Learning governing equations from observed solution data is a fundamental challenge in scientific machine learning \cite{bruntonDiscoveringGoverningEquations2016,kovachkiNeuralOperatorLearning2023,longPDENetLearningPDEs2018,rudyDatadrivenDiscoveryPartial2017,raonicConvolutionalNeuralOperators2023}, yet the theoretical conditions under which a ground-truth ODE can be uniquely and stably identified from multiple solution observations remain largely undeveloped, and no quantitative analysis of the sample complexity of such learning tasks exists in the literature. To address this gap, we introduce the Hausdorff distance on solution sets as the natural metric for comparing differential equations, since it captures the worst-case separation between two equations over all admissible initial conditions and thus encodes the minimax structure of the identification problem.
arXiv:2606. 12050v1 Announce Type: new Abstract: Physics-informed neural networks (PINNs) combine machine learning with physical laws to solve differential equations.
By Ismail Huseynov, Arzu Ahmadova, Agamirza Bashirov
arXiv:2606. 05191v1 Announce Type: new Abstract: Data-driven equation discovery is fundamentally an inverse problem that seeks to infer the governing differential equations of a system directly from time-series measurements.
By Federico J. Gonzalez
arXiv:2606. 24999v1 Announce Type: new Abstract: High-dimensional partial differential equations (PDEs) with unknown coefficients arise widely in scientific machine learning, including continuous-time reinforcement learning, yet solving them efficiently in a data-driven way remains challenging.
By Yanwei Jia, Du Ouyang, Huy\^en Pham, Xun Yu Zhou
arXiv:2506. 20181v2 Announce Type: replace Abstract: We study operator relevance in data-driven partial differential equation (PDE) discovery.
By Ronald Katende
arXiv:2606. 09276v1 Announce Type: new Abstract: Equation discovery aims to automate the discovery of scientific models in the form of mathematical equations from data.
By Paul Kahlmeyer, Henrik Voigt, Michael Habeck, Joachim Giesen