Reduced Order Modeling for Tsunami Forecasting with Bayesian Hierarchical Pooling

Abstract

Reduced-order models (ROMs) can represent spatiotemporal processes in significantly fewer dimensions and can often be solved many orders of magnitude faster than their governing partial differential equations (PDEs). For example, proper orthogonal decomposition yields a ROM in which the state is represented as a low-dimensional linear combination of fixed spatial modes and time-dependent coefficients, but this representation remains constrained by the process used to construct the basis. In this work, we explore a new type of ROM that is not restricted to a single fixed coefficient trajectory. Specifically, we consider a corrected Galerkin-projection ROM, formulated as an initial value problem that encodes the physics of the governing PDEs and is calibrated through operator corrections to more accurately reproduce the coefficient dynamics. By combining this corrected reduced model with a Bayesian hierarchical pooling framework over the initial reduced coefficients, we obtain new, statistically interpretable and physically grounded coefficient trajectories that generalize across related scenarios. When recombined with the spatial modes, these trajectories define a complete probabilistic physics surrogate, called a randPROM, for generating simulations that are distributionally consistent with a neighborhood of initial conditions near those used to construct the ROM. We apply the randPROM framework to tsunami modeling, a setting involving unpredictable, catastrophic, and strongly nonlinear dynamics, using both a synthetic case study near Fiji and the real-world 2011 Tohoku tsunami. We demonstrate that randPROMs can substantially reduce the number of full simulations required while providing statistically calibrated and physically defensible predictions of tsunami wave arrival times and heights.