通过投影保持物理约束，实现长时间动态系统精准建模
Semi-Explicit Neural DAEs: Learning Long-Horizon Dynamical Systems with Algebraic Constraints
- 将微分方程每步投影到约束流形上，显式满足代数约束
- 六组基准测试中约束误差低于10⁻¹⁰，且运行更快
- 适用于需严格遵守守恒律的物理系统建模
尽管科学机器学习（SciML）结合数据驱动与机理建模前景广阔，但现有神经微分方程（NDEs）在引入硬约束时存在显著局限：可扩展性差且数值性质不佳，难以建模具有复杂守恒律的物理系统。本文提出流形投影神经微分方程（PNODEs），通过将每一步OED解投影至约束流形，显式强制代数约束。该框架源于半显式微分代数方程（DAEs），包含一种稳健的迭代变体和一种仅需一次雅可比分解的快速近似方法。我们进一步证明，以往的松弛方法是本方法的特例。PNODEs在六组基准测试中一致优于基线，平均约束违反误差低于10⁻¹⁰，且在相同误差容忍度下运行时间更短。结果表明，约束投影为学习物理一致的长时程动态提供了一种简单有效策略。
原文摘要 · Abstract (English)
Despite the promise of scientific machine learning (SciML) in combining data-driven techniques with mechanistic modeling, existing approaches for incorporating hard constraints in neural differential equations (NDEs) face significant limitations. Scalability issues and poor numerical properties prevent these neural models from being used for modeling physical systems with complicated conservation laws. We propose Manifold-Projected Neural ODEs (PNODEs), a method that explicitly enforces algebraic constraints by projecting each ODE step onto the constraint manifold. This framework arises naturally from semi-explicit differential-algebraic equations (DAEs), and includes both a robust iterative variant and a fast approximation requiring a single Jacobian factorization. We further demonstrate that prior works on relaxation methods are special cases of our approach. PNODEs consistently outperform baselines across six benchmark problems achieving a mean constraint violation error below $10^{-10}$. Additionally, PNODEs consistently achieve lower runtime compared to other methods for a given level of error tolerance. These results show that constraint projection offers a simple strategy for learning physically consistent long-horizon dynamics.
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