AI 中文总结
研究提出基于非标准有限差分离散化的保结构神经ODE框架，其动力学由非负速率参数化，能无条件保正性与一阶一致性，扩展可保守恒律，SIR模型实验显示该方法在生成轨迹、抗粗离散化及保定性结构上优于传统NODEs。
AI 中文摘要
尽管神经常微分方程（NODEs）是学习连续时间动力学的强大框架，但通常不保诸如正性等基本定性性质。我们提出基于非标准有限差分（NSFD）离散化的保结构神经ODE框架。学习到的动力学由非负产生和破坏率参数化，产生可明确、可微更新并无缝集成到标准自动微分管道。证明该方案对任意时间步长无条件保正性且保持一阶一致性。基于Patankar型离散化的扩展能精确保守恒律。SIR疫情模型数值实验表明，该方法生成有物理意义轨迹，粗离散化下稳健，在保学习动力学定性结构方面优于传统NODEs。
英文摘要
Although neural ordinary differential equations (NODEs) are a powerful framework for learning continuous-time dynamics, they generally do not preserve essential qualitative properties, such as positivity. We propose a structure-preserving Neural ODE framework based on nonstandard finite difference (NSFD) discretization. The learned dynamics are parameterized by nonnegative production and destruction rates, yielding an explicit, differentiable update that integrates seamlessly into standard automatic differentiation pipelines. We prove that the resulting scheme unconditionally preserves positivity for arbitrary time-step sizes while retaining first-order consistency. We outline an extension based on Patankar-type discretizations that preserves conservation laws exactly. Numerical experiments on an SIR epidemic model show that our approach generates physically meaningful trajectories, remains robust under coarse discretizations, and outperforms conventional NODEs in preserving the qualitative structure of the learned dynamics.
Comments8 pages, 2 tables, 1 figure
