Reduced Order Modeling for Tsunami Forecasting with Bayesian Hierarchical Pooling

TL;DR

This work tackles the challenge of real-time, uncertainty-aware tsunami forecasting by developing randPROM, a reduced-order surrogate that blends neural Galerkin‑projection ROMs with Bayesian hierarchical pooling. The pipeline couples shallow water equation dynamics, POD mode reduction, GP-ROM coefficient evolution, neural ODE–based stabilization (nGP), and probabilistic calibration to learn distributions over initial coefficients from sparse sensor data, yielding posterior predictive distributions for arrival times and wave heights. Key contributions include the neural Galerkin extension of GP-ROMs with quadratic coefficient optimization, the randPROM integration with hierarchical pooling, and demonstrations on both synthetic Fiji perturbations and the real 2011 Tohoku event, illustrating improved efficiency and uncertainty quantification with limited simulations. The approach enables near real‑time, physically grounded tsunami risk assessment by leveraging offline neighborhood training and sparse observations to interpolate to unseen scenarios, offering a pathway toward rapid, credible forecasts in operational settings.

Abstract

Reduced order models (ROM) can represent spatiotemporal processes in significantly fewer dimensions and can be solved many orders faster than their governing partial differential equations (PDEs). For example, using a proper orthogonal decomposition produces a ROM that is a small linear combination of fixed features and weights, but that is constrained to the given process it models. In this work, we explore a new type of ROM that is not constrained to fixed weights, based on neural Galerkin-Projections, which is an initial value problem that encodes the physics of the governing PDEs, calibrated via neural networks to accurately model the trajectory of these weights. Then using a statistical hierarchical pooling technique to learn a distribution on the initial values of the temporal weights, we can create new, statistically interpretable and physically justified weights that are generalized to many similar problems. When recombined with the spatial features, we form a complete physics surrogate, called a randPROM, for generating simulations that are consistent in distribution to a neighborhood of initial conditions close to those used to construct the ROM. We apply the randPROM technique to the study of tsunamis, which are unpredictable, catastrophic, and highly-detailed non-linear problems, modeling both a synthetic case of tsunamis near Fiji and the real-world Tohoku 2011 disaster. We demonstrate that randPROMs may enable us to significantly reduce the number of simulations needed to generate a statistically calibrated and physically defensible prediction model for arrival time and height of tsunami waves.

Figures (17)

• Figure 1: Frames of a tsunami simulation and sample sensor readings. Increments of every three hours of a tsunami simulation, emanating just south of Fiji with a source at 174$^\circ$E, 21$^\circ$S, at an approximated magnitude of 7. By 10-12 hours, the primary tsunami wave will have reached every Pacific coast, driving waves often exceeding 1 m in height. In this example, artificial buoy sensors (shown as colored dots) record the passing tsunami wave height.
• Figure 2: Modes and coefficients of the Fiji tsunami simulation. Selected spatial modes $\bm{\phi}_i$ of the reference simulation shown in Figure \ref{['fig:frames']} (we show only the height component of these modal vectors.) In the second row, the corresponding temporal coefficients show the generally observed quasi-periodic time evolution of these modes.
• Figure 3: Relative informational content of the modes. By selecting 16 modes we capture roughly 80% of the information. After 24 modes, about 95% is captured.
• Figure 4: Creating new activations. The first column shows the GP-ROM with its inherent instability, the second shows significantly improved stability after the nGP calibration, and the third shows the randPROM, employing 8 posterior samples drawn from a learned distribution. The first three rows show activations $a_2$, $a_4$, and $a_{16}$ (in color), respectively, against the true SVD activations (in gray). The final row shows an example reconstruction of a sensor, a key result.
• Figure 5: Training the nGP. The input to the neural GP network are the POD coefficients. Each training iteration first guesses a correction to the linear and quadratic coefficient matrices $L$ and $Q$, respectively, then solves corresponding ODE. The output of the ODE solver are predicted coefficients, which are then evaluated via a mean-square error metric in the loss function. An additional regularizing term ensures that the coefficients are stable at longer time scales.