Geometry-Agnostic Data-Driven Discovery of Extreme Scenarios

Abstract

We introduce a two-stage data-driven neural framework for discovering dynamically admissible extreme scenarios. Rather than assuming prior knowledge of the governing equations, the first stage infers the effective dynamical constraints directly from observational data. When the underlying physical system possesses suitable symmetries, the learned dynamics can be represented in a geometry-agnostic form, independent of any fixed grid, coordinate system, dimensionality, and intrinsic curvature. In the second stage, a physics-informed neural-search uses this learned dynamics to uncover rare, high-impact coherent structures that may not have appeared in the original observations. We demonstrate this framework through representative examples of increasing dimensionality and geometric complexity. Starting from a traditionally learned Duffing dynamics, we find a homoclinic pulse as a plausible localized extreme scenario. Next, since the same equation also appears in the stationary nonlinear Schrodinger system, with the coordinate interpreted as space instead of time, this pulse becomes a standing soliton. We then retrain the first stage using a geometry-agnostic neural representation, enabling the learned dynamics to be lifted to higher-dimensional space to obtain higher-dimensional solitary profile, known as Townes soliton. Finally, with a Liouville system, we demonstrate that dynamics learned from data on a (1+1)-dimensional Lorentzian quasi-manifold can be transferred to a (2+0)-dimensional Euclidean manifold to identify an instanton. Our approach can be applied to other coherent spatiotemporal structures, including kinks, rogue waves, compactons, peakons, and potentially even singularities, with applications in risk analysis, nonlinear forecasting, and anomaly detection.