Schur-Neural KF: Learned Schur-Consistent Corrections to the Extended Kalman Filter

Min Kim; Lianghao Cao; Soon-Jo Chung; Andrew M. Stuart

eess.SYSep 26, 2026

Schur-Neural KF: Learned Schur-Consistent Corrections to the Extended Kalman Filter

Authors: Min Kim, Lianghao Cao, Soon-Jo Chung, Andrew M. Stuart

Organizations: The authors are with the California Institute of Technology (Caltech), Pasadena, CA 91125 USA.

Abstract

We present Schur-Neural KF (SN-KF), a learning-based correction to the extended Kalman filter (EKF) that preserves the probabilistic conditioning interpretation of the EKF. The method perturbs the predictive state-measurement cross-covariance and the Cholesky factor of the measurement noise covariance so that the resulting joint predictive covariance is always positive semidefinite. The positive semidefiniteness is ensured by a Schur complement-based parametrization. We instantiate the parametrization with a recurrent neural architecture whose matrix outputs are modulated by amplitude gates. We prove that incorporating a measurement does not increase the filter's state uncertainty, and show that no measurement can induce an arbitrarily large state correction relative to its statistical surprise. We also present a perturbative analysis suggesting SN-KF's structural strength in the data-scarce regime. We provide two numerical experiments to illustrate the practical benefits of SN-KF. In a two-radar experiment, enforcing Schur-consistency provides a much broader failure-free hyperparameter region and reduces RMSE for small training subsets, consistent with our theoretical analysis in the data-scarce regime. In the unicycle experiment, SN-KF achieves the best precision, recall, false alarm rate, and gated RMSE under innovation-based sensor-fault rejection.

Figures & tables

Method	Initial Succ.	Val. RMSE	Test RMSE	Local Succ.	Valid
SN-KF	108/108	1.5393	1.7006	27/27	9/9
No-Schur	66/108	1.5359	1.6913	7/27	0/9
Gain correction	18/33	1.6222	1.7708	19/27	2/9
EKF (infl.-tuned)	–	1.7993	1.9045	–	–

TABLE I: Numerical stability of training for the two-radar benchmark. “Initial Succ.” counts successful runs over the initial grids before refinement. For neural methods, “Val. RMSE” and “Test RMSE” are three-seed means at the final selected scales, trained using 30 training epochs. A local configuration is valid only if all three 30-epoch runs complete successfully. The EKF uses validation-selected inflation

\gamma=0.905

Method	Success	Test RMSE	Worst RMSE
SN-KF	100/100	1.873(30)	1.952
No-Schur	48/100	1.908(43)	2.017
Gain correction	61/100	1.909(20)	2.023
EKF (infl.-tuned)	–	1.904(0)	1.904

TABLE II: Results from training on 100 different small subsets. Parentheses in “Test RMSE” give the sample standard deviation. The EKF is fixed across subsets, so the corresponding standard deviation is zero.

Method	Proj. NIS	C95	C99
SN-KF	1.945	0.9539	0.9912
No-Schur	1.330	0.9866	0.9984
Gain correction	1.419	0.9832	0.9975
EKF (infl.-tuned)	1.493	0.9808	0.9972

TABLE III: Projected NIS over all 50 steps of healthy trajectories. The

\chi^{2}_{2}

reference values are

2/0.95/0.99

Method	Precision	Recall	FAR	Gated RMSE
SN-KF	0.808	0.937	0.0249	0.2104
No-Schur	0.534	0.692	0.0677	0.2523
Gain correction	0.523	0.814	0.0829	0.2800
EKF (infl.-tuned)	0.713	0.839	0.0377	0.2229

TABLE IV: Innovation-based gating for

[9,11]\sigma

common bearing faults, where

\sigma=0.03

rad; FAR = False Alarm Rate. Gated RMSE is computed over the full 50-step trajectory.

Kalman Filtering

Kalman filtering recursively estimates hidden states by balancing model predictions with noisy measurements.

See all 73 papers

Unscented KalmanNet: Structure-Preserving Deep Learning with Calibrated Posterior Uncertainty under Incomplete Physics and Unknown Noise
Aug 4, 2026Minhyeok Ko, Abdollah ShafieezadehKalman Filtering Unscented Kalman Filtering
Department of Civil and Construction Engineering and Management, The University of Texas at Tyler, Tyler, TX, 75799, USA · Department of Civil, Environmental, and Geodetic Engineering, The Ohio State University, Columbus, OH, 43210, USA
Learned Memory Attenuation in Sage-Husa Kalman Filters for Robust UAV State Estimation
May 18, 2026Kenan Majewski, Marcin ŻugajKalman Filtering UAV Navigation
Institute of Aeronautics and Applied Mechanics, Warsaw University of Technology, Nowowiejska 24, Warsaw, 00-665, Poland
Design Guidelines for Nonlinear Kalman Filters via Covariance Compensation
Mar 24, 2026Shida Jiang, Jaewoong Lee, Shengyu Tao +1Kalman Filtering Extended Kalman Filtering
Department of Civil and Environmental Engineering, University of California, Berkeley, CA 94720, USA.

Aug 4, 2026J/K move · Enter open · S save

Unscented KalmanNet: Structure-Preserving Deep Learning with Calibrated Posterior Uncertainty under Incomplete Physics and Unknown Noise

Minhyeok Ko, Abdollah Shafieezadeh

Department of Civil and Construction Engineering and Management, The University of Texas at Tyler, Tyler, TX, 75799, USA · Department of Civil, Environmental, and Geodetic Engineering, The Ohio State University, Columbus, OH, 43210, USA

Kalman Filtering Unscented Kalman Filtering Covariance Estimation Robot State Estimation Uncertainty Calibration

Open paper AskPDF arXiv

Nonlinear state estimation requires sequentially fusing model-based predictions with noisy measurements. Under imperfect dynamics and unknown, time-varying noise statistics, this fusion can degrade in both accuracy and statistical consistency. Existing learning-aided filters largely treat accuracy and uncertainty estimation separately, limiting their ability to correct model-mismatch-induced bias while retaining an explicit, calibrated posterior covariance. This paper introduces Unscented KalmanNet (UKN), a model-based deep learning architecture that extends the Unscented Kalman Filter (UKF) with learned mechanisms for these two sources of filtering error while preserving explicit posterior covariance propagation. NoiseNet learns time-varying process and measurement covariances as bounded multiplicative corrections to baseline covariances, guaranteeing positive definiteness, while GainNet learns a bounded residual correction to the analytical UKF gain to compensate for model-mismatch-induced bias. A calibration-aware training objective couples state error with posterior covariance and innovation consistency terms through adaptive weighting, jointly optimizing accuracy and calibration. UKN is benchmarked against UKF, KalmanNet, and Bayesian KalmanNet on three synthetic systems and real-flight UZH-FPV data. It achieves the lowest state-estimation error in all four examples and reduces RMSE by 26.4-49.7% compared with UKF in the synthetic cases. Leave-one-sequence-out cross-validation over 11 flights shows 22.4% and 34.3% reductions in mean position and velocity RMSE, respectively. UKN also yields the lowest fold-to-fold variability, with dimension-normalized NEES and empirical coverage closest to nominal values among covariance-reporting filters. These results show that structured learned adaptation improves estimation accuracy while retaining calibrated uncertainty.

Abstract

Kalman Filtering

Explore similar work

Unscented KalmanNet: Structure-Preserving Deep Learning with Calibrated Posterior Uncertainty under Incomplete Physics and Unknown Noise

Learned Memory Attenuation in Sage-Husa Kalman Filters for Robust UAV State Estimation

Design Guidelines for Nonlinear Kalman Filters via Covariance Compensation