让神经微分方程精确控制特定点的零速度，同时保持强大表达能力。
Exact Fixed-Point Constraints in Neural-ODEs with Provable Universality
- 通过显式构造使指定点速度为零，约束训练过程。
- 在任意局部速度约束下证明模型具有通用逼近能力。
- 适用于需精确固定点的物理系统建模，如力学与动力学模拟。
我们提出一种技术，使神经微分方程（Neural-ODE）能够以预先设定的不动点来近似任意速度场。具体而言，该方法可明确地在多维空间中指定有限个点，使这些点处的速度场严格等于零。在此框架下，基于梯度的训练被严格约束于预设的假设类中，同时不削弱神经微分方程的表达能力。我们严格证明了在任意局部速度场约束条件下，神经微分方程仍具备通用逼近性，并提供了一种计算上简便的不动点施加方式。该方法在两个典型的物理模型上进行了验证。
原文摘要 · Abstract (English)
We introduce a technique that enables Neural-ODEs to approximate arbitrary velocity fields with a priori planted fixed-points. Specifically, a recipe is given to explicitly accommodate for a finite collection of points in the reference multi-dimensional space of the Neural-ODE where the velocity field is exactly equal to zero. In this way, the gradient-based training is rigorously constrained inside the prescribed hypothesis class while leaving the expressive power of the Neural-ODE unaltered. We rigorously prove the universality of the Neural-ODE under any local constraints in the velocity field and give a computationally convenient way of imposing the fixed points. Our method is then tested on two paradigmatic physical models.
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