[论文解读] Physics-informed machine learning for reconstruction of dynamical systems with invariant measure score matching

Chen, Yongsheng, Suddhasattwa Das+2 more|arXiv (Cornell University)|Jan 19, 2026

Model Reduction and Neural Networks被引用 0

一句话总结

他们提出 PINN-IMSM，这是一个无网格框架，从未标记数据学习不变分布的分数并通过受约束的分数基 FP 方程重构高维动力系统。

ABSTRACT

In this paper, we develop a novel mesh-free framework, termed physics-informed neural networks with invariant measure score matching (PINN-IMSM), for reconstructing dynamical systems from unlabeled point-cloud data that capture the system's invariant measure. The invariant density satisfies the steady-state Fokker-Planck (FP) equation. We reformulate this equation in terms of its score function (the gradient of the log-density), which is estimated directly from data via denoising score matching, thereby bypassing explicit density estimation. This learned score is then embedded into a physics-informed neural network (PINN) to reconstruct the drift velocity field under the resulting score-based FP equation. The mesh-free nature of PINNs allows the framework to scale to higher dimensions, avoiding the curse of dimensionality inherent in mesh-based methods. To address the ill-posedness of high-dimensional inverse problems, we recast the problem as a PDE-constrained optimization that seeks the minimal-energy velocity field. Under suitable conditions, we prove that this problem admits a unique solution that depends continuously on the score function. The constrained formulation is solved using a stochastic augmented Lagrangian method. Numerical experiments on representative dynamical systems, including the Van der Pol oscillator, an active swimmer in an anharmonic trap, and the chaotic Lorenz-63 and Lorenz-96 systems, demonstrate that PINN-IMSM accurately recovers invariant measures and reconstructs faithful dynamical behavior for problems in up to five dimensions.

研究动机与目标

• 从点云数据在没有时间标签的情况下重构动力系统。
• 利用不变分布和 Fokker-Planck 方程来推断速度场。
• 开发一种在更高维度（最高到五维）仍然有效的鲁棒、可扩展方法。
• 就速度重构的良好性与稳定性提供理论保证。

提出的方法

• 将稳态 Fokker-Planck 方程重新表述为不变度量的分数函数的形式。
• 使用多尺度去噪分数匹配从未标记数据学习分数 s(x)=∇log ρ(x)。
• 将学习到的分数嵌入到物理信息神经网络中，通过分数基 FP 约束重构速度场 v(x)。
• 将速度恢复重新表述为带有 FP 约束的 PDE-约束优化问题，最小化速度能量。
• 用随机增强拉格朗日方法求解受约束的问题。
• 给出理论结果，确保速度对分数函数的唯一性和连续依赖性。

Figure 3.1: Velocity field reconstruction from trajectory data without explicit time labels. From trajectory measurements (left), we first reconstruct the score function via denoising score matching (middle). This reconstructed score is then integrated into the PINN framework to infer the velocity f

实验结果

研究问题

• RQ1 是否能从未标记的点云数据通过不变密度的分数来恢复动力系统的不变度量？
• RQ2 在保证良好性和可行性的前提下，是否可以通过基于分数的 FP 方程构建无网格 PINN 框架来重构漂移场？
• RQ3 学习到的分数如何在五维及以下维度中影响重构速度场的准确性与稳定性？
• RQ4 在给定近似分数的情况下，关于速度重构的唯一性与稳定性存在哪些理论保障？
• RQ5 数值实验（Van der Pol、主动游动者、Lorenz-63、Lorenz-96）是否验证了不变性度量和动力学的准确性？

主要发现

• PINN-IMSM 能从未标记数据准确地恢复不变度量并重构忠实的动力学。
• 基于分数的重新表述实现了网格自由的重构，适用于更高维度（最高到五维）。
• 带有 PDE 约束的优化给出唯一的、能量最小的速度场，并对分数函数连续依赖。
• 理论结果显示速度重构对分数扰动具有唯一性和 Lipschitz 稳定性。
• 数值实验在典型系统（包括混沌的 Lorenz-63 和 Lorenz-96）上证明了忠实的恢复和动力学。

Figure 4.1: Van der Pol oscillator ( 4.2 ) with $D=0.05$ . Top: ground-truth velocity field (left), reference invariant density (middle), and noisy trajectory samples (right). Bottom: PINN-IMSM reconstructed velocity field ${\bm{v}}_{\theta_{2}^{*}}$ (left), learned invariant density (middle), and s