Geometry-Agnostic Data-Driven Discovery of Extreme Scenarios
Authors: Angeline Hu, Ava K. Tse, Kaitlyn S. Yasumura, Luciana Sarabia, Malcolm Yeung, Kavin K. Prasanna, Kai Wong, William Hammond, Ioannis G. Keverkidis, Shengkai Li, Trung V. Phan
Organizations: Claremont McKenna College, Claremont, CA 91711, USA. · Scripps College, Claremont, CA 91711, USA. · Pitzer College, Claremont, CA 91711, USA. · Department of Chemical and Biomolecular Engineering, John Hopkins University, Baltimore, MD 21218, USA · Department of Physics, Princeton University, Princeton, NJ 08544, USA · Department of Natural Sciences, Scripps and Pitzer Colleges, Claremont Colleges Consortium, Claremont, CA 91711, USA
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.
Figures & tables
Figure 1: Overview of the data-driven two-stage neural framework for extremescenario discovery. Observational data from simple or experimentally accessible settings are first used to train a dynamics-learning neural network that infers the underlying governing dynamics. The learned dynamics are then passed to a scenario-searching neural network, which performs a physics-informed search for dynamically admissible coherent structures. This second stage can identify plausible extreme scenarios that need not appear in the original observations, including localized events in higher-dimensional spacetime or in different geometric settings such as curved spaces.
Figure 2: Full two-stage pipeline for learning dynamics and searching for a homoclinic pulse. The comparison between the Duffing (A1,A2) and simple harmonic systems (B1,B2) provides a positive and negative test of the framework: the search should recover a pulse when the learned dynamics support one, and fail when they do not. (A1,B1) The representative trajectories of data used to train the dynamics-learning neural network in the first stage. (A2,B2) The result of the scenario-searching neural network in the second stage, with pinning conditions explicitly shown. For the Duffing system (A2) , the search recovers a localized homoclinic pulse in good agreement with the analytical solution. For the simple harmonic system (B2) , the same search constraints do not produce a dynamically admissible pulse.
Figure 3: Geometry-agnostic neural-search result for stationary soliton profiles in the nonlinear Schrödinger system. Those solutions are known as the Townes solitons. (A) The scenario-searching neural network recovers the one-dimensional stationary soliton profile, equivalent to the Duffing homoclinic pulse with the coordinate interpreted as space. (B1) Cross sections of the two-dimensional neural-found profile along the x~ and y~ . (B2) Surface plots of the neural-found normalized Townes soliton and the high-precision numerical reference. (C) Three orthogonal cross sections of the three-dimensional soliton profile in the positive octant.
Figure 4: Geometry-agnostic transfer from Lorentzian Liouville dynamics to a Euclidean instanton search. The geometry-agnostic formation of the dynamics learned in real spacetime can be used, after Wick rotation, to identify a localized instanton in imaginary time. (A) Lorentzian and Euclidean proper-length nullsets. (B) Representative field configurations used to train the dynamics-learning neural network on a (1+1) -dimensional Lorentzian quasi-manifold. (C1) Cross sections of the neural-found instanton along the x~ and t~E axes agree with the analytical Liouville instanton profile. (C2) Surface plots of the neural-found normalized instanton and the analytical solution show good agreement over the Euclidean domain. (D) AdS 2 and Euclidean AdS 2 proper-length nullsets. (E) Surface plots of the neural-found normalized instanton and the analytical solution also show reasonable agreement over the Euclidean AdS 2 domain.
Appendix figures & tables2 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 5: Neural network architectures used in this work. All architectures are standard feedforward MLPs.
Search
Hidden widths
Collocation points
Epochs
ηθ
ηs
Duffing / SHO
32
103
5×104
10−4
10−4
SNL, 1D
32
103
5×104
10−4
10−4
SNL, 2D
32,32
2×104
5×103
10−3
10−2
SNL, 3D
32,32,32
2×104
5×103
10−2
10−1
Liouville, Euclidean flat
32,32
2×104
1.5×104
10−3
10−2
Liouville, EAdS 2
32,32,32
2×104
3×103
10−3
10−3
Appendix
Table 1: Scenario-search settings. Hidden widths exclude input and output layers. ηθ and ηs denote learning rates for network and rescaling parameters, respectively.
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