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:2609. 18901v1 Announce Type: cross Abstract: In physics-informed machine learning, a target function $u^*$ is learned from noisy value observations $y_i=u^*(x_i)+ \varepsilon_i$, together with differential information, given either by noisy observations $d_j=(Du^*)(z_j)+\xi_j$ or by a known physical constraint $Du^*=v$.
By Luc Brogat-Motte, Joachim Bona-Pellissier, Giacomo Meanti, Lorenzo Rosasco
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
The paper presents a finite‑sample learning‑to‑control framework for geometrically supervised latent models of nonlinear deterministic systems. It introduces an encoder‑only local–global metric hinge that ensures directional resolution and state discrimination, and proves that any approximate empirical minimizer is pointwise co‑Lipschitz and uniformly approximately semiconjugate to the true dynamics under regularity assumptions. The results provide explicit bounds on approximation, sampling, and optimization errors, and demonstrate through controlled experiments that restoring metric resolution improves control performance.
By Alain Bensoussan, Minh-Nhat Phung, Minh-Binh Tran
arXiv:2609. 03129v1 Announce Type: cross Abstract: Several classical machine-learning methods, such as KRRs and SVRs, are both computationally and analytically tractable since their estimators either admit closed-form expressions or are obtained by minimizing convex training objectives; neither feature is generally available for deep neural networks.
By Ruiyang Hong, Hrad Ghoukasian, Anastasis Kratsios
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
The paper develops a rigorous framework for computing the Normalized Maximum Likelihood (NML) codelength for regular path‑differentiable Lipschitz (PDL) estimators, which include non‑smooth models such as Lasso and Sparse SVMs. By leveraging geometric measure theory and a novel Propose‑and‑Project Metropolis‑Hastings sampler, the authors provide a method to exactly evaluate the stochastic complexity for these non‑smooth estimators and demonstrate its scalability to high‑dimensional settings. The study shows that the exact NML criterion can match cross‑validation performance while being more data‑efficient, offering a theoretically grounded alternative for model selection in modern machine learning.
By Trenton Lau, Gary P. T. Choi
The paper investigates how many linear samples are needed to learn Lipschitz operators under Gaussian measures. It establishes both lower and upper bounds on the Hermite polynomial approximation error and shows that the minimal worst‑case error cannot converge algebraically with the number of samples. However, if the covariance operator of the Gaussian measure decays rapidly, convergence rates arbitrarily close to any algebraic rate can be achieved.
By Ben Adcock, Michael Griebel, Gregor Maier
arXiv:2609.06546v1 Announce Type: cross
Abstract: Although machine learning can be used to predict the evolution of physical systems from data, a formulation that learns only the system state at each...
By Sai Siddharth, Maniarasu Ravi
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