Title: Entropy Structure Informed Learning for Inverse XDE Problems Show Chinese title

Title: 熵结构信息学习用于反向XDE问题

Abstract: Entropy, since its first discovery by Ludwig Boltzmann in 1877, has been widely applied in diverse disciplines, including thermodynamics, continuum mechanics, mathematical analysis, machine learning, etc. In this paper, we propose a new method for solving the inverse XDE (ODE, PDE, SDE) problems by utilizing the entropy balance equation instead of the original differential equations. This distinguishing feature constitutes a major difference between our current method and other previous classical methods (e.g. SINDy). Despite concerns about the potential information loss during the compression procedure from the original XDEs to single entropy balance equation, various examples from MM reactions, Schlogl model and chemical Lorenz equations in the form of ODEs to nonlinear porous medium equation and Fokker-Planck equation with a double-well potential in the PDE form all well confirm the accuracy, robustness and reliability of our method, as well as its comparable performance with respect to SINDy. Show Chinese abstract

Abstract: 熵自1877年由路德维希·玻尔兹曼首次发现以来，已被广泛应用于多个学科，包括热力学、连续介质力学、数学分析、机器学习等。 在本文中，我们提出了一种新的方法来解决逆XDE（ODE、PDE、SDE）问题，该方法利用熵平衡方程而不是原始微分方程。 这一显著特点构成了我们当前方法与其他以前的经典方法（例如SINDy）之间的主要差异。 尽管在从原始XDEs压缩到单一熵平衡方程的过程中可能存在信息丢失的担忧，但来自MM反应、Schlogl模型和化学洛伦兹方程的多种例子，以及以ODE形式到非线性多孔介质方程和具有双势阱的Fokker-Planck方程的PDE形式，都充分证实了我们方法的准确性、鲁棒性和可靠性，以及其与SINDy相当的性能。