让神经微分方程自动遵守物理约束，提升模型稳定性与泛化能力。
Projected Neural Differential Equations for Learning Constrained Dynamics
- 通过投影向量场到约束流形的切空间，强制模型满足已知物理规律。
- 在混沌系统和电力网络模型中表现优于现有方法，且超参数更少。
- 适合需要高精度与可靠性的复杂动态系统建模，如能源、航天领域。
神经微分方程为从数据中学习动力学提供了强大工具，但通常不施加应被模型遵守的已知约束。众所周知，在代理模型中施加约束可提升其泛化能力和数值稳定性。本文提出投影神经微分方程（PNDEs），一种基于将学习到的向量场投影到约束流形切空间的新方法，以约束神经微分方程。在多个具有挑战性的例子中测试，包括混沌动力系统和最先进的电力网络模型，PNDEs 在性能上超越现有方法，同时所需超参数更少。该方法在增强受限动力系统建模方面展现出显著潜力，尤其适用于对准确性和可靠性要求极高的复杂领域。
原文摘要 · Abstract (English)
Neural differential equations offer a powerful approach for learning dynamics from data. However, they do not impose known constraints that should be obeyed by the learned model. It is well-known that enforcing constraints in surrogate models can enhance their generalizability and numerical stability. In this paper, we introduce projected neural differential equations (PNDEs), a new method for constraining neural differential equations based on projection of the learned vector field to the tangent space of the constraint manifold. In tests on several challenging examples, including chaotic dynamical systems and state-of-the-art power grid models, PNDEs outperform existing methods while requiring fewer hyperparameters. The proposed approach demonstrates significant potential for enhancing the modeling of constrained dynamical systems, particularly in complex domains where accuracy and reliability are essential.
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