bathymetry for the Shinnecock Bay.
• Figure 3: The model predictions are evaluated at three sensor locations that exhibit distinct flow dynamics. Panel (a) shows the locations of the sensor with different colors: sensor 1 (blue) - inner bay, sensor 2 (yellow) - inlet, and sensor 3 (green) - bay. Panel (b) shows a close-up of the sensor locations in the inlet and inner bay area.
• Figure 4: Box-and-whiskers plot of the $RMSE$ for all $r$ values between days 5 and 60 for $H$ (Row 1), $U$ (Row 2), and $V$ (Row 3). The range of values for each hydrodynamic variable across the spatiotemporal domain is presented in the legend of each subplot.
• Figure 5: Time series of the $RMSE$ of MITONet predictions for all test $r$ values between days 5 and 60 for $H$ (Row 1), $U$ (Row 2), and $V$ (Row 3). The dashed lines at days 15 and 30 represent the endpoints of the time series used in the training data.