Statistical Parameter Calibration via the Generalized Fluctuation Dissipation Theorem and Generative Modeling

Abstract

We introduce a response-theoretic framework that recasts parameter calibration of ergodic stochastic differential equations as a fluctuation-dissipation problem. Our central result is that the full Jacobian of any stationary observable with respect to drift and diffusion parameters admits an exact linear-response representation as a time-correlation integral evaluated along unperturbed dynamics alone, without perturbed simulations, adjoint derivations, or tangent-linear models. The key idea is to interpret infinitesimal parameter variations as causal perturbations of the dynamics, thereby bringing the Generalized Fluctuation-Dissipation Theorem to inverse problems. The resulting kernels couple observables to the score of the invariant density, for which modern score-estimation methods provide practical non-Gaussian estimators. We validate the framework across a hierarchy of models, from analytically tractable processes to stochastic parameterization in the chaotic Lorenz-96 system, and show that a single baseline trajectory can recover parameter sensitivities and calibration updates with accuracy comparable to finite-difference approaches. More broadly, the framework opens a new route to model calibration, statistical inverse problems, and uncertainty quantification when the quantities of interest are long-time statistics of complex dynamical systems.

Figures (4)

• Figure 1: Parameter Dependence of Statistics in Quartic Potential System. By varying the coefficient in front of the quadratic term (the $\alpha_3$ parameter) we get different values for the mean (left), second moment (middle), and the probability that $x > 2.0$ (right). The GFDT obtains the tangent line to this curve (blue) for the choice of parameters $\vec{\alpha} = (0,1,0,-1)$.
• Figure 2: Reduced 1D Calibration. Left panel: active-observable mismatch norm on a logarithmic scale versus iteration. Top-right panel: normalized observable deviations for $\mathcal{A}_i\in\{\langle x\rangle,\,\langle x^2\rangle,\,P(x\le x_{\mathrm{th}})\}$, each rescaled so that the initial deviation equals $\pm 1$. Bottom-right panel: normalized parameter errors for the tuned parameters $(F_0,a,\sigma)$. Colored curves correspond to the Jacobian surrogate used inside the Gauss--Newton step: Analytical (pink), Finite differences (grey), GFDT: Quasi-Gaussian (orange), and GFDT: DSM (blue). The DSM and analytical Jacobians yield rapid, accurate convergence and agree with the finite-difference baseline, while the quasi-Gaussian closure exhibits systematic bias and substantially slower mismatch reduction.
• Figure 3: ENSO Calibration. Left panel: active-observable mismatch norm on a logarithmic scale versus iteration. Top-right panel: normalized observable deviations for the six second-order observables $\{u_1^2,\,u_2^2,\,u_1u_2,\,\tau^2,\,u_1\tau,\,u_2\tau\}$, each rescaled so that the initial deviation equals $\pm 1$. Bottom-right panel: normalized parameter errors for $(d_u,\omega,d_\tau,\sigma_1,\sigma_2,\sigma_3)$. Colored curves: Finite differences (grey), GFDT: Quasi-Gaussian (orange), and GFDT: DSM (blue). The DSM-based Jacobians steer the iteration rapidly to the correct parameters and statistics, whereas the quasi-Gaussian closure shows systematic errors, especially in the mismatch norm and the $\tau$-related moments.
• Figure 4: Lorenz--96 Calibration. Left panel: active-observable mismatch norm on a logarithmic scale versus iteration. Top-right panel: normalized observable deviations for the five matched statistics $(\phi_1,\phi_2,\phi_3,\phi_4,\phi_5)=(\mu,V,\operatorname{Sk},\operatorname{Ku},C_1)$, each rescaled by its initial deviation from the target. Bottom-right panel: normalized parameter deviations for the five closure parameters $(\alpha_0,\alpha_1,\alpha_2,\alpha_3,\sigma)$, expressed as relative changes from the initial values. Colored curves: Finite differences (grey), GFDT: Quasi-Gaussian (orange), and GFDT: DSM (blue). The DSM-based GFDT and finite-difference branches exhibit comparable mismatch reduction and observable convergence, whereas the quasi-Gaussian branch decreases more slowly and retains larger residual bias, particularly in $\phi_1$ (mean) and $\phi_3$ (skewness).