同时发现微分方程中的未知参数与函数，解决传统方法解不唯一问题。
A Unified Framework for Simultaneous Parameter and Function Discovery in Differential Equations
- 构建统一框架，确保参数与函数同时求解时解的唯一性。
- 在生物系统与生态动力学中验证，结果准确且可解释。
- 适合需联合学习方程参数与非线性项的科学建模场景。
涉及微分方程的逆问题通常需要从数据中识别未知参数或函数。现有方法如物理信息神经网络（PINNs）、通用微分方程（UDEs）和通用物理信息神经网络（UPINNs）虽能有效分离参数或函数，但在同时求解时因解的不唯一性面临挑战。本文提出一个新框架，通过建立唯一解的保证条件，克服上述限制。以生物系统和生态动力学为例进行演示，结果准确且具有可解释性。该方法显著提升了机器学习在科学与工程复杂系统建模中的应用潜力。
原文摘要 · Abstract (English)
Inverse problems involving differential equations often require identifying unknown parameters or functions from data. Existing approaches, such as Physics-Informed Neural Networks (PINNs), Universal Differential Equations (UDEs) and Universal Physics-Informed Neural Networks (UPINNs), are effective at isolating either parameters or functions but can face challenges when applied simultaneously due to solution non-uniqueness. In this work, we introduce a framework that addresses these limitations by establishing conditions under which unique solutions can be guaranteed. To illustrate, we apply it to examples from biological systems and ecological dynamics, demonstrating accurate and interpretable results. Our approach significantly enhances the potential of machine learning techniques in modeling complex systems in science and engineering.
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