Learning Effective Soliton Dynamics from Scattering Data

The inverse scattering transform (IST) provides the standard theoretical framework for deriving soliton dynamics. Traditionally, such derivations have been of an analytical, rather than data-driven, nature. In this paper, we combine the conceptual framework of the IST with weak-form system identification methods to discover effective soliton dynamics directly from observed scattering data, without assuming prior knowledge of the scattering equations. Our method avoids parameterizing solitary waves via ad hoc curve-fitting by working in the scattering domain, yielding interpretable low-dimensional models that remain valid in perturbed and near-integrable regimes. We demonstrate the performance of the proposed approach on synthetic and experimental data governed by shallow-water equations of Korteweg--de Vries-type and recover models that are consistent with canonical IST theory.

Weak Form Learning for Mean-Field Partial Differential Equations: an Application to Insect Movement

Insect species subject to infection, predation, and anisotropic environmental conditions may exhibit preferential movement patterns. Given the innate stochasticity of exogenous factors driving these patterns over short timescales, individual insect trajectories typically obey overdamped stochastic dynamics. In practice, data-driven modeling approaches designed to learn the underlying Fokker-Planck equations from observed insect distributions serve as ideal tools for understanding and predicting such behavior. Understanding dispersal dynamics of crop and silvicultural pests can lead to a better forecasting of outbreak intensity and location, which can result in better pest management. In this work, we extend weak-form equation learning techniques, coupled with kernel density estimation, to learn effective models for lepidopteran larval population movement from highly sparse experimental data. Galerkin methods such as the Weak form Sparse Identification of Nonlinear Dynamics (WSINDy) algorithm have recently proven useful for learning governing equations in several scientific contexts. We demonstrate the utility of the method on a sparse dataset of position measurements of fall armyworms (Spodoptera frugiperda) obtained in simulated agricultural conditions with varied plant resources and infection status.

Learning Weather Models from Data with WSINDy

The multiscale and turbulent nature of Earth's atmosphere has historically rendered accurate weather modeling a hard problem. Recently, there has been an explosion of interest surrounding data-driven approaches to weather modeling, which in many cases show improved forecasting accuracy and computational efficiency when compared to traditional methods. However, many of the current data-driven approaches employ highly parameterized neural networks, often resulting in uninterpretable models and limited gains in scientific understanding. In this work, we address the interpretability problem by explicitly discovering partial differential equations governing various weather phenomena, identifying symbolic mathematical models with direct physical interpretations. The purpose of this paper is to demonstrate that, in particular, the Weak form Sparse Identification of Nonlinear Dynamics (WSINDy) algorithm can learn effective weather models from both simulated and assimilated data. Our approach adapts the standard WSINDy algorithm to work with high-dimensional fluid data of arbitrary spatial dimension. Moreover, we develop an approach for handling terms that are not integrable-by-parts, such as advection operators.