Machine learning methods are widely used in the natural sciences to model and predict physical systems from observation data. Yet, they are often used as poorly understood "black boxes," disregarding existing mathematical structure and invariants of the problem. Recently, the proposal of Hamiltonian Neural Networks (HNNs) took a first step towards a unified "gray box" approach, using physical insight to improve performance for Hamiltonian systems. In this paper, we explore a significantly improved training method for HNNs, exploiting the symplectic structure of Hamiltonian systems with a different loss function. This frees the loss from an artificial lower bound. We mathematically guarantee the existence of an exact Hamiltonian function which the HNN can learn. This allows us to prove and numerically analyze the errors made by HNNs which, in turn, renders them fully explainable. Finally, we present a novel post-training correction to obtain the true Hamiltonian only from discretized observation data, up to an arbitrary order.

Hilbert's tenth problem, posed in 1900 by David Hilbert, asks for a general algorithm to determine the solvability of any given Diophantine equation. In 1970, Yuri Matiyasevich proved the DPRM theorem which implies such an algorithm cannot exist. This paper will outline our attempt to formally state the DPRM theorem and verify Matiyasevich's proof using the proof assistant Isabelle/HOL.

No abstract

How difficult are interactive theorem provers to use? We respond by reviewing the formalization of Hilbert's tenth problem in Isabelle/HOL carried out by an undergraduate research group at Jacobs University Bremen. We argue that, as demonstrated by our example, proof assistants are feasible for beginners to formalize mathematics. With the aim to make the field more accessible, we also survey hurdles that arise when learning an interactive theorem prover. Broadly, we advocate for an increased adoption of interactive theorem provers in mathematical research and curricula.

ResumeAla difference des groupes ouest-africaines evoquees dans la litterature sur l'economie domestique, les Kpelle (femmes et hommes) mettent l'accent sur la communaute des biens entre epoux et sur le fait que les maris, consideres comme les pourvoyeurs essentiels dans le menage, controlent les revenus communs. Get article declare la necessite de ne passe con tenter de l'explication fonctionnelle pour justifier la division des responsabilites budgetaires entre les sexes. Il nous demande de considerer plutot le role des ideologies sexuelles et les systemes de production pour expliquer la nouvelle distribution des responsabilites et la valeur accordee aux contributions economiques et monetaires des hommes et des femmes. La division des spheres de production entre hommes et femmes dans la societe Kpelle ainsi qu'une ideologie generalisee basee sur la superiorite masculine sont les deux elements determinants les plus importants des relations economiques entre epoux et de l'independance economique et financiere des femmes. En outre, les ideologies soulignant les responsabilites economiques et financieres des hommes et des femmes au sein du menage mettent en contraste les veritables contributions des hommes et des femmes ainsi que leur appreciation.