用物理约束扩散模型精准重建混沌系统的稀疏观测轨迹。
PIDM-DP: Physics-Informed Diffusion with Dormand-Prince Integration for Chaotic System Identification and State Reconstruction across Multiple Dynamical Regimes
- 将可微分的龙格-库塔积分器嵌入扩散模型反演过程，保证轨迹满足动力学方程。
- 在10%采样率、噪声σ=0.05下，重构误差比无约束模型降低15.4倍。
- 适合处理高维刚性混沌系统，尤其对分布外数据鲁棒性强。
从稀疏、含噪观测中重构混沌动力系统的连续状态轨迹，仍是非线性科学中的核心难题。本文提出物理信息扩散模型结合多诺姆-普赖斯积分（PIDM-DP），将全可微的五阶龙格-库塔（DP-RK45）常微分方程积分器直接嵌入去噪扩散概率模型（DDPM）的反向采样流程中。每一步去噪时，通过自动微分回传物理残差，确保生成轨迹满足系统控制方程至五阶精度。采用线性调度引导机制，使物理权重在高噪声阶段从零逐渐提升至接近纯净数据极限，有效防止因雅可比特征值量级达 $O(10^3)$ 的刚性系统中梯度爆炸问题。在五个基准系统上评估：3D Lorenz、3D Rössler、5D 超混沌系统、20D Lorenz-96 及刚性 3D Rabinovich-Fabrikant，观测密度为10%，添加高斯噪声（σ=0.05）。相比无约束扩散基线，PIDM-DP 最大实现15.4倍的均方根误差（RMSE）改善；在刚性系统上显著优于集合卡尔曼滤波（EnKF），后者因协方差坍塌而失效。在Rabinovich-Fabrikant分布外测试中，PIDM-DP 的 RMSE 为 $0.1097 \pm 0.0269$，远优于无约束扩散（$0.9443 \pm 0.5288$，差8.6倍）与 EnKF（$0.3561 \pm 0.3040$，差3.2倍），配对威尔科克斯检验 $p<0.001$（N=30）。通过Rosenstein李雅普诺夫估计器进行拓扑验证，确认PIDM-DP 保持了混沌不变测度。
原文摘要 · Abstract (English)
Reconstructing continuous state trajectories of chaotic dynamical systems from sparse, noisy observations remains a fundamental open problem in nonlinear science. We introduce the Physics-Informed Diffusion Model with Dormand-Prince Integration (PIDM-DP), which embeds a fully differentiable 5th-order Dormand-Prince (DP-RK45) ODE integrator directly into the reverse sampling loop of a Denoising Diffusion Probabilistic Model (DDPM). At each denoising step, physics residuals are back-propagated via automatic differentiation, constraining every generated trajectory to satisfy the system's governing equations to 5th-order accuracy. A linear-scheduled guidance mechanism that ramps the physics weight from zero at high noise levels to its full value near the clean-data limit prevents the gradient explosions that cause naive physics-informed approaches to fail on stiff systems with Jacobian eigenvalues of order $O(10^3)$. Evaluated across five benchmark systems of increasing complexity 3D Lorenz, 3D Rössler, 5D Hyperchaotic, 20D Lorenz-96, and the stiff 3D Rabinovich-Fabrikant at 10% observation density with additive Gaussian noise ($σ=0.05$), PIDM-DP achieves reconstruction RMSE improvements of up to $15.4\times$ over an unconstrained diffusion baseline and decisively outperforms the Ensemble Kalman Filter on stiff systems where ensemble covariance collapses. On the Rabinovich-Fabrikant out-of-distribution benchmark, PIDM-DP attains RMSE $0.1097 \pm 0.0269$ versus $0.9443 \pm 0.5288$ (unconstrained diffusion, $8.6\times$ worse) and $0.3561 \pm 0.3040$ (EnKF, $3.2\times$ worse), with $p<0.001$ in paired Wilcoxon tests ($N = 30$). Topological validation via the Rosenstein Lyapunov estimator confirms that PIDM-DP preserves the chaotic invariant measure.
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