PIDM-DP: Physics-Informed Diffusion with Dormand-Prince Integration for Chaotic System Identification and State Reconstruction across Multiple Dynamical Regimes

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 R\"ossler, 5D Hyperchaotic, 20D Lorenz-96, and the stiff 3D Rabinovich-Fabrikant at 10% observation density with additive Gaussian noise ($\sigma=0.05$), 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.