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Geometry-Agnostic Data-Driven Discovery of Extreme Scenarios

This paper introduces a two-stage, geometry-agnostic neural framework that infers dynamical constraints directly from observational data to discover and transfer rare, high-impact coherent structures—such as solitons and instantons—across varying dimensions and geometric manifolds without requiring prior knowledge of the governing equations.

Original 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

Published 2026-10-21

📖 5 min read🧠 Deep dive

CC BY 4.0

Original 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

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Nature often surprises us with moments of extreme intensity. In the ocean, a single rogue wave can tower over a fleet of ships; in the atmosphere, a sudden storm can upend a climate model; in a financial market, a crash can erase years of growth in a single day. These events are rare, often localized, and difficult to predict because they do not follow the average behavior of the system. They are outliers that, by their very nature, rarely appear in the data we collect. If a scientist only watches a system under normal conditions, they might never see the moment when everything goes wrong. This creates a fundamental problem: how can we understand or prepare for events that our observations have never shown us?

A team of researchers has developed a new way to answer this question by teaching computers to imagine these missing scenarios. Instead of trying to write down the exact laws of physics that govern a system—a task that is often impossible for complex, real-world phenomena—they trained a computer to learn those laws directly from ordinary observations. Once the computer understood the basic rules of the system, they asked it to search for a specific kind of rare event: a sudden, intense burst of energy that fits within those rules but has never been seen before. The result is a method that can predict extreme events in complex systems, even in spaces or dimensions where no human has ever been able to take a measurement.

The researchers tested their approach using a series of increasingly complex examples, starting with a simple system that behaves like a swinging pendulum with a twist. In this first test, they fed the computer data from a standard, predictable motion. The computer learned the underlying rules of the motion without being told what the rules were. Then, the researchers asked the computer to find a specific, rare pattern within those rules: a pulse that starts from nothing, spikes to a high peak, and then returns to nothing. This is known as a homoclinic pulse, a type of extreme event that appears in many physical systems. The computer successfully found this pulse, matching the known mathematical solution with high precision. Crucially, when the researchers tried the same process on a system that simply could not support such a pulse, the computer correctly failed to find one, proving that it was not just guessing but truly understanding the limits of the system.

The true power of this method, however, lies in its ability to step beyond the geometry of the data it was given. In the physical world, the laws of nature often look the same regardless of how you measure them or how many directions you are looking in. The researchers built their computer model to respect this fact, teaching it to learn rules that are independent of any specific grid, coordinate system, or number of dimensions. To demonstrate this, they took the rules learned from a one-dimensional line and asked the computer to find a similar extreme event in a two-dimensional space, like a flat sheet. The computer successfully identified a stable, self-focusing wave pattern, known as a Townes soliton, which exists in two dimensions but was never seen in the one-dimensional data used to train the system. This showed that the computer had learned the fundamental essence of the physics, allowing it to project that knowledge into a higher-dimensional world.

In their most ambitious test, the researchers applied this technique to a system that involves the fabric of spacetime itself. They trained the computer on data from a system with one dimension of space and one dimension of time, a setting that resembles our everyday experience of moving forward in time. They then asked the computer to find a solution in a completely different geometric setting: a system with two dimensions of space and no time at all. This is a mathematical transformation often used in theoretical physics to study phenomena that are impossible to observe directly. The computer successfully transferred the rules it learned from the time-based system to the space-only system and identified a specific, rare structure known as an instanton. This structure is a solution that exists in a realm that is experimentally inaccessible, yet the computer found it by relying solely on the geometric consistency of the laws it had learned.

This work suggests that we do not need to observe an extreme event to understand it or predict its existence. By teaching machines to learn the deep, invariant rules of a system from ordinary data, we can use those machines to explore the boundaries of what is possible. The method does not require prior knowledge of the governing equations, nor does it need to see the rare event to know it is there. It simply learns the language of the system and then asks what sentences can be written that are grammatically correct but have never been spoken. This capability opens a new path for risk analysis and forecasting, allowing scientists to hunt for the most consequential scenarios in fields ranging from fluid dynamics to financial markets, even when those scenarios live in dimensions or geometries that are beyond our direct reach.

Technical Summary: Geometry-Agnostic Data-Driven Discovery of Extreme Scenarios

Problem Statement
Extreme events—rare, localized departures from typical behavior in nonlinear systems (e.g., rogue waves, market crashes, epidemic spikes)—are often absent from naturally sampled observational data, particularly in high-dimensional systems governed by ordinary differential equations (ODEs) or partial differential equations (PDEs). Traditional data-driven approaches struggle to predict these scenarios because they rely on the presence of such events in the training set. Furthermore, standard machine learning models for dynamical systems are often tied to specific coordinate systems, grid structures, or dimensionalities, limiting their ability to transfer learned dynamics to new geometric settings or higher dimensions where extreme events might occur.

Methodology
The authors propose a two-stage, data-driven neural framework designed to discover dynamically admissible extreme scenarios without prior knowledge of the governing equations.

Stage 1: Dynamics Learning (Geometry-Agnostic Inference)
- Objective: Infer effective dynamical constraints directly from observational data.
- Approach: Instead of learning derivatives specific to a coordinate system, the model learns from invariant geometric quantities (e.g., field values, gradient inner products, and Laplacians) defined on Riemann-Cartan manifolds (including quasi-manifolds with torsion).
- Representation: The dynamics are represented in a "geometry-agnostic" form. For scalar fields, the model learns a mapping from scalar invariants (0th-order features like field values and 2nd-order features like $\langle \nabla u, \nabla u \rangle$ and $\nabla^2 u$ ) to the evolution of the field. This formulation is independent of fixed grids, coordinate systems, dimensionality, and intrinsic curvature.
- Architecture: A multi-layer perceptron (MLP) with tanh activation functions is used to approximate the dynamical law.
Stage 2: Scenario Search (Physics-Informed Neural Search)
- Objective: Uncover rare, high-impact coherent structures (extreme scenarios) that may not have appeared in the original observations.
- Approach: A second neural network acts as a scenario searcher, representing a candidate solution profile. It is optimized to satisfy the learned dynamical constraints from Stage 1, along with specific boundary and localization conditions (e.g., pinning conditions for peak amplitude and shoulder values).
- Mechanism: The search network outputs a normalized profile. The system simultaneously optimizes the network parameters and physical scaling factors (amplitude and time/space scales) to minimize a loss function composed of the learned dynamical residual and structural constraints. No labeled examples of the target extreme profile are provided; the network discovers the shape and scale purely from the dynamics.

Key Contributions and Results
The paper demonstrates the framework through three examples of increasing complexity:

Duffing Homoclinic Pulse (1D ODE):
- Starting with synthetic trajectory data from a Duffing oscillator, the first stage learns the governing ODE. The second stage successfully recovers a localized homoclinic pulse solution with high precision (relative L2 error $\approx 0.05\%$ ), matching the analytical solution.
- A control case using a simple harmonic oscillator (which lacks homoclinic pulses) demonstrates that the search correctly fails to produce a valid pulse, confirming the method relies on the underlying dynamics rather than forcing a solution.
Townes Soliton (Dimensional Lifting):
- The authors reinterpret the Duffing equation as the stationary reduction of the Nonlinear Schrödinger (NLS) equation in 1D space.
- By retraining Stage 1 with a geometry-agnostic representation, the learned dynamics are "lifted" to higher dimensions.
- The framework successfully identifies a 2D and 3D Townes soliton profile (a self-focusing stationary soliton) using only data trained on 1D configurations. This proves that the learned dynamics are not bound to the dimensionality of the training data.
Instanton Discovery (Geometric Transfer):
- Using the Liouville relativistic PDE, the authors demonstrate transfer across geometric signatures.
- Dynamics learned from data on a $(1+1)$ -dimensional Lorentzian quasi-manifold (real-time) are transferred to a $(2+0)$ -dimensional Euclidean manifold (imaginary-time, obtained via Wick rotation).
- The framework successfully identifies an instanton solution in the Euclidean setting, a structure that is experimentally inaccessible in the original Lorentzian setting.

Significance and Claims
The paper claims that its primary contribution is a formulation that allows the first stage of learning to be geometry-agnostic. This enables:

Transferability: Learned dynamics can be aggregated from different settings and applied to predict behavior in completely new geometric settings, dimensions, or coordinate systems.
Discovery of Unobserved Scenarios: The framework can identify plausible extreme scenarios (such as instantons or higher-dimensional solitons) that do not exist in the original observational data but are dynamically admissible under the learned laws.
Minimal Complexity: The approach achieves these results using simple MLPs with minimal architectural complexity, avoiding the need for complex structure-preserving architectures in the initial demonstration.

The authors emphasize that while the framework constructs candidate structures and determines their physical scales, it does not automatically calculate the probability of occurrence or statistical significance of these events; those remain subjects for future statistical analysis. The method is presented as a tool for risk analysis, nonlinear forecasting, and anomaly detection by generating plausible extreme scenarios for further investigation.

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