提出可保证收敛的神经微分方程，提升复杂动态系统建模能力。
ControlSynth Neural ODEs: Modeling Dynamical Systems with Guaranteed Convergence
- 引入控制项捕捉多尺度动态，增强模型灵活性。
- 通过线性不等式确保高度非线性模型的收敛性。
- 在物理动力系统上表现优于主流神经网络。
神经微分方程（NODEs）是连续时间神经网络，能突破离散时间间隔限制，适用于复杂真实动态系统的建模与理解。以往工作多聚焦于简化的节点形式，而许多物理系统属于更复杂的准类，需具备高可扩展性和灵活性的通用神经微分方程。本文提出可控合成神经微分方程（CSODEs），尽管具有高度非线性特性，仍可通过可处理的线性不等式保证收敛性。在模型设计中引入额外控制项，用于学习不同尺度下动态的并行捕获，对偏微分方程描述的系统尤为有效。在多种代表性物理动力系统上，对比多个典型神经网络，结果表明在引入CSODE诱导偏置条件下，其学习与预测性能更优。
原文摘要 · Abstract (English)
Neural ODEs (NODEs) are continuous-time neural networks (NNs) that can process data without the limitation of time intervals. They have advantages in learning and understanding the evolution of complex real dynamics. Many previous works have focused on NODEs in concise forms, while numerous physical systems taking straightforward forms, in fact, belong to their more complex quasi-classes, thus appealing to a class of general NODEs with high scalability and flexibility to model those systems. This, however, may result in intricate nonlinear properties. In this paper, we introduce ControlSynth Neural ODEs (CSODEs). We show that despite their highly nonlinear nature, convergence can be guaranteed via tractable linear inequalities. In the composition of CSODEs, we introduce an extra control term for learning the potential simultaneous capture of dynamics at different scales, which could be particularly useful for partial differential equation-formulated systems. Finally, we compare several representative NNs with CSODEs on important physical dynamics under the inductive biases of CSODEs, and illustrate that CSODEs have better learning and predictive abilities in these settings.
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