Potential-energy gating for robust state estimation in bistable stochastic systems

TL;DR

The paper tackles robust state estimation in bistable stochastic systems by introducing potential-energy gating, a physics-informed mechanism that modulates observation reliability based on the energy landscape. By replacing the constant observation noise with a state-dependent covariance $R(x)=R_0[1+gV(x)]$ and adding a Boltzmann-inspired regularization term $\lambda V(x)$, the authors derive Hessian-based posterior covariances and extend the approach across multiple filter architectures. Across synthetic bistable dynamics with outliers and an empirical Dansgaard-Oeschger application to NGRIP data, potential-energy gating yields substantial RMSE improvements (up to ~80%) with robustness to parameter misspecification and greater gains during energy-driven transitions. The study demonstrates that incorporating a known energy landscape into the observation model can outperform purely statistical gating and simple topological baselines, offering a practical path for physics-informed real-time state estimation in systems with metastable states.

Abstract

We introduce potential-energy gating, a method for robust state estimation in systems governed by double-well stochastic dynamics. The observation noise covariance of a Bayesian filter is modulated by the local value of a known or assumed potential energy function: observations are trusted when the state is near a potential minimum and progressively discounted as it approaches the barrier separating metastable wells. This physics-based mechanism differs from purely statistical robust filters, which treat all regions of state space identically, and from constrained filters, which impose hard bounds on states rather than modulating observation trust. We implement the gating within Extended, Unscented, Ensemble, and Adaptive Kalman filters and particle filters, requiring only two additional hyperparameters. Synthetic benchmarks on a Ginzburg-Landau double-well process with 10% outlier contamination and Monte Carlo validation over 100 replications show 57-80% RMSE improvement over the standard Extended Kalman Filter, all statistically significant (p < 10^{-15}, Wilcoxon signed-rank test). A naive topological baseline using only distance to the nearest well achieves 57%, confirming that the continuous energy landscape adds an additional ~21 percentage points. The method is robust to misspecification: even when assumed potential parameters deviate by 50% from their true values, improvement never falls below 47%. Comparing externally forced and spontaneous Kramers-type transitions, gating retains 68% improvement under noise-induced transitions whereas the naive baseline degrades to 30%. As an empirical illustration, we apply the framework to Dansgaard-Oeschger events in the NGRIP delta-18O ice-core record, estimating asymmetry parameter gamma = -0.109 (bootstrap 95% CI: [-0.220, -0.011], excluding zero) and demonstrating that outlier fraction explains 91% of the variance in filter improvement.

Figures (8)

• Figure 1: The Ginzburg-Landau potential $V(x)$ for different parameter regimes. The symmetric case ($\gamma=0$) exhibits two degenerate wells at $x_{\pm} = \pm\sqrt{\alpha/\beta}$ separated by a barrier of height $\Delta V = \alpha^2/(4\beta)$. The asymmetric case ($\gamma \neq 0$) tilts the landscape, making one well deeper than the other. In the potential-energy gating framework, observations are trusted near the well minima (low $V$) and discounted near the barrier (high $V$).
• Figure 2: Distribution of RMSE across $100$ Monte Carlo replications for all twelve filters ($N=150$, $10\%$ outlier contamination). Violin bodies show kernel density estimates; horizontal bars mark means. The potential-gated filters (PG-PF through PG-EnKF) cluster at low RMSE, well separated from their standard counterparts. The NT-EKF (naive topological baseline) sits between the gated and standard groups, capturing $57\%$ of the improvement without using the full potential profile. Note the long upper tail of PG-UKF (mean $= 0.414$, median $= 0.285$), reflecting occasional numerical instability when sigma points fall in the barrier region where $V"(0) < 0$.
• Figure 3: Sensitivity of PG-EKF performance to hyperparameters $\lambda$ and $g$ across four potential topologies ($\alpha = 0.6, 1.0, 1.41, 2.0$). A broad "Goldilocks zone" ($\lambda \in [0.05, 0.5]$, $g \in [5, 30]$, shaded region) provides near-optimal RMSE reduction regardless of the specific potential shape. The robustness of this plateau means that approximate parameter knowledge is sufficient for effective gating.
• Figure 4: Model misspecification analysis. Left: PG-EKF RMSE as a function of assumed potential parameters $(\alpha_{\mathrm{assumed}}, \beta_{\mathrm{assumed}})$ when the true parameters are $\alpha^* = \beta^* = 1.0$ (starred cell). Right: corresponding improvement over EKF Std. The improvement never falls below $47\%$ across the entire grid, even at the most extreme misspecification ($\alpha_{\mathrm{assumed}} = 1.5$, $\beta_{\mathrm{assumed}} = 0.5$). This robustness arises because the GL potential preserves the correct topology (two wells separated by a barrier) regardless of the precise parameter values.
• Figure 5: RMSE distributions for EKF Std, PG-EKF, and NT-EKF under forced transitions (left, $T=300$, $N_{\mathrm{rep}}=20$) and spontaneous Kramers transitions (right, $\sigma=0.50$, $T=2000$, $N_{\mathrm{rep}}=9$ with $\geq 2$ zero-crossings). Under Kramers transitions, the PG-EKF retains most of its advantage ($+67.7\%$), while the NT-EKF degrades sharply ($+30.3\%$), demonstrating that the continuous potential profile is most valuable precisely when transitions follow the physical energy landscape rather than external forcing.